COMPUTING ILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, USING A STABILIZED EXPLICIT FINITE DIFFERENCE SCHEME MARCHING BACKWARD IN TIME

Abstract

This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on (−∆)^p, with real p > 2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier-Stokes initial value problems.

Several reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on 256 × 256 pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T > 0. Successful backward recovery is shown to be possible at parameter values exceeding expectations.

1. Introduction.

Significant applications in science and engineering have generated considerable interest in the development of effective computational techniques for solving ill-posed time-reversed evolution equations. In Environmental Forensics, much success has been achieved using backward parabolic equations to locate sources of both groundwater and atmospheric contaminant plumes [1–3]. In Geophysics, the inverse problem of thermal evolution of the Earth interior is of major interest [4,5]. In Image Science, time-reversed fractional and logarithmic diffusion equations have been successfully applied to deblur nanoscale scanning electron micrographs, as well as galactic scale Hubble space telescope imagery [6–8]. In Aerodynamics, backward Burgers’ equations play a central role in questions of inverse design associated with minimizing aircraft sonic boom [9–11]. Backward Burgers’ equations are also of prime interest in data assimilation studies in Geophysical Fluid Dynamics [12,13].

Continuing the line of work developed in [14–19], the present self-contained paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed time-reversed 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. Eventually, this leads to a distortion away from the true solution. However, in many interesting cases, the cumulative effect of these errors is sufficiently small to allow for useful results. Effective smoothing operators based on (−∆)^p, with real p > 2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in various other ill-posed equations, [14–19]. As was the case in these papers, the analysis of numerical stabilty given in Sections 5 and 6 below, is restricted to a related linear problem. However, extensive numerical experiments indicate that these linear stability results remain valid when the explicit scheme is applied to a significant class of nonlinear problems.

Indeed, in the documented experiments discussed in Section 7, it is noteworthy that notwithstanding its limitations, the stabilized explicit scheme can produce useful reconstructions at parameter values far exceeding what would seem feasible, based on well-known uncertainty estimates for backward in time Navier-Stokes solutions.

In recent years, important new discretization methods have been developed for solving forward 2D incompressible Navier-Stokes equations, as well as other dissipative problems. See e.g., [20–28] and the references therein. Provided appropriate stabilizing operators can be constructed and applied at every time step t_n = n∆t, several such methods may possibly be marched backward in time, and produce valuable new computational tools for solving ill-posed time-reversed evolution equations.

2. 2D Navier-Stokes equations.

Let Ω be the unit square in R² with boundary ∂Ω. Let <, > and ‖ ‖₂, respectively denote the scalar product and norm on ${\mathcal{L}}^2(\Omega )$. We shall study the 2D problem in stream function-vorticity formulation [20,21]. Here, the flow is governed by a stream function ψ(x, y, t) which defines the velocity components $u\left(x,y,t\right)\equiv {\psi }_y$, $v\left(x,y,t\right)\equiv -{\psi }_x$, together with the vorticity $w\left(x,y,t\right)\equiv v_x-u_y=-\Delta \psi$. In this exploratory initial paper, we restrict attention to stream functions ψ defined on the unit square Ω, which, for the limited duration 0 ≤ t ≤ T, of the flow, vanish together with all their derivatives near the boundary ∂Ω. However, ψ(x, y, t) will generally have large spatial derivatives away from the boundary. Given such a ψ(x, y,0) and a kinematic viscosity ν > 0, the forward Navier-Stokes initial value problem takes the form

$$\begin{gathered} {\omega }_t+u{w}_x+v{w}_y-\nu \Delta w=0,\hspace{1em}\hspace{0.17em}\Delta \psi =-w,\hspace{1em}\hspace{0.17em}\hspace{0.17em}\left(x,y,t\right)\in \Omega \times \left(0,T\right], \\ u\left(x,y,0\right)={\psi }_y,\hspace{1em}v\left(x,y,0\right)=-{\psi }_x,\hspace{1em}w\left(x,y,0\right)=-\Delta \psi , \\ u\left(x,y,t\right)=v\left(x,y,t\right)=w\left(x,y,t\right)=0,\hspace{1em}\hspace{0.17em}\hspace{0.17em}\left(x,y,t\right)\in \partial \Omega \times \left[0,T\right]. \end{gathered}$$ (2.1)

Solving the system in Eq. (2.1) may be visualized as a marching computation as follows. Given the initial values, the evolution equation for ω allows advancing ω forward one time step ∆t > 0. The Poisson equation ∆ψ = −ω is then solved to obtain ψ(x, y, ∆t), which leads to u(x, y, ∆t), v(x, y, ∆t). This now allows advancing ω to the next time step through the evolution equation, and so on.

With A the area of the flow domain, define U_max and the Reynolds number RE as follows

$$\begin{gathered} U_{max}={\sup }_{\Omega }{\left\{u^2\left(x,y,0\right)+v^2\left(x,y,0\right)\right\}}^{1/2}, \\ RE=\left(\sqrt{A}/\nu \right)U_{max}. \end{gathered}$$ (2.2)

As in [14–19], our numerical experiments will involve 256×256 pixel gray scale images. These images, defined by highly non-smooth underlying intensity data, present significant challenges to any ill-posed reconstruction algorithm. At the same time, using images of recognizable objects is highly instructive, as it enables immediate visual appreciation of the amount of distortion produced by the forward evolution, together with the subsequent partial undoing of that distortion by marching backward in time.

In [14–19], image intensity data were used directly to represent actual primary variables such as displacement, velocity, temperature, etc. In the present paper, image data are used to construct non-smooth stream functions ψ(x, y), and the primary variables u, v, ω, in Eq. (2.1) are derived by taking first and second derivatives of ψ. In particular, larger values of the Reynolds number RE are to be expected.

3. Feasibility of backward in time Navier-Stokes reconstruction.

In dissipative evolution equations, there is a necessary uncertainty in backward in time reconstructions from imprecise data given at some positive time T. That uncertainty is a function of how quickly the given evolution equation forgets the past as time advances. This is a deep subject, explored in such publications as [30–38], and the references therein.

We now contemplate the feasibility of solving the system in Eq. (2.1), backward in time, from approximately known values. Let $\left\{\tilde{u}\left(x,y,t\right),\hspace{0.17em}\tilde{v}\left(x,y,t\right),\tilde{w}\left(x,y,t\right)\right\}$ be the solution on Ω × [0,T) corresponding to approximate data $\left\{\tilde{u}\left(\cdot ,T\right),\tilde{v}\left(\cdot ,T\right),\tilde{w}\left(\cdot ,T\right)\right\}$ at a given positive time T, and let {u(x, y, t), v(x, y, t), ω(x, y, t)} be the unique solution associated with the true data {u(·,T), v(·,T), ω(·,T)} at time T. Define w(x, y, t) on Ω × [0,T], and F(t) on [0,T], as follows

$$\begin{gathered} {\left(w\left(x,y,t\right)\right)}^2={\left\{u\left(x,y,t\right)-\tilde{u}\left(x,y,t\right)\right\}}^2+{\left\{v\left(x,y,t\right)-\tilde{v}\left(x,y,t\right)\right\}}^2, \\ F\left(t\right)={\Vert w\left(\cdot ,t\right)\Vert }_2^2={\int }_{\Omega }{\left(w\left(x,y,t\right)\right)}^{2}dxdy. \end{gathered}$$ (3.1)

For a given small ϵ > 0, assume that |w(x, y, T)| is small enough that F(T) ≤ ϵ What constraints must be placed on the solutions of Eq. (2.1) to ensure that F(t) will remain small for 0 ≤ t < T ? This is the backward stability problem for the Navier-Stokes equations in the ${\mathcal{L}}^2(\Omega )$ norm. The best known results in this direction were obtained by Knops and Payne using logarithmic convexity arguments [32,33,35].

Let P, Q, be prescribed positive constants. A velocity field {u(x, y, t), v(x, y, t)} is said to belong to the class $\mathcal{P}$ if

$$\underset{\left(x,t\right)\in \Omega \times \left[0,T\right]}{\sup }\left\{\left(u^2+v^2\right)\right\}\le P^2,$$ (3.2)

while it belongs to the class $\mathcal{Q}$ if

$$\underset{\left(x,t\right)\in \Omega \times \left[0,T\right]}{\sup }\left\{\left(u^2+v^2\right)+w^2+\left(u_t^2+v_t^2\right)\right\}\le Q^2.$$ (3.3)

Define a, b, c, µ(t), K, as follows,

$$\begin{gathered} a=2\left(P^2+1\right)/\nu ,\hspace{1em}\hspace{0.17em}b=Q^2\left(1+a/\nu \right), \\ c=b/a,\hspace{1em}\hspace{1em}\mu \left(t\right)=\left(e^{at}-1\right)/\left(e^{aT}-1\right),\hspace{1em}0\le t\le T, \\ K\equiv K\left(P,Q,\nu ,t,T\right)=\exp \left\{c\left(t-\mu \left(t\right)T\right)\right\},\hspace{1em}0\le t\le T. \end{gathered}$$ (3.4)

If $\left\{u\left(x,y,t\right),v\left(x,y,t\right)\right\}\in \mathcal{P}$ and $\left\{\tilde{u}\left(x,y,t\right),\tilde{v}\left(x,y,t\right)\right\}\in \mathcal{Q}$, it is shown in [32,33] that then

$$F\left(t\right)F^{″}\left(t\right)-{\left(F^{\prime }\left(t\right)\right)}^2\ge aF\left(t\right)F^{\prime }\left(t\right)-b{F}^2\left(t\right),$$ (3.5)

from which, as will be shown below, it follows that

$$F\left(t\right)\le e^{\left\{c\left(t-\mu \left(t\right)T\right)\right\}}{\left\{F\left(0\right)\right\}}^{1-\mu \left(t\right)}{\left\{F\left(T\right)\right\}}^{\mu \left(t\right)},\hspace{1em}0\le t\le T.$$ (3.6)

With log x denoting the natural logarithm of x > 0, we establish Eq. (3.6) by reproducing the convexity arguments developed in [30, p. 18] and [31, p. 34]. We may assume that F(t) > 0, 0 ≤ t ≤ T, since, as was shown in [29], if F(t^†) = 0 for any t^† in [0,T], then $F\left(t\right)\equiv 0$ on [0,T], and Eq. (3.6) is trivially true in that case. Multiplying both sides of Eq. (3.5) by e^−at/F²(t), we find

$${\left(e^{-at}{F}^{\prime }/F\right)}^{\prime }\ge -b{e}^{-at}.$$ (3.7)

Let s = e^at, and change variables from t to s in Eq. (3.7). With G(s) = F(t), we then have $F^{\prime }\left(t\right)=\left(dG/ds\right)\left(ds/dt\right)=as\left(dG/ds\right)$. Hence, from Eq. (3.7),

$${\left(e^{-at}{F}^{\prime }/F\right)}^{\prime }=s{a}^{2}d/ds\left\{\left(dG/ds\right)/G\right\}\ge -b/s.$$ (3.8)

Therefore, with s₁ = 1, s₂ = e^aT,

$$\begin{gathered} d^2/d{s}^2\left\{\log G\left(s\right)\right\}\ge -b/\left(a^2{s}^2\right)=-d^2/d{s}^2\left\{\log \left(s^{-b/a^2}\right)\right\},\hspace{1em}\hspace{1em}{s}_1\le s\le s_2, \\ {d}^2/d{s}^2\left\{\log \left(G\left(s\right)s^{-b/a^2}\right)\right\}\ge 0,\hspace{1em}\hspace{0.17em}{s}_1\le s\le s_2. \end{gathered}$$ (3.9)

The last inequality implies that $\log \left(G\left(s\right)s^{-b/a^2}\right)$ is a convex function of s on [s₁,s₂]. Hence, on that same interval,

$$G\left(s\right)s^{-b/a^2}\le {\left\{G\left(s_1\right)s_1^{-b/a^2}\right\}}^{\left(s_2-s\right)/\left(s_2-s_1\right)}{\left\{G\left(s_2\right)s_2^{-b/a^2}\right\}}^{\left(s-s_1\right)/\left(s_2-s_1\right)}.$$ (3.10)

Returning to the original t variable, Eq. (3.10) translates into Eq. (3.6). QED.

Next, in Eq. (3.1), let M be an a-priori bound for $F\left(0\right)={\Vert \omega \left(\cdot ,0\right)\Vert }_2^2$. With F(T) ≤ ϵ, we then obtain from Eq. (3.6)

$${\Vert w\left(\cdot ,t\right)\Vert }_2^2=F\left(t\right)\le e^{\left\{c\left(t-\mu (t\right)T)\right\}}{M}^{1-\mu \left(t\right)}{ϵ}^{\mu \left(t\right)},\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le t\le T.$$ (3.11)

However, as the example below indicates, even with small ϵ > 0, || w(., t) ||₂ need not be small for t < T, because the factor $K=\exp \left\{c\left(t-\mu \left(t\right)T\right)\right\}$ may be extremely large, unless T is extremely small.

3.1. An example: the Joan Crawford stream function.

The 256 × 256 pixel gray-scale Joan Crawford image, shown in Figure 3.1, is defined by highly non smooth intensity data, with integer values ranging between 0 and 255. Here, these values are multiplied by 0.0025 to create an array ψ(x, y) of non negative numbers ranging from 0 to 0.6375. We now consider ψ(x, y) as a candidate initial stream function, defined on the unit square Ω. Contour plots of ψ(x, y,0) and of the associated vorticity ω(x, y,0) = −∆ψ, are also shown in Figure 3.1. However, even though 0 ≤ ψ(x, y,0) ≤ 0.6375, one must expect large values for its derivatives. Indeed, with U_max as in Eq. (2.2), we find

$$U_{max}\approx 114,\hspace{1em}\hspace{0.17em}\hspace{0.17em}\underset{\Omega }{\sup }\left\{\left|w\left(x,y,0\right)\right|\right\}\approx 1.0\times {10}^5.$$ (3.12)

Using ψ(x, y,0) in Figure 3.1 together with ν = 0.01 in the forward problem in Eq. (2.1), and using A = 1 in Eq. (2.2), leads to a Reynolds number RE ≈ 11400. Without the above rescaling, these critical values would be 400 times larger.

Fig. 3.1.

Non-smooth Joan Crawford data leads to challenging ill-posed Navier-Stokes reconstruction problem. Here, U_max = 114, RE = 11400, sup_Ω{|ω(x, y,0)|} = 1.0 × 10⁵. Theoretical uncertainty estimates indicate backward recovery of above data from distorted values at T > 0, would be feasible provided T ≤ 1.0 × 10⁻⁹.

We now assess the feasibility of reconstructing the initial values shown in Figure 1, from the data at time T. From Eqs. (3.2, 3.3, 3.4), we have

$$\begin{gathered} P^2>U_{max}^2\approx 13000,\hspace{1em}\hspace{0.17em}\hspace{0.17em}{Q}^2>1.0\times {10}^{10}, \\ a\approx 2.6\times {10}^6,\hspace{1em}b\approx 2.6\times {10}^{18},\hspace{1em}c\approx 1.0\times {10}^{12}, \\ \mu \left(t\right)=\left(e^{at}-1\right)/\left(e^{aT}-1\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}K=e^{\left\{c\left(t-\mu \left(t\right)T\right)\right\}}. \end{gathered}$$ (3.13)

With T = 1.0 × 10⁻⁹, and t = T/2, we find µ(T/2) = 0.499675, and K ≈ 1.4. However, with T = 1.0 × 10⁻⁸, and t = T/2, we now find µ(T/2) = 0.49675, and K ≈ 1.3 × 10¹⁴. Thus, if T ≥ 1.0 × 10⁻⁸, the reconstruction uncertainty at t = T/2, given in Eq. (3.11), is much too large for any realistic value of the data error ϵ at time T.

Rigorous uncertainty estimates, such as the Knops-Payne result in Eq. (3.11), must necessarily reflect worse-case error propagation scenarios, and may be too pessimistic in some cases. In Section 7 below, several backward recovery experiments are discussed and documented. In these experiments, the kinematic viscosity ν = 0.01, and the associated Reynolds numbers RE, vorticities |ω|, and speeds U_max, are of magnitudes comparable to those found in the Joan Crawford stream function in Figure 3.1, resulting in parameter values very similar to those displayed in Eqs. (3.12) and (3.13). Nevertheless, using the stabilized explicit scheme discussed in Eq. (4.8) below, remarkably good recovery at t = 0 is found possible from approximate data at values of T on the order of 1.0 × 10⁻³. However, at larger values of T, the resulting image blur becomes too severe and precludes useful reconstruction.

4. Stabilized explicit schemes for the 2D Navier-Stokes Equations.

For well-posed initial value problems, explicit stepwise marching schemes are quite useful, but often require restrictive Courant stability conditions. Some explicit schemes, such as Richardson’s leapfrog scheme, are unconditionally unstable for parabolic problems, [39, p. 190]. For ill-posed initial value problems, all consistent stepwise marching schemes, whether explicit or implicit, are necessarily unconditionally unstable [39, p. 59]. It is an important fact that one can stabilize explicit marching schemes, and render them unconditionally stable marching forward or backward in time, but at the cost of making them slightly inconsistent.

Consider the following 2D Navier-Stokes system for (x, y) ∈ Ω,

$$w_t=L^{†}w=\nu \Delta w-u{w}_x-v{w}_y,\hspace{1em}\hspace{0.17em}\Delta \psi =-w,\hspace{1em}\hspace{0.17em}0<t\le T,$$ (4.1)

together with homogeneous boundary conditions on ∂Ω, and the initial values

$$u\left(x,y,0\right)=u_0\left(x,y\right),\hspace{1em}v\left(x,y,0\right)=v_0\left(x,y\right),\hspace{1em}w\left(x,y,0\right)=w_0\left(x,y\right).$$ (4.2)

The well-posed forward initial value problem in Eq. (4.1) becomes ill-posed if the time direction is reversed, and one wishes to recover u(x, y,0)), v(x, y,0), ω(x, y,0), from given approximate values for u(x, y, T), v(x, y, T), ω(x, y, T). We contemplate such time-reversed computations by allowing for possible negative time steps ∆t in the explicit time-marching finite difference scheme described below. With a given positive integer N, let |∆t| = T/(N + 1) be the time step magnitude, and let ${\tilde{u}}^n\left(x,y\right)\equiv \tilde{u}\left(x,y,n\Delta t\right)$, $n=0,1,\cdots ,N+1$, denote the intended approximation to u(x, y, n∆t), and likewise for ${\tilde{v}}^n\left(x,y\right)$, ${\tilde{w}}^n\left(x,y\right)$. It is helpful to consider Fourier series expansions for ${\tilde{w}}^n\left(x,y\right)$, on the unit square Ω,

$${\tilde{w}}^n\left(x,y\right)=\sum _{j,k=-\infty }^{\infty }{\tilde{w}}_{j,k}^n\exp \left\{2\pi i\left(jx+ky\right)\right\},$$ (4.3)

with Fourier coefficients $\left\{{\tilde{w}}_{j,k}^n\right\}$ given by

$${\tilde{w}}_{j,k}^n={\int }_{\Omega }{\tilde{w}}^n\left(x,y\right)\exp \left\{-2\pi i\left(jx+ky\right)\right\}dxdy,$$ (4.4)

Since ω satisfies Eq. (4.1), ∆ω exists and is continuous. Hence, the Fourier series in Eq. (4.3) is expected to be well-behaved. In the error analyses discussed below, ω_tt, ω_ttt, ω_tttt, will be assumed to exist and be continuous. From Eq. (4.1), this implies that

$${\Vert {\left(\nu \Delta \right)}^{m}w\Vert }_2<\infty ,\hspace{1em}m=1,4,\hspace{1em}\hspace{0.17em}\hspace{0.17em}0\le t\le T.$$ (4.5)

With given fixed γ > 0 and real p with 1 < p ≤ 3, define λ_{j, k}, σ_{j, k}, as follows

$${\lambda }_{j,k}=4{\pi }^2\nu \left(j^2+k^2\right),\hspace{1em}{\sigma }_{j,k}=\exp \left\{-2\gamma \left|\Delta t\right|{\lambda }_{j,k}^p\right\}.$$ (4.6)

For functions ${\tilde{w}}^n\left(x,y\right)$ satisfying Eq. (4.5), define the linear operators P and S as follows

$$\begin{gathered} P{\tilde{w}}^n={\sum }_{j,k=-\infty }^{\infty }{\lambda }_{j,k}^p{\tilde{w}}_{j,k}^n\exp \left\{2\pi i\left(jx+ky\right)\right\}, \\ S{\tilde{w}}^n={\sum }_{j,k=-\infty }^{\infty }{\sigma }_{j,k}{\tilde{w}}_{j,k}^n\exp \left\{2\pi i\left(jx+ky\right)\right\}. \end{gathered}$$ (4.7)

As in [14–19], the operator S is used as a stabilizing smoothing operator at each time step. With the operator L^† as in Eq (4.1), let $L^{†}{\tilde{w}}^n\equiv \nu \Delta {\tilde{w}}^n-{\tilde{u}}^n{\tilde{w}}_x^n-{\tilde{v}}^n{\tilde{w}}_y^n$. Consider the following explicit time-marching difference scheme for the system in Eq (4.1), in which only the time variable is discretized, while the space variables remain continuous,

$${\tilde{w}}^{n+1}=S{\tilde{w}}^n+\Delta tS\left(L^{†}{\tilde{w}}^n\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}\Delta {\tilde{\psi }}^{n+1}=-{\tilde{w}}^{n+1}.$$ (4.8)

As already noted, we allow for negative time steps ∆t in Eq. (4.8). The above semi-discrete problem is highly nonlinear. In the operator L^†, the coefficients ${\tilde{u}}^n={\tilde{\psi }}_y^n$, ${\tilde{v}}^n=-{\tilde{\psi }}_x^n$, are defined in terms of ${\tilde{w}}^n$, which is needed to obtain ${\tilde{\psi }}^n$ by solving the Poisson problem $\Delta {\tilde{\psi }}^n=-{\tilde{w}}^n$. The analysis presented in Sections 5 and 6 below, is relevant to the above semi-discrete problem. The Lemmas and Theorems presented in these two sections are results previously obtained in [19], in connection with 2D Burgers’ equation. They are reproduced here and adapted to 2D Navier-Stokes equations for the convenience of the reader. In Section 7, where actual numerical computations are discussed, the space variables are also discretized, and FFT algorithms are used to synthesize the smoothing operator S.

5. Fourier stability analysis in linearized problem.

As in [19], useful insight into the behavior of the nonlinear scheme in Eq. (4.8), can be gained by analyzing a related linear problem with constant coefficients. From a naive computational perspective, we may view L^† in Eq. (4.1), as a linear operator with time-dependent coefficients u(x, y, t), v(x, y, t), which need to be evaluated at every time step t = n∆t, through an independent subsidiary calculation. If |u(x, y, t)| ≤ a, |v((x, y, t)| ≤ b, with positive constants a, b, the numerical stability of the evolution equation ${\tilde{w}}^{n+1}=S{\tilde{w}}^n+\Delta tS\left(L^{†}{\tilde{w}}^n\right)$ in Eq. (4.8), may be investigated by considering the constant coefficient linear operator $L{\tilde{w}}^n\equiv \nu \Delta {\tilde{w}}^n-a{\tilde{w}}_x^n-b{\tilde{w}}_y^n$, in lieu of $L^{†}{\tilde{w}}^n$. Accordingly, we examine the following evolution equation

$$\begin{gathered} w_t=Lu\equiv \nu \Delta w-a{w}_x-b{w}_y,\hspace{1em}0<t\le T, \\ w\left(x,y,0\right)=w_0\left(x,y\right), \end{gathered}$$ (5.1)

together with periodic boundary conditions on ∂Ω. Unlike the case in Eq. (4.8), the stabilized marching scheme

$${\tilde{w}}^{n+1}=S{\tilde{w}}^n+\Delta tSL{\tilde{w}}^n,\hspace{1em}\hspace{0.17em}n=0,1,\cdots ,N,$$ (5.2)

with the linear operator L, is susceptible to Fourier analysis. If $L{\tilde{w}}^n=f^n\left(x,y\right)$, then the Fourier coefficients $\left\{f_{j,k}^n\right\}$ satisfy $f_{j,k}^n=g_{j,k}{\tilde{w}}_{j,k}^n$, where

$$g_{j,k}=-\left\{4{\pi }^2\nu \left(j^2+k^2\right)+2\pi i\left(aj+bk\right)\right\}.$$ (5.3)

Let R be the linear operator R = S + ∆tSL. Then,

$$\begin{gathered} \hspace{1em}\hspace{1em}{\tilde{w}}^{n+1}=R{\tilde{w}}^n=\sum _{j,k=-\infty }^{\infty }{\tilde{w}}_{j,k}^n\left\{1+\Delta t{g}_{j,k}\right\}{\sigma }_{j,k}, \\ {\Vert {\tilde{w}}^{n+1}\Vert }_2^2={\Vert R{\tilde{w}}^n\Vert }_2^2\le \sum _{j,k=-\infty }^{\infty }{\left|{\tilde{w}}_{j,k}^n\right|}^2{\left\{1+\left|\Delta t\right|\left|g_{j,k}\right|\right\}}^2{\sigma }_{j,k}^2, \end{gathered}$$ (5.4)

on using Parseval’s formula.

Lemma 1. Let λ_{j, k}, σ_{j, k}, be as in Eq. (4.6), and let g_j,k be as in Eq. (5.3). Choose a positive integer J such that if λ_J = 4π²νJ, we have

$$ma{x}_{\left(j^2+k^2\right)\le J}\left\{\left|g_{j,k}\right|\right\}\le 2{\lambda }_J,\hspace{1em}\hspace{0.17em}\left|g_{j,k}\right|\le 2{\lambda }_{j,k},\hspace{0.17em}\hspace{0.17em}\forall \left(j^2+k^2\right)>J.$$ (5.5)

With p > 1, choose γ ≥ (λ_J)^1−p in Eq. (4.6). Then

$${\sigma }_{j,k}\left(1+\left|\Delta t\right|\left|g_{j,k}\right|\right)\le 1+2\left|\Delta t\right|{\lambda }_J.$$ (5.6)

Hence, from Eq. (5.4),

$$\begin{gathered} {\Vert R\Vert }_2\equiv {\sup }_{\left\{0\ne {\Vert {\tilde{w}}^n\Vert }_2\right\}}\left\{{\Vert R{\tilde{w}}^n\Vert }_2/{\Vert {\tilde{w}}^n\Vert }_2\right\}\le 1+2\left|\Delta t\right|{\lambda }_J\le \exp \left\{2\left|\Delta t\right|{\lambda }_J\right\}, \\ {\Vert R^m\Vert }_2\le {\Vert R\Vert }_2^m\le \exp \left\{2m\left|\Delta t\right|{\lambda }_J\right\},\hspace{1em}\hspace{0.17em}m\ge 1, \end{gathered}$$ (5.7)

and, for n = 1,2,···, N + 1,

$${\Vert {\tilde{w}}^n\Vert }_2={\Vert R^n{w}_0\Vert }_2\le \exp \left\{2n\left|\Delta t\right|{\lambda }_J\right\}\hspace{0.17em}{\Vert w_0\Vert }_2.$$ (5.8)

Therefore, with this choice of (γ,p), the explicit linear scheme in Eq. (5.2) is unconditionally stable, marching forward or backward in time.

Proof : We first show how to find a positive integer J such that Eq. (5.5) is valid. We have ${\left|g_{j,k}\right|}^2={\lambda }_{j,k}^2+4{\pi }^2{\left(aj+bk\right)}^2\le {\lambda }_{j,k}^2+\left(2c^2/\nu \right){\lambda }_{j,k}$, where 0 < c = max(a,b). Choose a positive integer J such that ${\lambda }_J=4{\pi }^2\nu J>\left(2c^2/\nu \right)$. Then, $\forall \left(j,k\right)$, ${\left|g_{j,k}\right|}^2\le {\lambda }_{j,k}^2+{\lambda }_J{\lambda }_{j,k}$, which implies Eq. (5.5). Next, the inequality in Eq. (5.6) is valid whenever (j² + k²) ≤ J, since σ_j,k ≤ 1. For (j² + k²) > J, we have λ_J < λ_j,k and $\left|g_{j,k}\right|\le 2{\lambda }_{j,k}$. Hence

$${\sigma }_{j,k}=\exp \left\{-2\gamma \left|\Delta t\right|{\lambda }_{j,k}^p\right\}\le \exp \left\{-2\gamma \left|\Delta t\right|{\lambda }_{j,k}{\lambda }_J^{p-1}\right\}\le \exp \left\{-2\left|\Delta t\right|{\lambda }_{j,k}\right\},$$ (5.9)

since $\gamma {\lambda }_J^{p-1}\ge 1$. Also, exp{−2|∆t|λ_j,k} ≤ (1 + 2|∆t|λ_j,k)⁻¹, since 1+x ≤ e^x for real x. Hence, with |g_j,k| ≤ 2λ_j,k for (j²+k²) > J, we find σ_j,k (1 + |∆t||g_j,k|) ≤ 1. Thus, Eq. (5.6) is valid ∀ (j,k). Next, using Eq. (5.6) in Eq. (5.4), together with Parseval’s formula, leads to Eqs. (5.7) and (5.8). QED.

For functions h(x, y, t) on Ω × [0,T], define the norm ${\Vert \left|h\right|\Vert }_{2,\infty }$ as follows

$${\Vert \left|h\right|\Vert }_{2,\infty }\equiv \underset{0\le t\le T}{\sup }\left\{{\Vert h\left(\cdot ,t\right)\Vert }_2\right\}.$$ (5.10)

Lemma 2. Let ωⁿ(x, y) ≡ ω(x, y, n∆t), assumed to satisfy Eq. (4.5), be the exact solution in Eq. (5.1). Let γ, p, λ_{j, k}, σ_{j, k}, be as in Eq. (4.6). Let P and S be as in Eq. (4.7), and let L be the linear operator in Eq. (5.1). Then, ωⁿ⁺¹ = ωⁿ + ∆tLωⁿ + τⁿ, where τⁿ is the truncation error. With the norm definition in Eq (5.10), and 0 ≤ n ≤ N,

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em} {\Vert {\tau }^n\Vert }_2\le 1/2{\left(\Delta t\right)}^2{\Vert \left|L^{2}w\right|\Vert }_{2,\infty }, \\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} {\Vert w^n-S{w}^n\Vert }_2\le 2\gamma \left|\Delta t\right|{\Vert \left|Pw\right|\Vert }_{2,\infty }, \\ \left|\Delta t\right|{\Vert L{w}^n-SL{w}^n\Vert }_2\le 2\gamma {\left(\Delta t\right)}^2{\Vert PLw\Vert }_{2,\infty }. \end{gathered}$$ (5.11)

Proof : The inequality for the truncation error ${\tau }^n$ in Eq. (5.11) follows naturally from a truncated Taylor series expansion. Since the linear operator L in Eq. (5.1) has constant coefficients, and ω_t = Lω, we have ω_tt = Lω_t = L²ω. Hence,

$$\begin{gathered} w^{n+1}=w^n+\Delta t{w}_t^n+\left(1/2\right){\left(\Delta t\right)}^2{w}_{tt}^n+O{\left(\Delta t\right)}^3, \\ {w}^{n+1}=w^n+\Delta tL{w}^n+\left(1/2\right){\left(\Delta t\right)}^2\overline{\left(L^{2}w\right)}, \end{gathered}$$ (5.12)

where $\overline{\left(L^{2}w\right)}$ denotes the value of L²ω at an intermediate time t, lying between n∆t and (n + 1)∆t. This proves the first inequality in Eq. (5.11). Using the inequality 1 − e^−x ≤ x for all real x, together with Parseval’s formula, we have

$${\Vert w^n-S{w}^n\Vert }_2^2=\sum _{j,k=-\infty }^{\infty }{\left|w_{j,k}^n\right|}^2{\left(1-{\sigma }_{j,k}\right)}^2\le {\left(2\gamma \left|\Delta t\right|\right)}^2{\Vert \left|Pw\right|\Vert }_{2,\infty }^2.$$ (5.13)

This proves the second inequality in Eq. (5.11). The last inequality is a corollary of the second. QED.

In Lemma 1, the finite difference approximation ${\tilde{w}}^n\left(x,y\right)\equiv \tilde{w}\left(x,y,n\Delta t\right)$ satisfies Eq. (5.2), whereas the exact solution $w^n\left(x,y\right)\equiv w\left(x,y,n\Delta t\right)$ in Eq. (5.1), satisfies $w^{n+1}=w^n+\Delta tL{w}^n+{\tau }^n$, where ${\tau }^n$ is the truncation error. We need to estimate the error $er{r}^n\left(x,y\right)=w^n\left(x,y\right)-{\tilde{w}}^n\left(x,y\right)$, $n=0,1,\cdots ,N+1$.

Theorem 1. With ∆t > 0, let ωⁿ(x, y), assumed to satisfy Eq. (4.5), be the unique solution of Eq. (5.1) at t = n∆t. Let ${\tilde{w}}^n\left(x,y\right)$ be the corresponding solution of the forward explicit scheme in Eq. (5.2), and let p, λ_J, γ, be as in Lemma 1. If $er{r}^n\left(x,y\right)=w^n\left(x,y\right)-{\tilde{w}}^n\left(x,y\right)$, denotes the error at t = n∆t, n = 1,2,···, N +1, we have

$$\begin{gathered} \hspace{1em}{\Vert er{r}^n\Vert }_2\le e^{2t{\lambda }_J}{\Vert er{r}^0\Vert }_2+\left\{\gamma \left(e^{2t{\lambda }_J}-1\right)/{\lambda }_J\right\}{\Vert \left|Pw\right|\Vert }_{2,\infty } \\ +\left\{\left(e^{2t{\lambda }_J}-1\right)/2{\lambda }_J\right\}\left\{2\gamma \Delta t{\Vert \left|PLw\right|\Vert }_{2,\infty }+\left(\Delta t/2\right){\Vert \left|L^{2}w\right|\Vert }_{2,\infty }\right\}. \end{gathered}$$ (5.14)

Proof : Let $h^n={\tau }^n+\left(w^n-S{w}^n\right)+\Delta t\left(L{w}^n-SL{w}^n\right)$. Let R be the linear operator in Eq. (5.4). Then, ωⁿ⁺¹ = Rωⁿ + hⁿ, while ${\tilde{w}}^{n+1}=R{\tilde{w}}^n$. Therefore

$$er{r}^{n+1}=R{w}^n+h^n=R^{n+1}er{r}^0+\Delta t\sum _{j=0}^n{R}^{n-j}{h}^j/\left(\Delta t\right).$$ (5.15)

Let ${\Vert \left|h\right|\Vert }_{2,\infty }\equiv {\max }_{0\le n\le N+1}\left\{h^n\right\}$. Using Eq. (5.7) in Lemma 1, and letting t = (n + 1)∆t,

$${\Vert er{r}^{n+1}\Vert }_2\le e^{2t{\lambda }_J}{\Vert er{r}^0\Vert }_2+\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\Delta t\right\}\Delta t\sum _{j=0}^n{\Vert R^{n-j}\Vert }_2.$$ (5.16)

We have $\Delta t{\sum }_{j=0}^n{\Vert R^{n-j}\Vert }_2\le {\sum }_{j=0}^{n}Aj$, in Eq. (5.16), where, for each j, A_j = ∆t exp{2λ_J(n−j)∆t}. Notice that with t = (n+1)∆t, A₀ is less than the area under the curve z(s) = exp{2λ_J(t − s)} on the interval 0 ≤ s ≤ ∆t. Indeed, $\left({\sum }_{j=0}^n{A}_j\right)$ is less than the total area under that curve on the interval 0 ≤ s ≤ t. Hence, from Eq. (5.16),

$$\begin{gathered} {\Vert er{r}^{n+1}\Vert }_2\le e^{2t{\lambda }_J}{\Vert er{r}^0\Vert }_2+\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\Delta t\right\}{\int }_0^t{e}^{2{\lambda }_J\left(t-s\right)}ds \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em} =e^{2t{\lambda }_J}{\Vert er{r}^0\Vert }_2+\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\Delta t\right\}\left(e^{2t{\lambda }_J}-1\right)/2{\lambda }_J. \end{gathered}$$ (5.17)

Next, using Lemma 2 to estimate $\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\Delta t\right\}$, one obtains Eq. (5.14) from Eq. (5.17). QED.

6. The stabilization penalties in the forward and backward linearized problem in Eq. (5.1).

The stabilizing smoothing operator S in the explicit scheme in Eq. (5.2) leads to unconditional stability, but at the cost of introducing a small error at each time step. We now assess the cumulative effect of that error.

In the forward problem in Theorem 1, we may assume the given initial data ω₀(x, y) to be known with sufficiently high accuracy that one may set ${\Vert er{r}^0\Vert }_2=0$ in Eq. (5.14). Choosing γ = (λ_J)^1−p in Lemma 1, and putting t = n∆t ≤ T, Eq.(5.14) reduces to

$${\Vert er{r}^n\Vert }_2\le {\left({\lambda }_J\right)}^{-p}\left(e^{2t{\lambda }_J}-1\right){\Vert \left|Pw\right|\Vert }_{2,\infty }+O\left(\Delta t\right),\hspace{1em}n=1,2,\cdots ,N+1.$$ (6.1)

Therefore, when using the explicit scheme in Eq.(5.2), there remains the non-vanishing residual error ${\left({\lambda }_J\right)}^{-p}\left(e^{2t{\lambda }_J}-1\right){\Vert \left|Pw\right|\Vert }_{2,\infty }$, as ∆t ↓ 0. This is the stabilization penalty, which results from smoothing at each time step, and grows monotonically as t ↑ T. Recall that λ_J must be chosen large enough to satisfy Eq. (5.5) in Lemma 1. Clearly, if T = (N +1)∆t is large, the accumulated distortion may become unacceptably large as t ↑ T, and the stabilized explicit scheme is not useful in that case. On the other hand, if T is small, as is the case in problems involving small values of t, it may be possible to choose p > 2 and sufficiently large λ_J, yet with small enough λ_JT that ${\left({\lambda }_J\right)}^{-p}\left(e^{2{\lambda }_{J}T}-1\right)$ is quite small. In that case, the stabilization penalty remains acceptable on 0 ≤ t ≤ T. As an example, with T = 2.0 × 10⁻⁴, p = 3.25, and λ_J = 5.0 × 10⁴, we find ${\left({\lambda }_J\right)}^{-p}\left(e^{2{\lambda }_{J}T}-1\right)<2.6\times {10}^{-7}$. This often leads to an acceptable ${\mathcal{L}}^2$ relative error $\left({\Vert er{r}^n\Vert }_2/{\Vert w^n\Vert }_2\right)$.

For this important but limited class of problems, the absence of restrictive Courant conditions on the time step ∆t in the explicit scheme in Eq.(5.2), provides a significant advantage in well-posed forward computations of two dimensional problems on fine meshes.

However, there is an additional penalty in the ill-posed problem of marching backward from t = T, in that solutions exist only for a restricted class of data satisfying certain smoothness and other constraints. These data are seldom known with sufficient precision. We shall assume that the given data ${\tilde{\omega }}^0(x,y)$ at t = T, differ from such unknown exact data by small errors κ(x, y):

$${\tilde{\omega }}^0\left(x,y\right)=\omega \left(x,y,T\right)+\kappa \left(x,y\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}{\Vert \kappa \Vert }_2\le ϵ.$$ (6.2)

Theorem 2. With ∆t < 0, let ωⁿ(x, y), assumed to satisfy Eq. (4.5), be the unique solution of the forward well-posed problem in Eq. (5.1) at s = T − n|∆t|. Let ${\tilde{\omega }}^n\left(x,y\right)$ be the corresponding solution of the backward explicit scheme in Eq. (5.2), with initial data ${\tilde{\omega }}^0\left(x,y\right)=\omega \left(x,y,T\right)+\kappa \left(x,y\right)$ as in Eq. (6.2). Let p, λ_J, γ, be as in Lemma 1. If $er{r}^n\left(x,y\right)\equiv {\omega }^n\left(x,y\right)-{\tilde{w}}^n\left(x,y\right)$, denotes the error at s = T −n|∆t|, n = 0,1,2,···, N + 1, we have, with ϵ as in Eq.(6.2),

$$\begin{gathered} \hspace{1em}{\Vert er{r}^n\Vert }_2\le ϵe^{2n\left|\Delta t\right|{\lambda }_J}+\left(\gamma /{\lambda }_J\right)\left\{\left(e^{2n\left|\Delta t\right|{\lambda }_J}-1\right)\right\}{\Vert \left|P\omega \right|\Vert }_{2,\infty } \\ +\left\{\left(e^{2n\left|\Delta t\right|{\lambda }_J}-1\right)/{\lambda }_J\right\}\left\{\gamma \left|\Delta t\right|{\Vert \left|PL\omega \right|\Vert }_{2,\infty }+\left(\left|\Delta t\right|/2\right){\Vert \left|L^2\omega \right|\Vert }_{2,\infty }\right\}. \end{gathered}$$ (6.3)

Proof : Let hⁿ = τⁿ + (ωⁿ − Sωⁿ) + ∆t(Lωⁿ − SLω’ⁿ). Let R be the linear operator in Eq. (5.4). Then, ${\omega }^{n+1}=R{\omega }^n+h^n$, while ${\tilde{\omega }}^{n+1}=R{\tilde{\omega }}^n$. Therefore

$$er{r}^{n+1}=R\hspace{0.17em}er{r}^n+h^n=R^{n+1}er{r}^0+\left|\Delta t\right|\sum _{j=0}^n{R}^{n-j}{h}^j/\left(\left|\Delta t\right|\right).$$ (6.4)

Let ${\Vert \left|h\right|\Vert }_{2,\infty }\equiv {\max }_{0\le n\le N+1}\left\{h^n\right\}$. Using Lemma 1, and letting r = (n + 1)|∆t|, we find, as in the case of Eq. (5.16),

$$\begin{gathered} {\Vert er{r}^{n+1}\Vert }_2\le ϵe^{2r{\lambda }_J}+\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\left|\Delta t\right|\right\}\left|\Delta t\right|\sum _{j=0}^n{\Vert R^{n-j}\Vert }_2, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em} \le ϵe^{2r{\lambda }_J}+\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\left|\Delta t\right|\right\}{\int }_0^r{e}^{2{\lambda }_J\left(r-s\right)}ds, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}=ϵe^{2r{\lambda }_J}\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\left|\Delta t\right|\right\}\left\{e^{2r{\lambda }_J}-1\right\}/2{\lambda }_J. \end{gathered}$$ (6.5)

As in the preceding Theorem, we may now use Lemma 2 to estimate $\left\{{\Vert \left|h\right|\Vert }_{2,\infty }/\left|\Delta t\right|\right\}$ and obtain Eq. (6.3) from Eq.(6.5). QED.

It is instructive to compare the results in the well-posed case in Eq.(6.1), with the ill-posed results implied by Eq.(6.3). For this purpose, we must reevaluate Eq.(6.3) at the same t values that are used in Eq. (6.1). With ∆t > 0, t = k∆t, and ω^k(x, y) = ω(x, y, k∆t), let ${\tilde{\omega }}^k\left(x,y\right)$ now denote the previously computed backward solution evaluated at t = k∆t. With T = (N + 1)∆t, let $er{r}^k\left(x,y\right)={\omega }^k\left(x,y\right)-{\tilde{w}}^k\left(x,y\right)$, $k=0,1,2,\cdots ,N+1$, denote the error at t = k∆t. Again, choosing γ = (λ_J)^1−p, we get from Eq. (6.3),

$$\begin{gathered} {\Vert er{r}^k\Vert }_2\le {\left|\left({\lambda }_J\right)\right|}^{-p}\left\{\exp \left[2{\lambda }_J\left(T-t\right)\right]-1\right\}{\Vert \left|P\omega \right|\Vert }_{2,\infty } \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+ϵ\exp \left\{2{\lambda }_J\left(T-t\right)\right\}+O\left(\Delta t\right),\hspace{1em}0\le t\le T. \end{gathered}$$ (6.6)

Here, the stabilization penalty is augmented by an additional term, resulting from amplification of the errors κ(x, y) in the given data at t = T, as indicated in Eq. (6.2). Both of the first two terms on the right in Eq. (6.6) grow monotonically as t ↓ 0, reflecting backward in time marching from t = T.

Let the exact solution ω(x, y,0) at t = 0, satisfy a prescribed ${\mathcal{L}}^2$ bound,

$${\Vert {\omega }^0\Vert }_2\le M.$$ (6.7)

Again, with large T, and λ_J large enough to satisfy Eq. (5.5) in Lemma 1, the non-vanishing residuals in Eq. (6.6) lead to large errors, and the backward explicit scheme is not useful in that case. However, if T is small enough that

$$2{\lambda }_{J}T\le \log \left(M/ϵ\right),$$ (6.8)

with (ϵ, M) as in Eqs. (6.2) and (6.7), we find, with t = k∆t, and µ(t) = t/T,

$$\begin{gathered} {\Vert er{r}^k\Vert }_2\le {\left|\left({\lambda }_J\right)\right|}^{-p}\left\{\exp \left[2{\lambda }_J\left(T-t\right)\right]-1\right\}{\Vert \left|P\omega \right|\Vert }_{2,\infty } \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+M^{1-\mu \left(t\right)}{ϵ}^{\mu \left(t\right)}+O\left(\Delta t\right),\hspace{1em}0\le t\le T. \end{gathered}$$ (6.9)

The second term on the right in Eq. (6.9) represents the fundamental uncertainty in ill-posed backward continuation from noisy data, for solutions satisfying the prescribed bounds (ϵ, M) in Eqs. (6.2) and (6.7). That uncertainty is known to be best-possible in the case of autonomous selfadjoint problems. Therefore, in a limited but potentially significant class of problems, the stabilized backward explicit scheme for the linearized problem in Eq. (5.1), can produce results differing from what is best-possible only by a small stabilization penalty as ∆t ↓ 0.

For example, with parameter values such as T = 10^‒3; M = 10⁴, ϵ = 10^‒3, we have M/ϵ = 10⁷ = exp{2λ_JT}, and λ_J ≥ 8059. Hence, with p = 3.0, we find (λ_J)^−p < 1.91 × 10⁻¹². With t = k∆t, we would then obtain from Eq. (6.9),

$$\begin{gathered} {\Vert er{r}^k\Vert }_2\le M^{1-\mu \left(t\right)}{ϵ}^{\mu \left(t\right)} \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(1.91\times {10}^{-5}\right){\Vert P\omega \Vert }_{2,\infty }+O\left(\Delta t\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le t\le T, \end{gathered}$$ (6.10)

which may lead to an acceptable ${\mathcal{L}}^2$ relative error $\left({\Vert er{r}^k\Vert }_2/{\Vert {\omega }^k\Vert }_2\right)$.

Remark. In most applications of time-reversed problems, the values of M and ϵ in Eq. (6.10), are seldom known accurately. In most cases, interactive adjustment of the parameter pair (γ,p) in Eqs. (4.6, 4.7), in the definition of the smoothing operator S, based on prior knowledge about the exact solution, is necessary to obtain useful reconstructions. This process is similar to the manual tuning of an FM station, or the manual focusing of binoculars, and likewise requires user recognition of a ‘correct’ solution. There may be several possible good solutions. Typical values of (γ,p) lie in the ranges 10⁻¹² ≤ γ ≤ 10⁻⁷, 2.5 ≤ p ≤ 3.5.

7. Behavior in the nonlinear stabilized explicit scheme in Eq. (4.8).

It remains to be seen whether the linear analysis in Sections 5 and 6 is indicative of actual numerical behavior in the nonlinear system in Eq. (4.8). Four instructive backward recovery experiments will be presented and discussed. With the operator L^† as in Eq (4.1), we now use second order accurate centered finite differencing for the space variables in the fully discrete non linear scheme

$${\tilde{\omega }}^{n+1}=S{\tilde{\omega }}^n+\Delta tS\left(L^{†}{\tilde{\omega }}^n\right),\hspace{1em}\hspace{0.17em}\hspace{0.17em}\Delta {\tilde{\psi }}^{n+1}=-{\tilde{\omega }}^{n+1}.$$ (7.1)

Fast Fourier Transform (FFT) algorithms are used to synthesize the smoothing operator S defined in Eq. (4.7), and an efficient multigrid Poisson solver is used at every time step to solve $\Delta {\tilde{\psi }}^{n+1}=-{\tilde{\omega }}^{n+1}$.

All four experiments involve 256 × 256 pixel gray scale images, defined by non-smooth underlying intensity data, with integer values ranging between zero and 255. These images have zero intensities on and near the boundary. In each case, the image intensity data are multiplied by 0.0025 to create the initial stream function ψ(x, y,0), with 0 ≤ ψ ≤ 0.6375. The kinematic viscosity ν = 0.01 in all four cases, and U_max, RE, are as defined in Eq. (2.2). The initial values for the forward problem, u(x, y,0), v(x, y,0), ω(x, y,0), are obtained by numerical differentiation of ψ(x, y,0). All of these quantities are assumed defined on the unit square Ω, and to vanish on ∂Ω for the duration of the flow.

USAF Resolution Chart.

This is illustrated in Figure 8.1, and documented in Table 1. The USAF Chart initial data are shown in the leftmost column of Figure 8.1 as the original sharp image, contour plots of the velocity component u(x, y,0), and contour plots of the vorticity ω(x, y,0). We have U_max = 115, RE = 11500, and sup_Ω{|ω(x, y,0)|} = 1.25 × 10⁵. With ∆t = 1.0 × 10⁻⁶, the fully discrete scheme in Eq. (7.1) was solved forward in time for 1000 time steps to create the data at time T = 1.0×10⁻³, shown in the middle column of Figure 8.1. In this forward calculation, the operator S was chosen as the identity operator. No compensation was required, as the explicit scheme was found stable with such small ν∆t.

Fig. 8.1.

Backward recovery in USAF Resolution Chart image, with ν = 0.01, U_max = 115, RE = 11500, sup_Ω{|ω(x, y,0)|} = 1.25 × 10⁵, T = 1.0 × 10⁻³. Using data shown in middle column, stabilized explicit scheme seeks to reconstruct initial data shown in leftmost column. Actually recovered data, shown in rightmost column, are documented in Table 1.

TABLE 1.

USAF Chart image at RE=11500 ${\mathcal{L}}^2(\Omega )$-norm behavior in backward recovery from data at T = 1 × 10⁻³

Variable	${\mathcal{L}}^2$ norm at 0	${\mathcal{L}}^2$ norm at T	Recovered at 0	${\mathcal{L}}^2$ rel err at 0
u = ψy	23.6	14.6	23.6	11.9 %
v = −ψx	21.2	14.0	21.2	13.1 %
ω = −∆ψ	15882	4009	15774	13.9%

Evidently, considerable change has occurred at time T = 1.0×10⁻³. Substantial erosion and disorganization of the sharp features in the leftmost column, is now apparent in the middle column of Figure 8.1. Moreover, as Table 1 indicates, the ${\mathcal{L}}^2(\Omega )$ norms of the velocity components u, v, are reduced to about 2/3 of their initial values, while the ${\mathcal{L}}^2$ norm of the vorticity ω is reduced to about 1/4 of its initial value.

The computed data at time T are affected by accumulated discretization and roundoff errors, exacerbated by repeated application of the elliptic solver. Using these data, and solving the fully discrete scheme in Eq. (7.1) backward in time for 1000 time steps, requires use of the smoothing operator S, with an appropriate choice of the parameters γ and p. Here, as is always the case in ill-posed reconstruction, a priori knowledge about the true solution is fundamental in arriving at useful values interactively. The explicit nature of the scheme in Eq. (7.1), together with its unconditional stability, allows for fast trial restorations using a relatively large time step |∆t|. Upon locating promising parameter values, a more refined search is undertaken with smaller |∆t|. With γ = 1.0 × 10⁻¹², p = 2.75, the stabilized explicit scheme produced the results shown in the rightmost column of Figure 8.1.

The recovery from the blurred data is surprisingly good. In Table 1, the restored values for the ${\mathcal{L}}^2$ norms of u, v, ω, are close to the original values. More significantly, the ${\mathcal{L}}^2$ relative errors range from 12% to 14%.

Significantly less successful reconstruction would result from more severely blurred data at a value of T three to five times larger.

Elizabeth Taylor.

Illustrated in Figure 8.2 and documented in Table 2, this experiment has T = 2.0×10⁻³, U_max = 103, RE = 10300, and sup_Ω{|ω(x, y,0)|} = 1.25 × 10⁵. As in Figure 8.1, the leftmost column contains the original sharp image, contour plots of the velocity component u(x, y,0), and contour plots of the vorticity ω(x, y,0). With ∆t = 1.0 × 10⁻⁶, the fully discrete scheme in Eq. (7.1) was solved forward in time for 2000 time steps to create the data at time T, shown in the middle column of Figure 8.2. As in the preceding experiment, S was chosen as the identity operator. More erosion of the initial data is now apparent in the middle column in Figure 8.2, and Table 2 indicates that the ${\mathcal{L}}^2$ norms of u, v, have been reduced to 55% of their initial values, while the ${\mathcal{L}}^2$ norm of ω was reduced to less than 12% of its initial value. Nevertheless, very good recovery is still possible with γ = 1.0 × 10⁻¹², p = 2.75, leading to ${\mathcal{L}}^2$ relative errors less than 5%. Larger values of T would lead to unsatisfactory reconstructions.

Fig. 8.2.

Backward recovery in Elizabeth Taylor image, with ν = 0.01, U_max = 103, RE = 10300, sup_Ω{|ω(x, y,0)|} = 1.25×10⁵, T = 2.0×10⁻³. Using data shown in middle column, stabilized explicit scheme seeks to reconstruct initial data shown in leftmost column. Actually recovered data, shown in rightmost column, are documented in Table 2.

TABLE 2.

Elizabeth Taylor image at RE=10300 ${\mathcal{L}}^2(\Omega )$-norm behavior in backward recovery from data at T = 2 × 10⁻³

Variable	${\mathcal{L}}^2$ norm at 0	${\mathcal{L}}^2$ norm at T	Recovered at 0	${\mathcal{L}}^2$ rel err at 0
u = ψy	9.6	5.3	9.5	3.3 %
v = −ψx	9.5	5.3	9.4	3.3 %
ω = −∆ψ	7806	895	7694	4.7 %

The previous two experiments used images of familiar objects, leading to easily grasped contour plots in Figures 8.1 and 8.2. The next two experiments involve images of complicated vortex flows actually found in nature. This results in the more complex contour plots shown in Figures 8.3 and 8.4.

Fig. 8.3.

Backward recovery in November 2 2015 NASA Aqua satellite image of Kármán Vortex Street in clouds off Heard Island. Here, ν = 0.01, U_max = 98, RE = 9800, sup_Ω{|ω(x, y,0)|} = 1.11 × 10⁵, T = 2.0 × 10⁻³. Using data shown in middle column, stabilized explicit scheme seeks to reconstruct initial data shown in leftmost column. Actually recovered data, shown in rightmost column, are documented in Table 3.

Fig. 8.4.

Backward recovery in August 31 2016 NASA Suomi weather satellite image of twin hurricanes Madeline and Lester over Hawaii. Here, ν = 0.01, U_max = 85, RE = 8500, sup_Ω{|ω(x, y,0)|} = 1.14 × 10⁵, T = 2.0 × 10⁻³. Using data shown in middle column, stabilized explicit scheme seeks to reconstruct initial data shown in leftmost column. Actually recovered data, shown in rightmost column, are documented in Table 4.

Kármán Vortex Street in clouds off Heard Island.

This is illustrated in Figure 8.3, and documented in Table 3. Here, U_max = 98, RE = 9800, and sup_Ω{|ω(x, y,0)|} = 1.11×10⁵. With ∆t = 1.0×10⁻⁶, and S the identity operator, the fully discrete scheme in Eq. (7.1) was solved forward in time for 2000 time steps to create the data at time T = 2.0 × 10⁻³, shown in the middle column of Figure 8.3. Evidently, dissipation has caused significant disorganization and erosion of sharp features, with the ${\mathcal{L}}^2$ norms of u, v, reduced to half their initial values, while the ${\mathcal{L}}^2$ norm of ω was reduced to less than 10% of its original value. Remarkably, the stabilized explicit scheme run backward with γ = 1.0 × 10⁻¹², p = 2.75, produced high quality reconstructions, with ${\mathcal{L}}^2$ relative errors less than 6%. Significantly less accurate reconstructions would result from blurred data at larger values of T.

TABLE 3.

Kármán Vortex Street image at RE=9800 ${\mathcal{L}}^2(\Omega )$-norm behavior in backward recovery from data at T = 2 × 10⁻³

Variable	${\mathcal{L}}^2$ norm at 0	${\mathcal{L}}^2$ norm at T	Recovered at 0	${\mathcal{L}}^2$ rel err at 0
u = ψy	10.6	5.2	10.5	4.4 %
v = −ψx	11.5	5.5	11.4	4.3 %
ω = −∆ψ	10668	1005	10522	5.8 %

The original image in Figure 8.3 was taken by NASA’s Aqua satellite on November 2 2015. Heard Island is located in the far south Indian Ocean, close to Antarctica.

Twin Hurricanes over Hawaii.

Illustrated in Figure 8.4 and documented in Table 4, this experiment has T = 2.0 × 10⁻³, U_max = 85, RE = 8500, and sup_Ω{|ω(x, y,0)|} = 1.14 × 10⁵. As in the previous experiment, the leftmost column contains the original sharp image, contour plots of the velocity component u(x, y,0), and contour plots of the vorticity ω(x, y,0). With S the identity operator, and ∆t = 1.0 × 10⁻⁶, the fully discrete scheme in Eq. (7.1) was solved forward in time for 2000 time steps to create the data at time T, shown in the middle column of Figure 8.4. As before, there is considerable dissipation and substantial erosion of sharp features, with the ${\mathcal{L}}^2$ norms of u, v, reduced to 38% of their initial values, while the ${\mathcal{L}}^2$ norm of ω was reduced to less than 7% of its initial value. Again, successful reconstruction of u(x, y,0), v(x, y,0), ω(x, y,0), was found possible with γ = 1.0×10⁻¹², p = 2.75, in the stabilized explicit scheme in Eq. (7.1). Here, the ${\mathcal{L}}^2$ relative errors were less than 4%. At larger values of T, the blurring is too severe to permit useful recovery.

TABLE 4.

Twin Hurricanes image at RE=8500 ${\mathcal{L}}^2(\Omega )$-norm behavior in backward recovery from data at T = 2 × 10⁻³

Variable	${\mathcal{L}}^2$ norm at 0	${\mathcal{L}}^2$ norm at T	Recovered at 0	${\mathcal{L}}^2$ rel err at 0
u = ψy	12.3	4.7	12.1	3.7 %
v = −ψx	12.7	4.8	12.6	3.6 %
ω = −∆ψ	14301	965	14093	3.9 %

The original image in Figure 8.4 was obtained by NASA on August 31 2016, using the Suomi National Polar Orbiting Weather Satellite.

Reconstructions with additional noise.

In each of the above experiments, the stream functions ψ(x, y, T), computed using the forward Navier-Stokes equations, contain an appreciable amount of noise originating from accumulated discretization and roundoff errors. Such stream functions generated the data u, v, ω at time T that were used as input in the backward problem, and that are shown in the middle columns in Figures 8.1–8.4. However, further experiments were conducted with ψ(x, y, T) replaced by ψ^†(x, y, T) ≡ ψ(x, y, T)+0.02 ϕ(x, y, T), where ϕ(x, y, T) is the forward computed Navier-Stokes stream function associated with a different image. Quality reconstructions were still obtained with such noisier input data ψ^†(x, y, T), albeit with larger ${\mathcal{L}}^2$ relative errors than are reported in Tables 1–4.

8. Concluding remarks.

To the author’s knowledge, work on numerical algorithms for solving Navier-Stokes equations backward in time, has not previously appeared in the research literature. The primary aim in this paper was to open doors, and demonstrate the possibility of useful reconstruction in a class of problems generally considered intractable. Given the discussion in Section 3 following the results in Eqs. (3.12, 3.13), and the fact that U_max ≥ 85, sup_Ω{|ω(x, y,0)|} ≥ 1.0×10⁵, in all four experiments in Section 7, the successful recoveries in each of these experiments from data at T ≥ 1.0 × 10⁻³, were quite unexpected. Moreover, high quality reconstruction from similar T values was found possible in numerous other images.