EXPLICIT LEAST-DEGREE BOUNDARY FILTERS FOR DISCONTINUOUS GALERKIN

Abstract

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation.

The paper introduces a new class of PSIAC filters NP₀ that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP₀ filters outperform the more complex known boundary filters. NP₀ filters typically reduce the L^∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP₀ filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.

1. Introduction

The output of Discontinuous Galerkin (DG) computations often captures higher order moments of the true solution [ML78]. Therefore post-processing DG output by convolution with splines can improve both smoothness and accuracy [BS77, CLSS03, JvRV14]. In the interior of the domain of computation, symmetric smoothness increasing accuracy conserving (SIAC) spline filters have been demonstrated to provide optimal accuracy [CLSS03]. To accommodate non-periodic boundary data, such symmetric filters need to be complemented by one-sided filters. The point-wise error and stability, of the pioneering Ryan-Shu boundary filters [RS03] have been noticeably improved upon, during the last decade [SRV11, MRK12, RLKV15, LRKV16].

Most recently these boundary filters have been simplified and improved by replacing numerical approximation with symbolic formulas, both in the uniform symmetric case [MRK15] and in the general case [Pet15]. For general knot sequences, [NP16] introduced a factored symbolic characterization of spline filters that identified the existing boundary filters as position-dependent SIAC spline filters (PSIAC filters). While in earlier formulations, the rule for each convolved value had to be derived numerically by Gauss quadrature, PSIAC prototype filters can be precomputed and then easily scaled and shifted to fit the boundary region as required. PSIAC filtering then amounts to forming a linear combination of the local DG output with simple weights that are polynomial in the position. The coefficients of these polynomial expressions are rational numbers for rational knot sequences. Remarkably [NP16, Theorem 4.2] PSIAC-filtered DG output lies on a single polynomial over the boundary region, where the filter is deployed. RLKV-filtered and SRV-filtered DG output is therefore infinitely differentiable and derivatives of PSIAC-filtered DG output need not be approximated [Tho77, RSA05, RC09, LRKV16] but have an explicit, easily-evaluated expression.

This paper introduces a new piecewise constant PSIAC filter, that we will refer to as the NP₀ filter. On variants of the canonical hyperbolic test equation (2.2), the simple NP₀ filter outperforms, on different aspects, both the SRV and the RLKV filter. Applied to the DG output, NP₀ is more stable than the superconvergent, but larger-support SRV filter [SRV11] – SRV requires quadruple precision to produce reliable results when the DG polynomials are of degree d > 2 [RLKV15]; and NP₀-filtered output converges faster than that of the small-support, stable but suboptimal RLKV filter [RLKV15]. Fig. 1 shows results for four different x-mesh spacings. The RLKV-error in the boundary region exceeds the error in the interior where the symmetric SIAC filter is applied; and for large mesh spacing, the pointwise RLKV error in Fig. 1a even exceeds that of the unfiltered DG error in Fig. 1c. By contrast Fig. 1b shows NP₀ consistently reducing the error in [0‥λ⁺], the boundary regionaugmented by the transition region to symmetric SIAC filter, even below the error of the symmetric SIAC filter. This message is condensed in Fig. 1d. Here the maximal error of the symmetric SIAC filter of degree d in the interior is displayed as dashed lines and the error near the left and right boundary by ▷, respectively ◁.

Fig. 1.

Pointwise errors e(x, T = 2/3) of the filtered output (a), (b) of the DG solution of degree d = 3 for the canonical partial differential equation u_τ + u_x = 0 computed up to fixed final time T = 2/3 (cf. (2.2)). The unfiltered error is shown in (c) and (d) summarizes the result. Each subfigure shows in order, from top to bottom, the graph for mesh-size (red), (blue), (green) and (purple). (a, b) The growing interior, bordered by vertical dashed lines of the color of the graph, is post-processed by the symmetric SIAC filter. The graph for the interior is therefore identical in (a) and (b). The left boundary region [0‥λ] is to the left of the left dashed line and and the right boundary region is to the right of the right line. The RLKV and NP₀ error are computed on ever smaller boundary regions [0‥λ] and [1 − λ‥1] since λ shrinks with h (see (a)). (a) Errors after convolving the data of the boundary regions with the one-sided RLKV filters [RLKV15] and the interior with the symmetric SIAC filter. This graph is identical to [RLKV15, Figure 5, top-right] but is computed with the symbolic formulation. (b) Errors after convolving the data of the boundary regions with the new least-degree PSIAC filters NP₀ and the interior with the symmetric SIAC filter. (c) Original errors of the degree d = 3 DG output. (d) The error in [0‥λ⁺], i.e. the left boundary region augmented by the transition to the inner symmetric SIAC filter region, is indicated by ▷. The error on the right by ◁. The error of the symmetric SIAC filter applied in the interior is indicated by −−. The error of the new PSIAC filter NP₀ is lower or on par with the optimally superconvergent symmetric SIAC filter in the interior.

Let d be the polynomial degree of the DG output and k the degree of the spline filters. The theory of [JvRV14] guarantees a convergence order of d+1+k for the NP_k filter. However, extensive numerical experiments, presented in detail in the Appendix B, indicate optimal super-convergence of order 2d+1 for the NP₀ filter (k = 0) applied to the canonical hyperbolic equation u_τ + u_x = 0 for a range of final times T of the DG computation. Yet, this paper does not aim to establish this super-convergence or sharp theoretical convergence bounds. Rather it establishes properties and practical formulas that make it easy to apply the NP₀ filter, e.g. explicit formulas for stably computing the convolution with the data based on a symbolic formula provided by [NP16]. PSIAC-filtering of DG output at a point x in the left boundary region [0‥λ] amounts to forming a linear combination of the vector of DG coefficients u_[x] whose functions have support at x:

$$u_{\text{filtered}}(x)≔{\mathbf{u}}_{[x]}·{\mathit{𝒱}}_x.$$ (1.1)

By Eq. (2.11) of Theorem 2.5 of this paper,

$${\mathit{𝒱}}_x≔Q_{\lambda }^h{\mathit{\lambda }}_x,{\mathit{\lambda }}_x≔{[{(\lambda -x)}^0\cdots {(\lambda -x)}^{3d}]}^t,$$ (1.2)

where $Q_{\lambda }^h$ is a matrix with rational entries that depend only on the space of DG polynomials and on the space of filter kernels, not on x. For efficient implementation, the local DG coefficient vector is therefore multiplied with the matrix $Q_{\lambda }^h$ in advance yielding a vector of size 3d + 1 to be multiplied with λ_x. The central four entries 𝒱̅_x of 𝒱_x are its largest in absolute value. For the NP₀ and the SRV filter (in PSIAC form), d = 3 and x = 0, the four entries are respectively

$${\overline{\mathit{𝒱}}}_0({\text{NP}}_0)={10080}^{-1}[\begin{array}{cccc}70381\hfill & 70381\hfill & -56627\hfill & -56627\hfill \end{array}]$$ (1.3)

$${\overline{\mathit{𝒱}}}_0(\text{SRV})={15256200960000}^{-1}[\begin{array}{cccc}3549982809648204\hfill & 9809076669570393\hfill & 11473452075703833\hfill & 6592494198365004\hfill \end{array}].$$

While the fractions of integers have at most 5 digits for NP₀, the fractions of the SRV filters have up to 17 digits and their absolute value is much larger: Fig. 2 shows that for d = 3 the alternating coefficients of the SRV filter are two orders of magnitude larger than those of the NP₀ filter.

Fig. 2.

Size of the entries of $\mathit{𝒱}=Q_{\lambda }^{h=1}\mathit{\lambda }$ for filtering d = 3 DG output at a boundary point x = 0. Note the 100-fold difference in scale between (a) and (b).

In summary,

• ▷applying the PSIAC boundary filter NP₀ is efficient: one multiplies the boundary DG data with the matrix Q_λ and then forms, for each x, hence λ_x, the dot product between the two short vectors;
• ▷numerical experiments in double floating point precision on the canonical wave equation indicate that NP₀ has a convergence rate of 2d+1, where d is the piecewise degree of the DG output.

Organization

Section 2 introduces notation, spline convolution and the canonical test equations, followed by a review of the literature in more detail. Section 3 focuses on the new NP₀ filter and compares it to the existing SRV and RLKV filters in their improved symbolic PSIAC form. Equation (3.11) and Appendix A provide explicit Q_λ for degree d ∈ {1, 2, 3}. Section 4 compares the SRV, RLKV, NP₀ and the symmetric SIAC filters numerically on three variants of the canonical hyperbolic equation.

2. Notation and Definitions

This section establishes the notation for filters and DG output, exhibits the canonical test problem, the DG method, and reproducing filters and reviews one-sided and position-dependent SIAC filters in the literature.

We denote by f * g the convolution of a function f with a function g, i.e.

$$(f\ast g)(x)≔{\int }_{ℝ}f(t)g(x-t)\mathrm{d}t=(g\ast f)(x),$$

for every x where the integral exists. Filtering means convolving a function f with a kernel g.

2.1. Sequences, Splines, and Reproduction

The goal of SIAC filtering is to spatially smooth out the DG output u(x, τ) by convolution in x with a linear combination of B-splines. Typically filtering is applied after the last time step when τ = T. Specifically, we will focus on piecewise polynomial SIAC spline kernels f : ℝ → ℝ such that convolution of f with monomials (·)^δ reproduces the monomials up to degree r. Let 𝒥 ≔ (0, …, j_r) be a sequence of strictly increasing integers between 0 and j_r, abbreviate the sequences of consecutive integers as

$$i:j≔\{\begin{array}{cc}\begin{array}{c}(i,i+1,\dots ,j-1,j),\text{if }i\le j,\hfill \\ (i,i-1,\dots ,j+1,j),\text{if }i>j,\hfill \end{array}\hfill & s_{i:j}≔(s_i,\dots ,s_j).\hfill \end{array}$$

Let B(x|t_j:j+k+1) denote the unit integral B-spline with (non-decreasing) knot sequence t_j:j+k+1 (see [dB02]) related to the recursively defined B-spline N(t|t_i:i+k+1) by $N(t|t_{i:i+k+1})=\frac{t_{i+k+1}-t_i}{k+1}B(t|t_{i:i+k+1})$. Then a SIAC spline kernel of degree k and reproduction degree r with index sequence 𝒥 and knot sequence t_0:n is a spline

$$f(x)≔\sum _{j\in 𝒥}{f}_{j}B(x|t_{j:j+k+1}),$$

of degree k with coefficients f_j chosen so that

$$(\sum _{j\in 𝒥}{f}_{j}B(·|t_{j:j+k+1})\ast {(-·)}^{\delta })(x)={(-x)}^{\delta },\delta =0:r.$$ (2.1)

We reserve the following symbols:

d	degree of the DG output;
m	number of intervals of the DG output;
s0:m	prototype increasing break point sequence, typically integers;
	the break sequence of the DG output is hs0:m;
k	degree of the filter kernel;
r + 1	number of filter coefficients
	for reproduction of polynomials up to degree r;
𝒥 ≔ (0, …, jr)	index sequence;
	if the B-splines of the filter are consecutive, then jr = r;
n	number of knot intervals spanned by the filter;
	n = jr + k + 1;
t ≔ t0:n	prototype (integer) knot sequence of the filter;
	the input knot sequence of the filter is ht0:n + ξ
	where ξ is the shift and h scales.

This notation is illustrated by the following example.

Example 2.1

A linear DG output sequence on 200 uniform segments of the interval [−1‥1] implies d = 1, m = 200, $h=\frac{1}{100}$ and s_0:m = −100 : 100. A degree-one spline filter defined over the knot sequence t ≔ 0 : 6 and associated with the index set 𝒥 ≔ {0, 3, 4} corresponds to k = 1, n = 6, r = 2 and j_r = 4. The two B-splines defined over the knot sequences 1 : 3 and 2 : 4 are skipped.

2.2. The canonical test problem and the Discontinuous Galerkin method

To demonstrate the performance of the filters on a concrete example, [RS03] used the following univariate hyperbolic partial differential wave equation:

$$\begin{array}{cc}\hfill \frac{\mathit{\text{du}}}{d\tau }+\frac{d}{\mathit{\text{dx}}}(\kappa (x,\tau )u)=\rho (x,\tau ),& \hfill x\in (a‥b),\tau \in (0‥T)\\ \hfill u(x,0)=u_0(x),& \hfill x\in [a‥b]\end{array}$$ (2.2)

subject to periodic boundary conditions, u(a, τ) = u(b, τ), or Dirichlet boundary conditions u(e, τ) = u₀(τ) where, depending on the sign of κ(x, τ), e is either a or b. Subsequent work [RS03, SRV11, RLKV15] adopted the same differential equation to test their new one-sided filters and to compare to the earlier work. Eq. (2.2) is therefore considered the canonical test problem. We note, however, that SIAC filters apply more widely, for example to FEM and elliptic equations [BS77].

In the DG method, the domain [a‥b] is partitioned into intervals by a sequence hs_0:m of break points a ≕ hs₀, …, hs_m ≔ b. Assuming that the sequence is rational, scaling by h will later allow us to consider a prototype sequence s_0:m of integers. Let ${\mathbb{P}}_h^d$ be the linear space of all piecewise polynomials with break points hs_0:m and of degree less than or equal to d. We use modal or nodal scalar-valued basis functions ϕ_i(․ ; hs_0:m) 0 ≤ i ≤ m of ${\mathbb{P}}_h^d$ that are linearly independent and satisfy the scaling relations

$${\varphi }_i(\mathit{\text{hx }};{\mathit{\text{hs}}}_{0:m})={\varphi }_i(x;s_{0:m}).$$ (2.3)

Relation (2.3) is typically used for refinement in FEM, DG or Iso-parametric PDE solvers. Examples of basis functions ϕ_i are Bernstein-Bézier basis functions [dB05], Lagrange polynomials dependent on Legendre-Gauss-Lobatto quadrature points [HW07], and Legendre polynomials.

The DG method approximates the time-dependent solution of Eq. (2.2) by

$$u(x,\tau )≔\sum _{i=0}^m{u}_i(\tau ){\varphi }_i(x;{\mathit{\text{hs}}}_{0:m}),{\varphi }_i\in {\mathbb{P}}_h^d.$$ (2.4)

Multiplying the two sides of Eq. (2.4) with a test function υ and integrating by parts yields the weak form of Eq. (2.2):

$${\int }_a^b(\frac{\mathit{\text{du}}}{d\tau }\upsilon -\kappa (x,\tau )u\frac{d\upsilon }{\mathit{\text{dx}}})\mathrm{d}x={\int }_a^b\rho (x,\tau )\upsilon \mathrm{d}x-{(\kappa (x,\tau )u(x,\tau )\upsilon (x))|}_{x=a}^{x=b}.$$ (2.5)

Substituting u on the left of Eq. (2.5) by (2.4), treating the rightmost, non-integral term of Eq. (2.5) as a numerical flux, and choosing υ(x) ≔ ϕ_j(x ; hs_0:m), yields a system of ordinary differential equations in τ with the coefficients u_i(τ), 0 ≤ i ≤ m, as unknowns. This system can be solved by, e.g., a standard fourth-order four stage explicit Runge-Kutta method (ERK) [HW07, Section 3.4].

2.3. A synopsis of DG filtering

Since convolution with a symmetric SIAC kernel of a function g at x requires g to be defined in a two-sided neighborhood of x, near boundaries, Ryan and Shu [RS03] proposed convolving the DG output with a kernel whose support is shifted to one side of the origin: for x near the left domain endpoint a, the one-sided SIAC kernel is defined over (x−a)+h(−(3d+1), −3d, …, 0) where d is the degree of the DG output. The Ryan-Shu x-position-dependent one-sided kernel yields optimal L²-convergence, but its point-wise error near a can be larger than that of the DG output.

In [SRV11], Slingerland-Ryan-Vuik improved the one-sided kernel by increasing its monomial reproduction from degree r = 2d to degree r = 4d. This one-sided kernel reduces the boundary error when d = 1 but the kernel support is increased by 2d additional knot intervals and numerical roundoff requires quadruple precision calculations to determine the kernel's coefficients. ([SRV11] additionally required quadruple precision for computing the DG output.) Indeed, the coefficients of the boundary filters [RS03, SRV11, MRK12, RLKV15, MRK15] are computed by inverting a matrix whose entries are determined by Gaussian quadrature; and, as pointed out in [RLKV15], SRV filter matrices are close to singular.

Ryan-Li-Kirby-Vuik [RLKV15] therefore suggested an alternative one-sided position-dependent kernel that has the same support size as the symmetric kernel and has reproduction degree higher by one, enriching the spline space by one B-spline. This RLKV kernel is stably computed, as has been verified numerically, in double precision, up to input data degree d = 4 and joins the symmetric SIAC filter, applied in the interior, without a jump in error. However, the error of the RLKV kernel at the boundaries can be higher than that of the symmetric SIAC filter and the L² and L^∞ superconvergence rates are sub-optimal [RLKV15] (c.f. Fig. 9,10,11). [LRKV16] additionally states that RLKV has a poorer derivative approximation than SRV filters.

Fig. 9.

Convergence rates Test 4.1 (constant wave-speed, periodic boundary condition). L²- and L^∞ convergence of the convolved DG data in the left, respectively right boundary region. Rows 1, 2, 3 show rates for degree d = 1, 2, 3. Upper, dotted grey: symmetric SIAC filter; lower dash-dotted: DG rate. The graphs have the same indicative colors and styles as those in Fig. 6, 7 and 8.

Fig. 10.

Convergence rates Test 4.2 (constant wave-speed, Dirichlet boundary condition). L²- and L^∞ convergence of the convolved DG data in the left, respectively right boundary region. Rows 1, 2, 3 show rates for degree d = 1, 2, 3. Upper, dotted grey: symmetric SIAC filter; lower dash-dotted: DG rate.

Fig. 11.

Convergence rates Test 4.3 (variable wave-speed, periodic boundary condition). L²- and L^∞ convergence of the convolved DG data in the left, respectively right boundary region. Rows 1, 2, 3 show rates for degree d = 1, 2, 3. Upper, dotted grey: symmetric SIAC filter; lower dash-dotted: DG rate.

[NP16] reinterprets the published one-sided filters in an explicit, symbolic form as position-dependent PSIAC spline filters. Symbolic expression of coefficients for spline filters have recently been developed in [MRK15] for uniform knot sequences and in [Pet15] for general knot sequences. Reinterpretation of the published filters in symbolic form improves their numerical stability.

2.4. Symbolic formulation of the filters

We split the DG data at any known discontinuities and treat the domains separately. Then convolution can be applied throughout a given closed interval [a‥b]. A SIAC spline kernel with knot sequence t_0:r+k+1 is symmetric (about the origin in ℝ) if t_ℓ+t_r+k+1−ℓ = 0 for ℓ = 0 : ⌈(r+k+ 1)/2⌉. Note that, unlike the (position-independent) classical symmetric SIAC filter, the position-dependent boundary kernel coefficients have to be determined afresh for each point x.

Lemma 2.1 (SIAC coefficients [NP16])

The vector f ≔ [f₀, …, f_r]^t ∈ ℝ^r+1 of B-spline coefficients of the SIAC filter with index sequence 𝒥 ≔ (0, …, j_r) and knot sequence t_0:n is

$$\mathbf{f}≔\mathit{\text{first column of }}{M}^{-1},M≔M_{t_{0:n},𝒥}={\left[\sum _{|\mathit{\omega }|=\delta }{t}_{j:j+k+1}^{\mathit{\omega }}\right]}_{\delta =0:r,j\in 𝒥}$$ (2.6)

$$\mathit{\text{where }}{t}_{0:p}^{\mathit{\omega }}≔t_0^{{\omega }_0}\dots t_p^{{\omega }_p}\mathit{\text{ and }}|\mathit{\omega }|≔\sum _{j=0}^n|{\omega }_j|$$

This characterization yields the following formula for the filter coefficients.

Theorem 2.2 (Scaled and shifted SIAC coefficients are polynomial [NP16])

The SIAC filter coefficients f_ξ;ℓ associated with the knot sequence ht_0:n+ξ are polynomials of degree r in ξ:

$${\mathbf{f}}_{\xi }≔{[f_{\xi ;\ell }]}_{\ell =0:r}=M_{t_{0:n},𝒥}^{-1}\text{ diag }({[{(-1)}^{\ell }(\begin{array}{c}\hfill \ell +k+1\hfill \\ \hfill \ell \hfill \end{array})]}_{\ell =0:r}){\left[{(\frac{\xi }{h})}^{0:r}\right]}^t.$$ (2.7)

The following corollary implies that the kernel coefficients f_ξ;ℓ, can be pre-computed stably, as scaled integers.

Corollary 2.3 (Coefficient polynomials f_ξ;ℓ have rational coefficients [NP16])

If the knots t_0:n are rational, then the filter coefficients f_ξ;ℓ are polynomials in ξ and h with rational coefficients.

We can now define the PSIAC kernel.

Definition 2.4 (PSIAC kernel)

A PSIAC kernel with index sequence 𝒥 = (0, …, j_r) and knot sequence ht_0:n + x has the form

$$f_x(s)≔\sum _{j\in 𝒥}{f}_{x;j}B(s|{\mathit{\text{ht}}}_{j:j+k+1}+x),s\in h[t_0,t_n]+x.$$ (2.8)

The DG output is convolved with a PSIAC kernel f_x−hλ(s) of reproduction degree r, associated with an index sequence 𝒥 and defined over shifted knots ht_0:n + x − hλ − where the constant hλ adjusts the filter kernel to the left or right boundary. Example 2.2 illustrates the simple explicit form of the PSIAC coefficients according to Theorem 2.2 and verifies that the corresponding PSIAC filter reproduces as predicted by the derivation.

Example 2.2 (Reproduction by PSIAC filtering)

Let h = 1, k = 0 and f_x be the least degree PSIAC filter with t_0:n = {−2, −1, 0}, r = 1, 𝒥 = (0, 1). According to (2.8) of Definition 2.4:

$$\begin{gathered} f_x(s)≔f_{x;0}B(s|\{-2,-1\}+x)+f_{x;1}B(s|\{-1,0\}+x) \\ =f_{x;0}{\chi }_{[-2,-1]+x}+f_{x;1}{\chi }_{[-1,0]+x},{\chi }_{[\alpha ,\beta ]}(s)≔\{\begin{array}{cc}1,\hfill & \mathit{\text{if s}}\in [\alpha ,\beta ];\hfill \\ 0,\hfill & \mathit{\text{else}}\hfill \end{array} \end{gathered}$$ (2.9)

where χ_[α,β] denotes the indicator function of the domain [α, β]. Equation (2.7) of Theorem 2.2 provides the formula

$$\left[\begin{array}{c}\hfill f_{x;0}\hfill \\ \hfill f_{x;1}\hfill \end{array}\right]≔{\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -2-1\hfill & \hfill -1+0\hfill \end{array}\right]}^{-1}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -2\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ \hfill x\hfill \end{array}\right]=\frac{1}{2}\left[\begin{array}{c}\hfill 2x-1\hfill \\ \hfill 3-2x\hfill \end{array}\right].$$ (2.10)

This choice of filter coefficients f_x;0 and f_x;1 satisfies Eq. (2.1). For δ = 0:

$$(f_x\ast 1)(x)={\int }_{ℝ}{f}_x(s)\mathrm{d}s=f_{x;0}{\int }_{ℝ}{\chi }_{[-2,-1]+x}(s)\mathrm{d}s+f_{x;1}{\int }_{ℝ}{\chi }_{[-1,0]+x}(s)\mathrm{d}s=1.$$

$$\text{For }\delta =1:(f_x\ast (-·))(x)={\int }_{ℝ}{f}_x(s)(s-x)\mathrm{d}s=f_{x;0}{\int }_{ℝ}{\chi }_{[-2,-1]+x}(s)(s-x)\mathrm{d}s+f_{x;1}{\int }_{ℝ}{\chi }_{[-1,0]+x}(s)(s-x)\mathrm{d}s=f_{x;0}{\int }_{-2+x}^{-1+x}(s-x)\mathrm{d}s+f_{x;1}{\int }_{-1+x}^{0+x}(s-x)\mathrm{d}s=f_{x;0}({\frac{s^2}{2}|}_{s=-2+x}^{s=-1+x}-x)+f_{x;1}({\frac{s^2}{2}|}_{s=-1+x}^{s=0+x}-x)=f_{x;0}(-f_{x;1}-x)+f_{x;1}(f_{x;0}-x)=-x.$$

Leveraging Theorem 2.2, we can efficiently compute the convolution as follows.

Theorem 2.5 (Efficient PSIAC filtering of DG output [NP16])

Let f_x(s) be a PSIAC kernel of reproduction degree r with index sequence 𝒥 = (0, …, j_r) and knot sequence ht_0:n + x − hλ. Let $u(x,\tau )≔{\sum }_{i=0}^m{u}_i(\tau ){\varphi }_i(x;{\mathit{\text{hs}}}_{0:m})$, x ∈ [a, b] and τ ≥ 0, be the DG output. Let ℐ be the set of indices of basis functions ϕ_i(․ ; hs_0:m) with support overlapping h[λ − t_n, λ − t₀]. Then the filtered DG approximation is a polynomial in x of degree r:

$$(u\ast f_x)(x)={\mathbf{u}}_{ℐ}{Q}_{\lambda }{\left[{(\frac{x}{h}-\lambda )}^{0:r}\right]}^t.$$ (2.11)

$${\mathbf{u}}_{ℐ}≔{[u_i(\tau )]}_{i\in ℐ},$$

$$Q_{\lambda }≔G_{\lambda }{\mathit{\text{ AM}}}_{0,\mathbf{t},𝒥}^{-1}\text{ diag}({[{(-1)}^{\ell }(\begin{array}{c}\hfill \ell +k+1\hfill \\ \hfill \ell \hfill \end{array})]}_{\ell =0:r}),$$

$$G_{\lambda }≔{[{\int }_{\lambda -t_n}^{\lambda -t_0}{\varphi }_i(s;s_{0:m})B(s|\lambda -t_{n-j:j_r-j})\mathrm{d}s]}_{i\in ℐ,j\in 𝒥}.$$ (2.12)

A is the reversal matrix with 1 on the antidiagonal and zero else.

The factored representation implies that instead of recomputing the filter coefficients afresh for each point x of the convolved output as was the practice prior to [NP16], we simply pre-compute the coefficients corresponding to one prototype knot sequence t and, at runtime, pre-multiply with the data and post-multiply with the vector of shifted monomials scaled by h according to Eq. (2.11).

Increased multiplicity of an inner knot of the symmetric, position-independent SIAC kernel reduces its smoothness, and this, in turn, reduces the smoothness of the filtered output. By contrast, Theorem 2.5 shows that when the PSIAC knots are shifted along evaluation points x then PSIAC convolution yields a polynomial, i.e. the representation near the boundary is infinitely smooth regardless of the knot multiplicity. That is, we may view position-dependent filtering as a form of polynomial approximation. For example, the RLKV-filtered output is a single polynomial over the boundary region where it applies.

Example 2.3 (Coefficients of the RLKV-filtered DG output polynomial)

Let d = k = 3. Consider the canonical partial differential equation u_τ + u_x = 0 of (2.2) for x ∈ [0, 1] at final time T = 1 for mesh-sizes h_i ≔ 2⁻ⁱ/10, i = 1, 2, 3. The analytical solution of this equation is u_e(x) = sin 2π(x − T). Let λ_L,i ≔ 5h_i, λ_R,i ≔ 1 − 5h_i and f_L,i,x be the RLKV filters with respect to the left boundary regions. Applying the RLKV-filter to the DG output u_hi computed for mesh size h_i yields a polynomial in x as predicted by Theorem 2.5:

$$P_{L,i}(x)≔(f_{L,i,x}\ast u_{h_i})(x)=\sum _{k=0}^7{a}_{k,i}{(x-{\lambda }_{L,i})}^k,x\in [0,{\lambda }_{L,i}],$$ (2.13)

where

$$\left[\begin{array}{ccc}\hfill a_{0,1}\hfill & \hfill \cdots \hfill & \hfill a_{3,1}\hfill \\ \hfill a_{4,1}\hfill & \hfill \cdots \hfill & \hfill a_{7,1}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 0.999999901374753\hfill & \hfill 0.000021494468508\hfill & \hfill -19.738996791744032\hfill & \hfill -0.008053345774016\hfill \\ \hfill 64.88630724285224\hfill & \hfill 0.623484670536888\hfill & \hfill -82.011046997856752\hfill & \hfill -11.875898409510508\hfill \end{array}\right]$$

$$\left[\begin{array}{ccc}\hfill a_{0,2}\hfill & \hfill \cdots \hfill & \hfill a_{3,2}\hfill \\ \hfill a_{4,2}\hfill & \hfill \cdots \hfill & \hfill a_{7,2}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 0.707106780904271\hfill & \hfill 4.442883051171333\hfill & \hfill -13.95772600673645\hfill & \hfill -29.233165736223864\hfill \\ \hfill 45.9166940244231\hfill & \hfill 57.754323004960845\hfill & \hfill -59.8089096658096\hfill & \hfill -57.790782092166637\hfill \end{array}\right]$$

$$\left[\begin{array}{ccc}\hfill a_{0,3}\hfill & \hfill \cdots \hfill & \hfill a_{3,3}\hfill \\ \hfill a_{4,3}\hfill & \hfill \cdots \hfill & \hfill a_{7,3}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 0.382683432364482\hfill & \hfill 5.804906304724222\hfill & \hfill -7.553868156289703\hfill & \hfill -38.194755175826380\hfill \\ \hfill 24.85114895244633\hfill & \hfill 75.39658368210462\hfill & \hfill -32.618972790201198\hfill & \hfill -71.729391408995241\hfill \end{array}\right].$$

Analogously, by symmetry, the filtered data of the right boundary region is

$$P_{R,i}(x)≔(f_{R,i,x}\ast u_{h_i})(x)=\sum _{k=0}^7{(-1)}^{k+1}{a}_{k,i}{(x-{\lambda }_{R,i})}^k,x\in [{\lambda }_{R,i},1].$$

Fig. 3a plots the polynomials P_L,i and P_R,i, the difference between the polynomials and the exact solution (note the scale 10⁻⁵ in Fig. 3b) and the error in log scale Fig. 3c. Fig. 3c matches the error graphs of [RLKV15, Figure 5, top-right] that were pointwise computed numerically.

Fig. 3.

RLKV-filtered boundary region of the DG output for d=3 of Example 2.3. The graphs in red, green, and blue correspond to i=1, 2, 3 respectively, i.e. to N=20, 40, 80 DG segments. (a) Three segments of degree 7 partially cover one another, representing the RLKV-filtered output, superimposed with the exact solution (black dashed). At this resolution all four are visually indistin-guishable where their domain overlaps (the x-domain of the boundary filter halves with doubling N). (b) Fine sampling of difference between the filtered output and the exact solution. (c) The canonical log scale display of the absolute error left: |P_L,i(x) − u_e(x)|, x∈[0, λ_L,i] and right: |P_R,i(x) − u_e(x)|, x∈[λ_R,i, 1] at computational resolution 6N. If the sampling were as fine as (b), the arches would reach down to zero.

Fig. 5.

Example graphs from Fig. 6a,b for L^∞ error ‖e(x, T)‖_∞,X vs final time for Test 4.1 (constant wave-speed, periodic boundary condition), d = 1, T = [0‥1]. The L^∞-error of PSIAC filters RLKV (red), SRV (green), NP₀ (blue) is computed only over the respective region: [0‥λ⁺] in (a) and [1 − λ⁺, 1] in (b). The top-most grey dash-dotted graphs in (a) and (b) are identical and represent e_[0‥1](T) for the raw DG output for N = 160 x-segments. The dotted grey graph in the middle is e_{[λ⁺‥1−λ⁺]}(T) for symmetric SIAC filter over the interior. Graphs for boundary filter errors are dashed when the overall partition is N = 80 and solid for N = 160. Each point on the abscissa therefore represents one final time 0 ≤ T ≤ 1 and each ordinate represents a maximal absolute error over the domain for that final time.

The polynomial characterization directly provides a symbolic expression for the derivatives of the convolved DG output.

Corollary 2.6 (Derivatives of PSIAC-filtered DG output [NP16])

$$\frac{{\mathrm{d}}^{\ell }}{\mathrm{d}{x}^{\ell }}(u\ast f_x)(x)={\mathbf{u}}_{ℐ}{Q}_{\lambda }\text{ diag}(h^{-(0:r)}){(\frac{{\mathrm{d}}^{\ell }}{\mathrm{d}{x}^{\ell }}{(x-h\lambda )}^{0:r})}^t.$$ (2.14)

2.5. Boundary filters as PSIAC filters

The symmetric knot sequence of the symmetric kernel of degree d is

$$\mathbf{t}≔h(-\mu ,-\mu +1,\dots ,\mu ),\mu ≔\frac{r+d+1}{2},r≔2d.$$ (2.15)

On [a‥b], this symmetric kernel can only be applied at evaluation points x where

$${\lambda }_{L,d}≔a+\mu ,\le x\le b-\mu ≕{\lambda }_{R,d}.$$ (2.16)

The boundary SIAC kernels RS [RS03] and SRV [SRV11] of reproduction degree r + 1 are of degree k = d, the degree of the DG output. Their index sequence 𝒥 is consecutive, and they are defined over the shifted knots

$${\mathit{t}}_{\ast ,d}(\xi )≔(-\mu ,-\mu +1,\dots ,\mu )+\xi -{\lambda }_{\ast ,d},\ast \in \{L,R\}$$ (2.17)

that form a symmetric support about the origin when ξ = λ_*,d. The two kernels differ in their degree: r(RS) = 2d and r(RV) = 4d. Explicit forms of the matrix G_λ, that is defined in Theorem 2.5 to efficiently construct the filter, are presented in [NP16].

The index sequence 𝒥 of the boundary kernel RLKV [RLKV15] is non-consecutive. The left and right kernels are of degree 2d + 1 and are defined over the shifted knots, symmetric about the origin:

$${\mathit{t}}_{L,d}(\xi )≔(-\mu ,\dots ,\mu -1,\underset{d+1\text{ times}}{\underbrace{\mu ,\dots ,\mu }})+\xi -{\lambda }_{L,d},$$ (2.18)

$${𝒥}_L≔\{1:(2d+1),3d+1\};$$

$${\mathit{t}}_{R,d}(\xi )≔(\underset{d+1\text{ times}}{\underbrace{-\mu ,\dots ,-\mu }},-\mu +1,\dots ,\mu ,)+\xi -{\lambda }_{R,d},$$ (2.19)

$${𝒥}_R≔\{1,d:(3d+1)\}.$$

Explicit forms of the matrix G_λ, that is defined in Theorem 2.5 to efficiently construct the filter, are presented in [NP16].

Theorem 2.5 shows that a PSIAC filter need not have the same degree as the sym- metric SIAC filter and it shows that PSIAC filters may have multiple knots without reducing the continuity of the filtered DG output. To illustrate this, [NP16] introduced filters with multiple interior knots. This class of filters is denoted NP_k and has knot sequences

$${\mathit{t}}_L≔x-{\lambda }_L+(-\mu ,\dots ,\mu -3,\mu -2,\underset{k+1\text{ times}}{\underbrace{\mu -1,\cdots ,\mu -1}},\underset{k+1\text{ times}}{\underbrace{\mu ,\cdots ,\mu }}),$$

$${\mathit{t}}_R≔x-{\lambda }_R+(\underset{k+1\text{ times}}{\underbrace{-\mu ,\cdots ,-\mu }},\underset{k+1\text{ times}}{\underbrace{-\mu +1,\cdots ,-\mu +1}},-\mu +2,-\mu +3\dots ,\mu ).$$ (2.20)

3. New least-degree DG filters

The symbolic formulation (2.11) applies to kernels of degree different from the degree d of the DG output. We may therefore consider a piecewise-constant (k = 0) PSIAC filter. The NP₀ filter has a consecutive index sequence 𝒥 and is defined over the shifted knots

$${\mathit{t}}_{\ast ,d}(\xi )≔(-\mu ,-\mu +1,\dots ,\mu )+\xi -{\lambda }_{\ast ,d},\mu =\frac{3d+1}{2},\ast \in \{L,R\}.$$ (3.1)

Therefore the piecewise constant NP₀ filter has reproduction degree r + 1 = 3d + 1 and the same support size as the symmetric kernel. (Numerical experiments show that a filter with 2d + 1 constant pieces still achieves optimal superconvergence – albeit with a larger error than using 3d + 1 pieces. We choose 3d + 1 since this is the number of pieces that the symmetric SIAC filter of the interior uses).

While the smoothness of the filtered output of position-independent filters, e.g. symmetric SIAC filters, depends on the filter degree, position-dependent PSIAC filters yield maximally smooth output regardless of their degree: by Theorem 2.5, the DG output filtered by a PSIAC filter is a polynomial over the respective boundary region independent of the degree or smoothness of the PSIAC filter. That is, even our piecewise constant NP₀ PSIAC filter increases the smoothness to infinity in the boundary region. Example 3.1 illustrates the remarkable fact that PSIAC filtering yields a single polynomial. This is in contrast to the finite smoothness at break points of data filtered with position-independent SIAC filters.

Example 3.1 (PSIAC-filtering yields polynomial output)

Let h = 1, χ_[α,β] be the indicator function of [α, β] and

$$u_{h_0}(x)≔{\chi }_{[0,1]}(x)+{\chi }_{[3,4]}(x),x\in [0,7]≕\mathrm{\Omega }.$$

the discontinuous DG output. Convolving u_h0 (x) at x in the interior region [2, 5] of Ω with the symmetric (position-independent) SIAC filter of reproduction degree r = 1,

$$𝒦(s)≔\frac{1}{2}B(s|-1:0)+\frac{1}{2}B(s|0:1)=\frac{1}{2}{\chi }_{[-1,1]}(s),$$

yields

$$(𝒦\ast u_{h_0})(x)=\frac{1}{2}B(s|2:4)+\frac{1}{2}B(s|3:5),x\in [2,5].$$ (3.2)

That is, convolving with 𝒦 yields a C⁰ output, as predicted by the SIAC theory developed by Ryan et al. [RS03, SRV11].

By contrast, at x in the left boundary region [0, 2] of Ω, convolving u_h0 (x) with the left-sided least-degree position-dependent PSIAC filter f_x defined in Example 2.2 yields

$$\text{for }x\in [0,2]:(f_x\ast u_{h_0})(x)={\int }_{ℝ}{f}_x(s)u_{h_0}(x-s)\mathrm{d}s\stackrel{\mathit{\text{by Eq}}.(2.9)}{=}{f}_{x;0}{\int }_{-2+x}^{-1+x}{u}_{h_0}(x-s)\mathrm{d}s+f_{x;1}{\int }_{-1+x}^{0+x}{u}_{h_0}(x-s)\mathrm{d}s\stackrel{0\le x-s\le 2}{=}{f}_{x;0}{\int }_{-2+x}^{-1+x}{\chi }_{[0,1]}(x-s)\mathrm{d}s+f_{x;1}{\int }_{-1+x}^{0+x}{\chi }_{[0,1]}(x-s)\mathrm{d}s\stackrel{\mathit{\text{change}}\mathit{:}t=x-s}{=}{f}_{x;0}{\int }_1^2{\chi }_{[0,1]}(t)\mathrm{d}t+f_{x;1}{\int }_0^1{\chi }_{[0,1]}(t)\mathrm{d}t=f_{x;1}\stackrel{\mathit{\text{by Eq}}.(2.10)}{=}\frac{1}{2}(3-2x).$$ (3.3)

The equality Eq. (3.3) holds because when 0 ≤ x ≤ 2 and −2 + x ≤ s ≤ 0 + x then 0 ≤ x − s ≤ 2. Hence u_h0 (x − s) = χ_[0,1](x − s). As Theorem 2.5 predicts $(f_x\ast u_{h_0})(x)=\frac{1}{2}(3-2x)$, x ∈ [0, 2], is a polynomial over the boundary region [0, 2].

Fig. 4 graphs instances of the SRV, RLKV and NP₀ kernels. Note that the NP₀ filter remains piecewise constant, while the degree of the other two filters increases with the degree d of the DG data.

Fig. 4.

Graphs of the three kernels defined at the left boundary x = a. By construction the degree of [SRV11] and [RLKV15] increases with d while the degree of the new NP₀ kernel remains piecewise constant. The NP₀ kernel (blue) has the same support as the RLKV kernel (red), shorter than the SRV kernel (green).

3.1. Symbolic form

We reduce the convolution of the DG data with the NP₀ kernels to an inner product of two short vectors. The inverse of the SIAC reproduction matrix M_t,𝒥 and the matrix G_λ*, of the formulation (2.11) in Theorem 2.5, are explicitly derived for NP₀ based on the following two propositions.

Proposition 3.1. (SIAC reproduction matrix for degree k = 0)

The SIAC reproduction matrix for least degree filters with index sequence 𝒥 ≔ (0, …, j_r) and knot sequence t_0:n is

$$M_{t_{0:n},𝒥}={\left[\frac{t_{j+1}^{\delta +1}-t_j^{\delta +1}}{t_{j+1}-t_j}\right]}_{\delta =0:r,j\in 𝒥}.$$ (3.4)

If the knot sequence t_0:n is uniform, i.e. h₁ ≔ t_j+1 − t_j, then

$$M_{t_{0:n},𝒥}=h_1^{-1}{[{(t_j+h_1)}^{\delta +1}-t_j^{\delta +1}]}_{\delta =0:r,j\in 𝒥}.$$ (3.5)

Proof

Since k = 0, each entry of M_t0:n,𝒥 given by Eq. (2.6) is the sum of all monomials in t_j+1 and t_j of total degree δ, and hence of the form (3.4). Eq. (3.5) is a direct consequence of Eq. (3.4).

Proposition 3.2 (G_λ* for degree k = 0 and uniform DG intervals)

Assume that the DG break point sequence s_0:m is uniform, hence after scaling consists of consecutive integers. Without loss of generality, the DG output on each interval [s_i, s_i+1] is defined in terms of Bernstein-Bézier polynomials $B_{\ell }^i$ of degree d, where the superscript i indicates the interval and ℓ = 0 : d, i.e.

$$B_{\ell }^i(x)≔\{\begin{array}{cc}(\begin{array}{c}\hfill d\hfill \\ \hfill \ell \hfill \end{array}){(x-s_i)}^{\ell }{(s_{i+1}-x)}^{d-\ell }\hfill & \mathit{\text{if }}x\in [s_i,s_{i+1}]\hfill \\ 0\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$ (3.6)

Let I be the 3d + 1 identity matrix and 1 the (d + 1) column vector of ones. The matrix G_λL for the left-sided kernel and the matrix G_λR for the right-sided kernel defined by Eq. (2.11) are

$$G_{{\lambda }_L}=G_{{\lambda }_R}=I\otimes \mathbf{1}=\frac{1}{d+1}{\left[\begin{array}{ccccccccccc}\hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill \end{array}\right]}^t\in {\mathbb{N}}^{(d+1)(3d+1)}\times {\mathbb{N}}^{(3d+1)}.$$ (3.7)

Proof

First we derive Eq. (3.7) for G_λL. In Eq. (2.12), we change to the variable t = s − λ_L + t_n. Since the B-splines are translation invariant and k = 0 (hence n = j_r + 1)

$$B(s|\lambda -t_{n-j:j_r-j})=B(t|t_n-t_{n-j:n-j-1})=B(t|j:j+1).$$ (3.8)

Consequently, Eq. (2.12) can be rewritten as

$$G_{{\lambda }_L}(i,j)={\int }_0^{3d+1}{\varphi }_i(t;s_{0:m}-{\lambda }_L+t_n)B(t|j:j+1)\mathrm{d}t.$$ (3.9)

Since $s_0=\frac{a}{h}$, the one-sided condition ${\lambda }_L=t_n+\frac{a}{h}$ implies that the first point of the sequence of translated DG break points s_0:m − λ_L + t_n equals 0, i.e., s₀ − λ_L + t_n = 0. Since the break points are consecutive integers starting from 0 and the B-splines B(t | j : j+1) are supported over [0, 3d+1], the relevant DG break points are 0 : 3d+1.

We re-write the basis functions ϕ_j(s ; s_0:m − λ_L + t_n), that are supported on an interval [i‥i + 1], in terms of DG output Bernstein-Bézier basis functions $B_{\ell }^i$, ℓ = 0‥d. Since each B-spline B(t | j : j + 1) is supported on [j, j + 1] and each $B_{\ell }^i$ is supported on [i, i + 1], the entries of G_λL are non-trivial only if i = j. Therefore

$$G_{{\lambda }_L}((d+1)i+\ell ,i)=\underset{i}{\overset{i+1}{\int }}{B}_{\ell }^i(t)B(t|i:i+1)\mathrm{d}t=\frac{1}{h}\times \frac{h}{d+1}=\frac{1}{d+1}.$$ (3.10)

The second last equality in Eq. (3.10) holds since $B(t{|i:i+1)|}_{[i,i+1]}\equiv \frac{1}{h}$ and the integral of $B_{\ell }^i(t)$ over [i, i + 1] equals $\frac{h}{d+1}$ [dB02]. Eq. (3.10) shares the entries of Eq. (3.7) for G_λL.

A similar argument for deriving G_λR but starting with the substitution t := −(s − λ_R + t₀) proves Eq. (3.7) for G_λR.

We note that G_λL = G_λR uniquely holds for the piecewise constant one-sided filter and not for any of the higher-degree filters in the literature.

Formula (3.7) of Proposition 3.2 makes it easy to explicitly list the matrix $Q_{\lambda }^{d,h}$ for computing the filtered DG output ${\mathbf{u}}_{[x]}·Q_{\lambda }^{d,h}{\mathit{\lambda }}_x$ for meshsize h and DG-degree d at x (cf. Eq. (1.1)). For d = 1,

$$Q_{\lambda }^{1,h}=Q_{\lambda }^{1,1}\text{ diag}([1,h^{-1},\dots ,h^{-r}]),Q_{\lambda }^{1,1}=\frac{1}{24}\left[\begin{array}{cccc}\hfill -1\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 7\hfill & \hfill -15\hfill & \hfill -3\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill -15\hfill & \hfill -3\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 15\hfill & \hfill -3\hfill & \hfill -6\hfill \\ \hfill 7\hfill & \hfill 15\hfill & \hfill -3\hfill & \hfill -6\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill -1\hfill & \hfill 3\hfill & \hfill 2\hfill \end{array}\right].$$ (3.11)

The Appendix A lists $Q_{\lambda }^{d,1}$ for d = 2 and d = 3.

3.2. Filter transition

Let ℒ_x * u_h denote the DG output u_h filtered by the left boundary PSIAC filter ℒ_x and 𝒦 * u_h the DG output filtered by the symmetric SIAC filter 𝒦 of the interior. These two filtered outputs overlap on the interval [a₁, a₂] = a + c_L + [0‥2h] where c_L separates the interior and left boundary. Without loss of generality, after substituting z ≔ (x − a₁)/(a₂ − a₁), we may assume that a₁ := 0, a₂ := 1. [SRV11] suggests a smoothness-preserving transition filtering scheme that we state succinctly as

$$u_h^{\star }(x)≔(1-\alpha (x))({ℒ}_x\ast u_h)(x)+\alpha (x)(𝒦\ast u_h)(x),$$ (3.12)

$$\alpha (x)≔\sum _{i=0}^{2\rho }{\alpha }_i{B}_i(x),{\alpha }_i≔\{\begin{array}{cc}0\hfill & \text{if }i\le \rho ;\hfill \\ 1\hfill & \text{else}.\hfill \end{array}$$

where $B_i(x)=(\begin{array}{c}\hfill 2\rho \hfill \\ \hfill i\hfill \end{array}){(1-x)}^{2\rho -1}{x}^i$ are Bernstein-Bézier polynomials of degree 2ρ defined over the unit interval [0‥1]. The Bernstein-Bézier representation guarantees that $u_h^{\star }$ Hermite-interpolates both filtered DG output up to order ρ. The degree 2ρ need not be twice the minimum degree of the boundary filter and the symmetric SIAC filter but can be chosen to further smooth out the transition. That is, one may choose ρ = 2 even though d = k = d_s = 1. The transition for ρ = 2 is less abrupt at 0 and 1 than for ρ = 1.

4. Numerical Comparison of the SRV, RLKV, NP₀ and the symmetric SIAC filter

This section compares numerical results on post-filtering DG output for (2.2) using the new NP₀ filters as well as the state-of-the-art filters, SRV [SRV11] and RLKV [RLKV15]. We implemented SRV and RLKV in their PSIAC form [NP16] and verified that the output of our RLKV implementation matches that of a numerical code provided by the authors [RLKV15]. Our measurements compare the absolute error e(x, T) as a function of domain points x and final times T. DG output is computed until time T before the filters are applied. Fig. 1a.b.c showed graphs for T-slices of e(x, T) for fixed T = 0.7 and x ∈ X = [0‥1]. The graphs of the Appendix display, as functions of T, the L^p norm of the slices

$$e_X(T)≔{\Vert e(x,T)\Vert }_{p,X},p\in \{2,\infty \},x\in X.$$

That is, each point on the graphs succinctly represents the max-norm or L²-norm error over the support X of the filter. Denote by [0‥λ⁺] the left boundary region [0‥λ] plus the transition region. Then for each boundary filter the graphs show e_[0‥λ⁺](T) on the left and e_{[1−λ⁺‥1]}(T) on the right and replicate in both left and right e_{[λ⁺‥1−λ⁺]}(T) for the symmetric SIAC filter and e_[0‥1](T) for DG. The graphs in the Appendix B summarize ca. 15,000 measurements, some of which are graphed as T-slices in the technical addendum [NPdf]. (Showing all T-slices would fill 100 pages in the compressed display format of [NPdf]).

The graphs plot errors for three instances of the canonical problem (2.2),

$$\frac{\mathit{\text{du}}}{d\tau }+\frac{d}{\mathit{\text{dx}}}(\kappa (x,\tau )u)=\rho (x,\tau )x\in (a‥b),\tau \in (0‥T).$$

The filters are applied at time T for final times T ∈ [0‥1] for Test 4.1 and for final times T ∈ [0‥2π] for the other two Test problems.

Test 4.1 (Constant wave-speed, periodic boundary conditions)

Consider the specializations of Eq. (2.2):

$$\kappa (x,\tau )\equiv 1,\rho (x,\tau )\equiv 0,0\le \tau \le T,$$ (4.1)

a ≔ 0, b ≔ 1, periodic boundary conditions, u₀(x) ≔ sin(2πx). The exact solution at the final time T is u(x, T) = sin(2π(x − T)).

Test 4.2 (Constant wave-speed, Dirichlet boundary conditions)

Consider Eq. (2.2) with specializations (4.1), but a ≔ 0, b ≔ 2π, Dirichlet boundary conditions, u₀(x) ≔ sin(x), u(0, τ) := −sin(τ). The exact solution is at the final time T is u(x, T) = sin(x − T).

Test 4.3 (Variable wave-speed, periodic boundary conditions)

Consider Eq. (2.2) with the specializations

$$\kappa (x,\tau )≔2+\text{sin}(x+\tau ),\rho (x,\tau )≔\text{cos}(x-\tau )+\text{sin}(2x),0\le \tau \le T,$$ (4.2)

a ≔ 0, b ≔ 2π, and periodic boundary conditions, u₀(x) ≔ sin(x). The exact solution at the final time T is u(x, T) = sin(x − T).

The comparison of the filters yields broadly the same qualitative results for all three test scenarios.

Fig. 6, 7 and 8 juxtapose the maximal point-wise errors ‖e(x, T)‖_∞,X for each final time T on the abcissa and resolutions of [0‥2π] into N = 80 segments (dashed) and N = 160 segments (solid) for d = 1, 2 and half the N-resolution for d = 3. The region X for the DG output is the whole domain, the left, respectively right boundary and transition regions for the boundary filters SRV, RLKV, NP₀, and the interior [λ⁺‥1 − λ⁺] for symmetric SIAC filter. In Fig. 5 (a copy of Fig. 6a,b) the filter error for Test 4.1 and the left boundary regions is plotted in (a) and the right boundary regions in (b). For example, for N = 160 and Test 4.2 solved up to final time T = 1/3 before filters were applied, we can read off in (a) the maximal absolute error of the left RLKV-filter as approximately 10⁻⁴, less that the top grey dotted DG-error and the error of the SRV-filter, the NP₀ and the symmetric SIAC filter as approximately 10⁻⁶. (The numbers for DG, RLKV and NP₀ match those of of the graphs in Section 1.1, Ex.1 of [NPdf] for fixed T = 1/3.).

Fig. 6.

L^∞ error vs final time Test 4.1 (constant wave-speed, periodic boundary condition). Abscissa: final time T at which the DG output was computed. Ordinate: maximum point-wise errors. Red (RLKV), green (SRV), blue (NP₀) dashed graphs correspond to N = 80 in (a−d) and to N = 40 in (e, f). All other graphs correspond to N = 160 in (a−d) and to N = 80 in (e, f), with top dot-dashed grey (DG output) and lower dotted grey (symmetric SIAC filter).

Fig. 7.

L^∞ error vs final time Test 4.2 (constant wave-speed, Dirichlet boundary condition). Abscissa: final time T at which the DG output was computed. Ordinate: maximum point-wise errors. Red (RLKV), green (SRV), blue (NP₀) dashed graphs correspond to N = 80 in (a−d) and to N = 40 in (e, f). All other graphs correspond to N = 160 in (a−d) and to N = 80 in (e, f), with top dot-dashed grey (DG output) and lower dotted grey (symmetric SIAC filter).

Fig. 8.

L^∞ error vs final time Test 4.3 (variable wave-speed, periodic boundary condition). Abscissa: final time T at which the DG output was computed. Ordinate: maximum point-wise errors. Red (RLKV), green (SRV), blue (NP₀) dashed graphs correspond to N = 80 in (a−d) and to N = 40 in (e, f). All other graphs correspond to N = 160 in (a−d) and to N = 80 in (e, f), with top dot-dashed grey (DG output) and lower dotted grey (symmetric SIAC filter).

Fig. 9, 10 and 11 display, for various final times T on the abscissa, the convergence rate

$$\rho =\rho (T,h)≔\text{ln }(\frac{e_X(T)\text{ for mesh size }2h}{e_X(T)\text{ for mesh size }h})/\text{ln }2.$$ (4.3)

for the L² and the L^∞ norm, for the left boundary region and the right boundary region, and for several h = 1/N where N is the sampling of the spatial domain. Fig. 9 and Fig. 10 each show ⧣(degrees×norms×boundary regions×final times×filter types or DG output ×refinements) = (3×2×2×50×5×2) = 6000 convergence rates. Fig. 11, with 30 final times, shows 3600 convergence rates. The graphs help us spot that convergence rates are higher for some isolated T than in general.

For d = 1, Fig. 6 of the Appendix B shows for all final times T that both SRV and NP₀ filtering result in lower errors than RLKV-filtering. For d = 2, 3, Fig. 7 and for d = 3, Fig. 8 point to the instability of the SRV filter at double precision: for certain final times T, the SRV-error even increases above that of the raw DG output. Instability also causes the SRV-error to exceed that of the NP₀ filter in Fig. 6. For d = 1, 2, the maximal point-wise errors of the RLKV filter are noticeably larger than those of the SRV filter and the NP₀ filter. The point-wise errors of the NP₀ filter are on par with the symmetric SIAC filter (in gray).

We note the consistency between L² and L^∞ convergence and therefore clearly higher rates for SRV, NP₀ and the symmetric SIAC filters over RLKV. When d = 1 and d = 2, the NP₀ and SRV filters show optimal L² and L^∞ super-convergence rates of order 2d + 1 in the boundary region. The convergence rate of 2 for the boundary regions for RLKV and d = 1 has been verified also with the original RLKV code. Fig. 12 in Appendix B compares the convergence orders for different end times T and DG input of degree d = 1, 2, 3.

Fig. 12.

L² error and convergence rates for the left boundary region for (top) end times T ∈ {1/3, 2/3} and periodic boundary constraints Test 4.1 and (bottom) T ∈ {2π/3, 4π/3} for Dirichlet constraints Test 4.2. The ordinate measures the L² error on the transition and boundary region, the abcissa counts the number of input DG pieces for the total interval [0‥1]. Only a constant fraction of these pieces is used by the boundary filters and hence to compute the L² norm. The grey downward lines in each graph indicate the slopes for optimal super-convergence order 2d + 1 that are attained by symmetric SIAC filter of DG degree d that is applied in the interior: from top to bottom they correspond to d = 1, 2, 3. The larger, black symbols indicate the (typically larger) error for RLKV: The smaller, red symbols indicate the error and rate for NP₀. The symbols are for d = 1 ‘◦’, for d = 2 ‘+’ and for d = 3 ‘∇’. The graphs for the L^∞ error look alike. Individual pointwise error, i.e. error-vs-x graphs are available at [NPdf].

For d = 3 the point-wise errors and convergence rates of all filters oscillate since calculations are close to machine precision. However, the point-wise errors and convergence rates of the NP₀ filter are on par with that of the symmetric SIAC filter, while the errors of the SRV filters are notably higher and convergence rates become sub-optimal, and even lower than d + 1 for some T when d = 3. Fig. 2 helps explain the instability of the SRV filters. Plotting the alternating entries of the convolution vector 𝒱 = Q_λλ at a boundary point x (e.g., x = 0 when a = 0), we find the entries of the SRV filters to be two orders of magnitude larger than those of NP₀.

5. Conclusion

PSIAC NP₀ filters enjoy three positive properties. First, explicit tabulated integer-based formulas make the computation of the NP₀ filters and their application for convolution more stable and efficient. Second, on canonical test equations, applying NP₀ filters reduces the errors of the DG output to that of optimal symmetric SIAC filters. Third, the support of the NP₀ filters matches that of the symmetric SIAC filter, making them naturally compatible.

Appendix A

Explicit filter matrix Q_λ for NP₀

For d = 1, $Q_{\lambda }^{1,1}$ is listed in (3.11). For d = 2,

$$Q_{\lambda }^{2,1}=\frac{1}{967680}[\begin{array}{ccccccc}\hfill -225\hfill & \hfill -7252\hfill & \hfill 3108\hfill & \hfill 7840\hfill & \hfill -2800\hfill & \hfill -1344\hfill & \hfill 448\hfill \\ \hfill -225\hfill & \hfill -7252\hfill & \hfill 3108\hfill & \hfill 7840\hfill & \hfill -2800\hfill & \hfill -1344\hfill & \hfill 448\hfill \\ \hfill -225\hfill & \hfill -7252\hfill & \hfill 3108\hfill & \hfill 7840\hfill & \hfill -2800\hfill & \hfill -1344\hfill & \hfill 448\hfill \\ \hfill 2862\hfill & \hfill 62608\hfill & \hfill -38808\hfill & \hfill -58240\hfill & \hfill 30240\hfill & \hfill 5376\hfill & \hfill -2688\hfill \\ \hfill 2862\hfill & \hfill 62608\hfill & \hfill -38808\hfill & \hfill -58240\hfill & \hfill 30240\hfill & \hfill 5376\hfill & \hfill -2688\hfill \\ \hfill 2862\hfill & \hfill 62608\hfill & \hfill -38808\hfill & \hfill -58240\hfill & \hfill 30240\hfill & \hfill 5376\hfill & \hfill -2688\hfill \\ \hfill -22863\hfill & \hfill -264740\hfill & \hfill 288540\hfill & \hfill 92960\hfill & \hfill -95760\hfill & \hfill -6720\hfill & \hfill 6720\hfill \\ \hfill -22863\hfill & \hfill -264740\hfill & \hfill 288540\hfill & \hfill 92960\hfill & \hfill -95760\hfill & \hfill -6720\hfill & \hfill 6720\hfill \\ \hfill -22863\hfill & \hfill -264740\hfill & \hfill 288540\hfill & \hfill 92960\hfill & \hfill -95760\hfill & \hfill -6720\hfill & \hfill 6720\hfill \\ \hfill 363012\hfill & \hfill 0\hfill & \hfill -505680\hfill & \hfill 0\hfill & \hfill 136640\hfill & \hfill 0\hfill & \hfill -8960\hfill \\ \hfill 363012\hfill & \hfill 0\hfill & \hfill -505680\hfill & \hfill 0\hfill & \hfill 136640\hfill & \hfill 0\hfill & \hfill -8960\hfill \\ \hfill 363012\hfill & \hfill 0\hfill & \hfill -505680\hfill & \hfill 0\hfill & \hfill 136640\hfill & \hfill 0\hfill & \hfill -8960\hfill \\ \hfill -22863\hfill & \hfill 264740\hfill & \hfill 288540\hfill & \hfill -92960\hfill & \hfill -95760\hfill & \hfill 6720\hfill & \hfill 6720\hfill \\ \hfill -22863\hfill & \hfill 264740\hfill & \hfill 288540\hfill & \hfill -92960\hfill & \hfill -95760\hfill & \hfill 6720\hfill & \hfill 6720\hfill \\ \hfill -22863\hfill & \hfill 264740\hfill & \hfill 288540\hfill & \hfill -92960\hfill & \hfill -95760\hfill & \hfill 6720\hfill & \hfill 6720\hfill \\ \hfill 2862\hfill & \hfill -62608\hfill & \hfill -38808\hfill & \hfill 58240\hfill & \hfill 30240\hfill & \hfill -5376\hfill & \hfill -2688\hfill \\ \hfill 2862\hfill & \hfill -62608\hfill & \hfill -38808\hfill & \hfill 58240\hfill & \hfill 30240\hfill & \hfill -5376\hfill & \hfill -2688\hfill \\ \hfill 2862\hfill & \hfill -62608\hfill & \hfill -38808\hfill & \hfill 58240\hfill & \hfill 30240\hfill & \hfill -5376\hfill & \hfill -2688\hfill \\ \hfill -225\hfill & \hfill 7252\hfill & \hfill 3108\hfill & \hfill -7840\hfill & \hfill -2800\hfill & \hfill 1344\hfill & \hfill 448\hfill \\ \hfill -225\hfill & \hfill 7252\hfill & \hfill 3108\hfill & \hfill -7840\hfill & \hfill -2800\hfill & \hfill 1344\hfill & \hfill 448\hfill \\ \hfill -225\hfill & \hfill 7252\hfill & \hfill 3108\hfill & \hfill -7840\hfill & \hfill -2800\hfill & \hfill 1344\hfill & \hfill 448\hfill \end{array}].$$

For d = 3,

$$Q_{\lambda }^{3,1}=\frac{1}{14515200}\left[\begin{array}{cccccccccc}\hfill 2880\hfill & \hfill -1152\hfill & \hfill -12300\hfill & \hfill 3280\hfill & \hfill 6825\hfill & \hfill -1638\hfill & \hfill -1050\hfill & \hfill 240\hfill & \hfill 45\hfill & \hfill -10\hfill \\ \hfill 2880\hfill & \hfill -1152\hfill & \hfill -12300\hfill & \hfill 3280\hfill & \hfill 6825\hfill & \hfill -1638\hfill & \hfill -1050\hfill & \hfill 240\hfill & \hfill 45\hfill & \hfill -10\hfill \\ \hfill 2880\hfill & \hfill -1152\hfill & \hfill -12300\hfill & \hfill 3280\hfill & \hfill 6825\hfill & \hfill -1638\hfill & \hfill -1050\hfill & \hfill 240\hfill & \hfill 45\hfill & \hfill -10\hfill \\ \hfill 2880\hfill & \hfill -1152\hfill & \hfill -12300\hfill & \hfill 3280\hfill & \hfill 6825\hfill & \hfill -1638\hfill & \hfill -1050\hfill & \hfill 240\hfill & \hfill 45\hfill & \hfill -10\hfill \\ \hfill -33120\hfill & \hfill 16848\hfill & \hfill 139020\hfill & \hfill -47160\hfill & \hfill -72975\hfill & \hfill 22302\hfill & \hfill 9870\hfill & \hfill -2880\hfill & \hfill -315\hfill & \hfill 90\hfill \\ \hfill -33120\hfill & \hfill 16848\hfill & \hfill 139020\hfill & \hfill -47160\hfill & \hfill -72975\hfill & \hfill 22302\hfill & \hfill 9870\hfill & \hfill -2880\hfill & \hfill -315\hfill & \hfill 90\hfill \\ \hfill -33120\hfill & \hfill 16848\hfill & \hfill 139020\hfill & \hfill -47160\hfill & \hfill -72975\hfill & \hfill 22302\hfill & \hfill 9870\hfill & \hfill -2880\hfill & \hfill -315\hfill & \hfill 90\hfill \\ \hfill -33120\hfill & \hfill 16848\hfill & \hfill 139020\hfill & \hfill -47160\hfill & \hfill -72975\hfill & \hfill 22302\hfill & \hfill 9870\hfill & \hfill -2880\hfill & \hfill -315\hfill & \hfill 90\hfill \\ \hfill 182880\hfill & \hfill -127152\hfill & \hfill -737400\hfill & \hfill 342360\hfill & \hfill 338100\hfill & \hfill -142128\hfill & \hfill -33600\hfill & \hfill 13680\hfill & \hfill 900\hfill & \hfill -360\hfill \\ \hfill 182880\hfill & \hfill -127152\hfill & \hfill -737400\hfill & \hfill 342360\hfill & \hfill 338100\hfill & \hfill -142128\hfill & \hfill -33600\hfill & \hfill 13680\hfill & \hfill 900\hfill & \hfill -360\hfill \\ \hfill 182880\hfill & \hfill -127152\hfill & \hfill -737400\hfill & \hfill 342360\hfill & \hfill 338100\hfill & \hfill -142128\hfill & \hfill -33600\hfill & \hfill 13680\hfill & \hfill 900\hfill & \hfill -360\hfill \\ \hfill 182880\hfill & \hfill -127152\hfill & \hfill -737400\hfill & \hfill 342360\hfill & \hfill 338100\hfill & \hfill -142128\hfill & \hfill -33600\hfill & \hfill 13680\hfill & \hfill 900\hfill & \hfill -360\hfill \\ \hfill -681120\hfill & \hfill 736848\hfill & \hfill 2408280\hfill & \hfill -1754760\hfill & \hfill -644700\hfill & \hfill 447552\hfill & \hfill 52080\hfill & \hfill -35280\hfill & \hfill -1260\hfill & \hfill 840\hfill \\ \hfill -681120\hfill & \hfill 736848\hfill & \hfill 2408280\hfill & \hfill -1754760\hfill & \hfill -644700\hfill & \hfill 447552\hfill & \hfill 52080\hfill & \hfill -35280\hfill & \hfill -1260\hfill & \hfill 840\hfill \\ \hfill -681120\hfill & \hfill 736848\hfill & \hfill 2408280\hfill & \hfill -1754760\hfill & \hfill -644700\hfill & \hfill 447552\hfill & \hfill 52080\hfill & \hfill -35280\hfill & \hfill -1260\hfill & \hfill 840\hfill \\ \hfill -681120\hfill & \hfill 736848\hfill & \hfill 2408280\hfill & \hfill -1754760\hfill & \hfill -644700\hfill & \hfill 447552\hfill & \hfill 52080\hfill & \hfill -35280\hfill & \hfill -1260\hfill & \hfill 840\hfill \\ \hfill 2342880\hfill & \hfill -5311152\hfill & \hfill -1797600\hfill & \hfill 3853080\hfill & \hfill 372750\hfill & \hfill -773388\hfill & \hfill -27300\hfill & \hfill 55440\hfill & \hfill 630\hfill & \hfill -1260\hfill \\ \hfill 2342880\hfill & \hfill -5311152\hfill & \hfill -1797600\hfill & \hfill 3853080\hfill & \hfill 372750\hfill & \hfill -773388\hfill & \hfill -27300\hfill & \hfill 55440\hfill & \hfill 630\hfill & \hfill -1260\hfill \\ \hfill 2342880\hfill & \hfill -5311152\hfill & \hfill -1797600\hfill & \hfill 3853080\hfill & \hfill 372750\hfill & \hfill -773388\hfill & \hfill -27300\hfill & \hfill 55440\hfill & \hfill 630\hfill & \hfill -1260\hfill \\ \hfill 2342880\hfill & \hfill -5311152\hfill & \hfill -1797600\hfill & \hfill 3853080\hfill & \hfill 372750\hfill & \hfill -773388\hfill & \hfill -27300\hfill & \hfill 55440\hfill & \hfill 630\hfill & \hfill -1260\hfill \\ \hfill 2342880\hfill & \hfill 5311152\hfill & \hfill -1797600\hfill & \hfill -3853080\hfill & \hfill 372750\hfill & \hfill 773388\hfill & \hfill -27300\hfill & \hfill -55440\hfill & \hfill 630\hfill & \hfill 1260\hfill \\ \hfill 2342880\hfill & \hfill 5311152\hfill & \hfill -1797600\hfill & \hfill -3853080\hfill & \hfill 372750\hfill & \hfill 773388\hfill & \hfill -27300\hfill & \hfill -55440\hfill & \hfill 630\hfill & \hfill 1260\hfill \\ \hfill 2342880\hfill & \hfill 5311152\hfill & \hfill -1797600\hfill & \hfill -3853080\hfill & \hfill 372750\hfill & \hfill 773388\hfill & \hfill -27300\hfill & \hfill -55440\hfill & \hfill 630\hfill & \hfill 1260\hfill \\ \hfill 2342880\hfill & \hfill 5311152\hfill & \hfill -1797600\hfill & \hfill -3853080\hfill & \hfill 372750\hfill & \hfill 773388\hfill & \hfill -27300\hfill & \hfill -55440\hfill & \hfill 630\hfill & \hfill 1260\hfill \\ \hfill -681120\hfill & \hfill -736848\hfill & \hfill 2408280\hfill & \hfill 1754760\hfill & \hfill -644700\hfill & \hfill -447552\hfill & \hfill 52080\hfill & \hfill 35280\hfill & \hfill -1260\hfill & \hfill -840\hfill \\ \hfill -681120\hfill & \hfill -736848\hfill & \hfill 2408280\hfill & \hfill 1754760\hfill & \hfill -644700\hfill & \hfill -447552\hfill & \hfill 52080\hfill & \hfill 35280\hfill & \hfill -1260\hfill & \hfill -840\hfill \\ \hfill -681120\hfill & \hfill -736848\hfill & \hfill 2408280\hfill & \hfill 1754760\hfill & \hfill -644700\hfill & \hfill -447552\hfill & \hfill 52080\hfill & \hfill 35280\hfill & \hfill -1260\hfill & \hfill -840\hfill \\ \hfill -681120\hfill & \hfill -736848\hfill & \hfill 2408280\hfill & \hfill 1754760\hfill & \hfill -644700\hfill & \hfill -447552\hfill & \hfill 52080\hfill & \hfill 35280\hfill & \hfill -1260\hfill & \hfill -840\hfill \\ \hfill 182880\hfill & \hfill 127152\hfill & \hfill -737400\hfill & \hfill -342360\hfill & \hfill 338100\hfill & \hfill 142128\hfill & \hfill -33600\hfill & \hfill -13680\hfill & \hfill 900\hfill & \hfill 360\hfill \\ \hfill 182880\hfill & \hfill 127152\hfill & \hfill -737400\hfill & \hfill -342360\hfill & \hfill 338100\hfill & \hfill 142128\hfill & \hfill -33600\hfill & \hfill -13680\hfill & \hfill 900\hfill & \hfill 360\hfill \\ \hfill 182880\hfill & \hfill 127152\hfill & \hfill -737400\hfill & \hfill -342360\hfill & \hfill 338100\hfill & \hfill 142128\hfill & \hfill -33600\hfill & \hfill -13680\hfill & \hfill 900\hfill & \hfill 360\hfill \\ \hfill 182880\hfill & \hfill 127152\hfill & \hfill -737400\hfill & \hfill -342360\hfill & \hfill 338100\hfill & \hfill 142128\hfill & \hfill -33600\hfill & \hfill -13680\hfill & \hfill 900\hfill & \hfill 360\hfill \\ \hfill -33120\hfill & \hfill -16848\hfill & \hfill 139020\hfill & \hfill 47160\hfill & \hfill -72975\hfill & \hfill -22302\hfill & \hfill 9870\hfill & \hfill 2880\hfill & \hfill -315\hfill & \hfill -90\hfill \\ \hfill -33120\hfill & \hfill -16848\hfill & \hfill 139020\hfill & \hfill 47160\hfill & \hfill -72975\hfill & \hfill -22302\hfill & \hfill 9870\hfill & \hfill 2880\hfill & \hfill -315\hfill & \hfill -90\hfill \\ \hfill -33120\hfill & \hfill -16848\hfill & \hfill 139020\hfill & \hfill 47160\hfill & \hfill -72975\hfill & \hfill -22302\hfill & \hfill 9870\hfill & \hfill 2880\hfill & \hfill -315\hfill & \hfill -90\hfill \\ \hfill -33120\hfill & \hfill -16848\hfill & \hfill 139020\hfill & \hfill 47160\hfill & \hfill -72975\hfill & \hfill -22302\hfill & \hfill 9870\hfill & \hfill 2880\hfill & \hfill -315\hfill & \hfill -90\hfill \\ \hfill 2880\hfill & \hfill 1152\hfill & \hfill -12300\hfill & \hfill -3280\hfill & \hfill 6825\hfill & \hfill 1638\hfill & \hfill -1050\hfill & \hfill -240\hfill & \hfill 45\hfill & \hfill 10\hfill \\ \hfill 2880\hfill & \hfill 1152\hfill & \hfill -12300\hfill & \hfill -3280\hfill & \hfill 6825\hfill & \hfill 1638\hfill & \hfill -1050\hfill & \hfill -240\hfill & \hfill 45\hfill & \hfill 10\hfill \\ \hfill 2880\hfill & \hfill 1152\hfill & \hfill -12300\hfill & \hfill -3280\hfill & \hfill 6825\hfill & \hfill 1638\hfill & \hfill -1050\hfill & \hfill -240\hfill & \hfill 45\hfill & \hfill 10\hfill \\ \hfill 2880\hfill & \hfill 1152\hfill & \hfill -12300\hfill & \hfill -3280\hfill & \hfill 6825\hfill & \hfill 1638\hfill & \hfill -1050\hfill & \hfill -240\hfill & \hfill 45\hfill & \hfill 10\hfill \end{array}\right].$$

Appendix B

Numerical experiments - Error Graphs and Convergence Rates

Our measurements compare the absolute error e(x, T) as a function of domain points x and final times T. DG output is computed until time T before the filters are applied. Fig. 1a.b.c showed graphs for T-slices of e(x, T) for fixed T = 0.7 and x ∈ X = [0‥1]. The graphs of the Appendix display, as functions of T, the L^p norm of the slices

$$e_X(T)≔{\Vert e(x,T)\Vert }_{p,X},p\in \{2,\infty \},x\in X.$$

That is, each point on the graphs succinctly represents the max-norm or L²-norm error over the support X of the filter. Denote by [0‥λ⁺] the left boundary region [0‥λ] plus the transition region. Then for each boundary filter the graphs show e_[0‥λ⁺](T) on the left and e_{[1−λ⁺‥1]}(T) on the right and replicate in both left and right e_{[λ⁺‥1−λ⁺]}(T) for the symmetric SIAC filter and e_[0‥1](T) for DG. Fig. 9, 10 and 11 show L^∞ and L² convergence rates of filtered DG output plotted against the final time T.

The errors in Fig. 6, 7, and 8 are computed for DG output of degree d = 1, 2, 3 and for uniform partitions of the x-domain into intervals of length h ≔ 1/N. A fixed number of points, depending on d but not N, in the boundary regions are the result of PSIAC-filtering with one of SRV, RLKV, NP₀. (We verified correctness of RLKV by comparison with a certified numerical RLKV implementation.) We emphasize that all boundary filters are measured only in the boundary regions inclusive of the transition region. Errors of the symmetric filter are computed only for the interior exclusive of the transition region. With increasing N the interior converges to the full domain and symmetric SIAC filter contributes an increasing number of values. The number of values generated by boundary filters does not increase with N but stays fixed. Regardless of the choice of boundary filter, the L² error over the full domain is dominated by the symmetric SIAC filter. Therefore errors and rates for the boundary filters are not combined with those of the symmetric filter.

The L^∞ errors in (Test 4.1, Test 4.2, Test 4.3) have been computed for (50, 50, 30) different final times T in ([0, 1], [0, 2π], [0, 2π]) respectively. (Due to scaling by 2π, the test problems have different final times T.) For example, the value of the RLKV error for Test 4.1 for N = 80 and DG element degree d = 1 (dashed red graph in Fig. 6a) at final time T = 0.4 is ca. 0.0002. This value is obtained by solving Test 4.1 using DG of degree d = 1 up to time T = 0.4 and then convolving with the RLKV filter to obtain s = 6 samples per boundary region interval for a total of 3s error samples. The maximum absolute sampled value is 0.0002. The error of the symmetric SIAC filter is computed from the remaining interior N − 2 * 3s samples. The s samples within the intervals are uniformly distributed. Sampling instead at Gauss-Legendre points does not noticeably change the outcome.

All graphs share the same color and style assignments. In Fig. 6, 7, and 8, the raw DG output for N = 160 in (a−d) and N = 80 in (e,f) is the top-most dash-dotted graph (largest error). In Fig. 9, 10 and 11 the DG convergence-rate is at the bottom. The dotted grey graph represents symmetric SIAC filter for N = 160 in (a−d) and N = 80 in (e,f). The boundary filters are RLKV (red), SRV(green) and NP₀ filter (blue). Dashed graphs correspond to N = 80 in (a−d) and N = 40 in (e,f) and solid graphs to N = 160 in (a−d) and N = 80 in (e,f), halving the interval length h.

Fig. 9 for Test 4.1 and Fig. 10 for Test 4.2, and Fig. 11 for Test 4.3 display the L^∞ and the L² convergence rates of the filtered DG outputs. All computations are performed in double precision.