Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

Abstract

We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

1. Introduction

As the imminent demands of seismological investigation and geophysical exploration, numerical simulation of wave propagation in heterogeneous geophysical materials has been intensively investigated by seismologists. Researchers are striving for efficient high-precision numerical modelling of acoustic and seismic wave propagations at large time with strong stability. Various numerical methods have been brought forward to model the wavefield through the acoustic or elastic wave equations, including the finite difference method [1–6], the finite-element [7] or spectral element method [8,9], the pseudo-spectral method [10,11], the discrete singular convolution method [12,13], the nearly analytical method [14–16], etc. All these aforementioned methods are mainly devoted to spatial discretizations for purposes of increasing the numerical precision, handling complex boundary conditions and improving their computational efficiency.

Comparing with spatial discrete methods, there are few discussions about temporal discretizations, especially for long-term simulations. Recently, by elaborately concatenating symplectic time integrations and spatial discretizations, structure-preserving algorithms such as the symplectic [12,17] and multi-symplectic [13] methods have been constructed. Further modifications combining the order condition and dispersion relation of the symplectic schemes are also investigated [18,19]. These methods can not only effectively reduce the numerical dispersion, but also maintain the stability during large time simulation. Therefore, structure-preserving methods have attracted increasing attention especially in modelling Earth's free oscillation and seismic wave propagation in high frequencies which demands high accuracy and large time stability of numerical algorithms.

Nevertheless, conventional structure-preserving methods are established for the wavefield in perfectly elastic medium which does not include attenuation. Under this assumption, the system corresponding to the acoustic or elastic wave equations without external force term can be formulated in the framework of Hamiltonian system [20,21], and thereby allows the application of symplectic or multi-symplectic method. However, in the realistic media of the earth, it preferentially attenuates seismic waves at higher frequencies by anelastic attenuation, resulting in energy dissipation and phase distortion. Simply introducing standard structure-preserving methods cannot guarantee the correct dissipation rate caused by the intrinsic attenuation. The numerical solution may be overdamped or underdamped, introducing large numerical dissipation and therefore leading to large time simulations that become physically meaningless.

Recently, a modified symplectic method has been applied to the two-dimensional attenuated acoustic wave equation and demonstrated the ability in capturing the dissipation of a wavefield [22]. The authors introduced an extra damping term to the standard acoustic wave equation to represent the intrinsic attenuation. In this work, we also consider the same model equation, but provide another efficient method to construct a dissipation-preserving scheme, which falls into a class of numerical methods by choosing proper spatial and temporal discrete methods. The basic idea is from the conformal symplectic [23–26] and multi-symplectic [27,28] methods, which can well preserve the dissipative properties for Hamiltonian ordinary differential equations (ODEs) and partial differential equation (PDEs) with linear dissipation. Such methods can effectively handle the seismic wavefield in realistic attenuating media. The conformal numerical approaches are established on the preservation of the conformal conservation laws being associated with the dissipative system, such as the conformal symplectic or multi-symplectic conservation law and the conformal energy or momentum conservation law. Instead of the standard symplectic method, authors demonstrate that splitting methods [24,25] can generally preserve the related conformal property for ODEs. Thereby, we extend the splitting method to the damped acoustic wave equation and derive a conformal symplectic scheme by concatenating a symplectic method for the conservative Hamiltonian part, in association with the exact flow map for the dissipative part, which guarantees a dissipation-preserving numerical scheme is consistent with the original dissipative wave system.

We construct an analytical solution for the attenuated acoustic wave equation to benchmark the proposed conformal symplectic method. Two additional non-conformal methods are presented for comparison on the wave phase and amplitude, which show an absolute advantage of the conformal symplectic method in both numerical phase and amplitude. Two heterogeneous cases are provided including the classic Marmousi model to test our method, which also confirm the effectiveness of the proposed method in minimization of the numerical dispersion and conservation of the dissipative amplitude. In addition, the perfect matched layer (PML) absorbing boundary condition (ABC) is discussed and successfully combined with the conformal symplectic scheme.

2. Conformal multi-symplectic structure

Consider the two-dimensional acoustic wave equation with damping term in an arbitrarily heterogeneous media

$$u_{tt}=c^2\mathrm{\Delta }u-a{u}_t,(x,z)\in \mathrm{\Omega },$$ 2.1

where Δ=∂²/∂x²+∂²/∂z² is the Laplace operator, u(x,z,t) is the displacement scalar of the wavefield, x,z represent the horizontal and depth coordinates and c(x,z) is the acoustic wave velocity. a(x,z)≥0 is the damping coefficient calculated by a=ωQ⁻¹, where ω is the dominant frequency of the wavefield and Q is the quality factor of attenuating media. By introducing auxiliary variables v=u_t, p=u_x and q=u_z, we can reform the acoustic wave equation (2.1) in the conformal multi-symplectic structure as

$$M{\mathit{w}}_t+K{\mathit{w}}_x+L{\mathit{w}}_z=\mathrm{\nabla }S(\mathit{w})-\frac{a}{2}M\mathit{w},$$ 2.2

where w=(u,v,p,q)^T, $S(\mathit{w})=\frac{1}{2}(auv/c+v^2-p^2-q^2)$ and

$$M=\left[\begin{array}{cccc}0& -\frac{1}{c}& 0& 0\\ \frac{1}{c}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],K=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],L=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 0\end{array}\right]$$

are skew-symmetric matrices, respectively. The merit of the conformal multi-symplectic structure (2.2) is that it naturally possesses the intrinsic conservation laws of the original system [28], for instance, the conformal multi-symplectic conservation law. For the attenuated acoustic wave equation (2.1), the corresponding form of the conformal multi-symplectic conservation law yields

$${\omega }_t+{\kappa }_x+{\xi }_z=-a\omega ,$$ 2.3

with

$$\omega =\frac{1}{c}\mathrm{d}u\wedge \mathrm{d}v,\kappa =\mathrm{d}p\wedge \mathrm{d}u,\xi =\mathrm{d}q\wedge \mathrm{d}u,$$

where ∧ is the wedge product. Under the proper boundary condition (e.g. periodic boundary condition), we can integrate the conservation law (2.3) in x- and z-directions to get the global conformal symplectic conservation law when a(x,z) is a constant

$${\overline{\omega }}_t=-a\overline{\omega },\text{or equivalently}\overline{\omega }(t)=\exp (-at)\overline{\omega }(0),$$ 2.4

where $\overline{\omega }$ is defined as ${\int }_{\mathrm{\Omega }}\omega \mathrm{d}x\mathrm{d}z$, which also admits the dissipative property as the amplitude of the wavefield does.

Mathematically, a numerical method is called conformal multi-symplectic if it can preserve a discrete version of the conservation law (2.3) or conformal symplectic if the method satisfies a discrete conformal symplectic conservation law (2.4). Although two conformal multi-symplectic methods have already been established for general Hamiltonian PDEs [27,28], their low order of accuracy restricts the direct application on modelling the acoustic wave equation (2.1). Herein, we choose to construct a conformal symplectic scheme based on the splitting of the original system, which can flexibly achieve high precision and is a generation of that for the dissipative ODEs in [25].

3. Conformal symplectic scheme by splitting method

We first introduce some notations. In order to verify the large time numerical behaviour of the proposed method, for convenience, a periodic boundary condition is applied in the following discussions. The modelling domain is denoted by Ω=[a_x,b_x]×[a_z,b_z] with uniform spatial grid. Let u=(u_1,1,…,u_Nx,Nz)^T, where u_i,j is the discrete value of the wavefield on grid, i.e. u_i,j≈u(a_x+iΔx,a_z+jΔz,t), i=1,…,N_x; j=1,…,N_z. Δx, Δz are space steps, and N_x, N_z are the sample indices in the x- and z-directions, respectively. uⁿ represents the vector wavefield at time level of t_n=nΔt. The rest of this section is devoted to the construction of the conformal symplectic scheme for the attenuated acoustic wave equation (2.1) by the splitting method.

To clarify, we briefly explain the splitting technique through the attenuated dynamical system

$${\mathit{w}}_t=f_1(\mathit{w})+f_2(\mathit{w}),$$ 3.1

where f₁ represents a Hamiltonian vector field, whereas f₂=Dw corresponds to the dissipative part with D being a matrix operator. To construct a conformal symplectic scheme, we divide the attenuated system into two subsystems. One is a conservative Hamiltonian system w_t=f₁(w), the other is a linear dissipative ODE system w_t=Dw. We solve the former system with a standard structure-preserving integrator, which is stated by wⁿ⁺¹=Ψ_Δt(wⁿ). While the latter one is solved exactly by the analytical flow ${\psi }_t(\mathit{w})=\exp (\mathit{D}t)\mathit{w}$. Thus, the attenuated dynamical system can be solved by composing these two parts of solutions, e.g. the Strang method ψ_Δt/2°Ψ_Δt°ψ_Δt/2 [29].

Following the above illustration, we split the attenuating acoustic wave equation (2.1) into two subsystems:

$$u_{tt}=c^2\mathrm{\Delta }u$$ 3.2

and

$$u_{tt}=-a{u}_t,$$ 3.3

which correspond to the conservative and dissipative terms, respectively. Note that the dissipative part (3.3) is actually equivalent to two simple linear ODEs

$$u_t=v,v_t=-av,$$ 3.4

and can be solved analytically with

$$v^{n+1}=\exp (-a\mathrm{\Delta }t)v^n,u^{n+1}=\frac{1}{a}(1-\exp (-a\mathrm{\Delta }t))v^n+u^n.$$ 3.5

We denote this exact flow $(u^n,v^n)\to (u^{n+1},v^{n+1})$ as ψ_Δt.

Obviously, (3.2) is a standard Hamiltonian wave equation in the two-dimensional case which admits both the symplectic structure and multi-symplectic structure. In order to derive a structure-preserving scheme with high-order temporal discretization, we choose to construct a symplectic scheme based on its symplectic structure instead of the multi-symplectic scheme based on its multi-symplectic structure of (3.2).

Using v=u_t, we can get the following infinite-dimensional Hamiltonian system that represents the symplectic structure of the wave equation (3.2) as

$${\left[\begin{array}{c}u\\ v\end{array}\right]}_t=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\hspace{0.17em}\left[\begin{array}{c}\frac{\delta \mathcal{H}}{\delta u}\\ \frac{\delta \mathcal{H}}{\delta v}\end{array}\right],$$ 3.6

where the energy functional is

$$\mathcal{H}={\int }_{\mathrm{\Omega }}[\frac{1}{2}(v^2+c^2(u_x^2+u_z^2))]\hspace{0.17em}\mathrm{d}x\hspace{0.17em}\mathrm{d}z,$$ 3.7

and $\delta \mathcal{H}/\delta u$, $\delta \mathcal{H}/\delta v$ are the functional derivatives calculated by

$$\frac{\delta \mathcal{H}}{\delta u}=-c^2(u_{xx}+u_{zz}),\frac{\delta \mathcal{H}}{\delta v}=v.$$

Substituting into the symplectic structure (3.6) and writing it component-wise, we have

$$\begin{array}{rl}{u}_t& =v\\ \text{and}{v}_t& =c^2(u_{xx}+u_{zz}).\end{array}\}$$ 3.8

Obviously, it becomes a separable ODE system after discretizing the spatial derivatives. Therefore, high-order temporal discrete approaches, for example, the Nyström method [30] can be easily applied to construct an explicit symplectic scheme for the Hamiltonian wave equation (3.2) directly.

However, the temporal integration must be put forward on the foundation of a proper spatial semi-discretization which is the crucial part for designing a symplectic scheme. Specifically, we first discretize the energy functional (3.7) instead of directly discretizing the system (3.8). Various numerical methods can be contributed for this procedure, such as the classic finite difference method, the high-order pseudo-spectral method, the wavelet method, even the finite-element or spectral element method, providing those methods can maintain the symmetry of any derivatives in the energy functional [31]. In this study, we simply adopt the pseudo-spectral method for the spatial discretization owing to its corresponding symmetric property and high-accuracy differential operator. Furthermore, fast Fourier transform (FFT) can be implemented to significantly reduce the computational cost. Note that the spectral element method exhibits high-precision and efficient computational superiority. In addition to its advantage of flexibly handling complex geometry, the method also delivers promising potential into the dissipation-preserving framework.

The pseudo-spectral method has been intensively studied by various authors [11,32]. We omit its detailed explanation and just present the corresponding first-order differential operator, which is denoted by D_x and D_z with regard to the horizontal and vertical directions, respectively. Thereafter, the symplectic structure (3.6) is discretized as

$${\left[\begin{array}{c}\mathit{u}\\ \mathit{v}\end{array}\right]}_t=\left[\begin{array}{cc}0& I\\ -I& 0\end{array}\right]\hspace{0.17em}\left[\begin{array}{c}{\mathrm{\nabla }}_{u}H\\ {\mathrm{\nabla }}_{v}H\end{array}\right],$$ 3.9

which is a finite-dimensional Hamiltonian ODE system with I being the identity matrix and the discrete energy

$$H=\frac{1}{2}({\mathit{v}}^T\mathit{v}-C^2({\mathit{u}}^T{D}_x^2\mathit{u}+{\mathit{u}}^T{D}_z^2\mathit{u})),$$ 3.10

where C=diag(c_1,1,c_2,1,…,c_Nx,Nz). Here, c_i,j are the discretizations of the wave velocity at the grid point (x_i,z_j). In the homogeneous case, both c_i,j are constants while in the heterogeneous case they turn to piecewise constants.

Taking the operator $D=C^2(D_x^2+D_z^2)$ for short, we rewrite the Hamiltonian system (3.9) in the equivalent form

$${\mathit{u}}_t=\mathit{v},{\mathit{v}}_t=D\mathit{u},$$ 3.11

which is apparently a separable ODE system. As a consequence, we use an explicit fourth-order Nyström method [30] for the time integration of (3.11):

$$\begin{array}{rl}{V}_1& =D({\mathit{u}}^n+d_1\mathrm{\Delta }t{\mathit{v}}^n),\\ V_2& =D({\mathit{u}}^n+d_2\mathrm{\Delta }t{\mathit{v}}^n+a_{21}{\mathrm{\Delta }t}^2{V}_1),\\ V_3& =D({\mathit{u}}^n+d_3\mathrm{\Delta }t{\mathit{v}}^n+a_{31}{\mathrm{\Delta }t}^2{V}_1+a_{32}{\mathrm{\Delta }t}^2{V}_2),\\ {\mathit{u}}^{n+1}& ={\mathit{u}}^n+\mathrm{\Delta }t{\mathit{v}}^n+{\mathrm{\Delta }t}^2({\overline{b}}_1{V}_1+{\overline{b}}_2{V}_2+{\overline{b}}_3{V}_3)\\ \text{and}{\mathit{v}}^{n+1}& ={\mathit{v}}^n+\mathrm{\Delta }t(b_1{V}_1+b_2{V}_2+b_3{V}_3),\end{array}\}$$ 3.12

where

$$\begin{array}{rl}{d}_1& =\frac{3+\sqrt{3}}{6},d_2=\frac{3-\sqrt{3}}{6},d_3=\frac{3+\sqrt{3}}{6},\\ {\overline{b}}_1& =\frac{5-3\sqrt{3}}{24},{\overline{b}}_2=\frac{3+\sqrt{3}}{12},{\overline{b}}_3=\frac{1+\sqrt{3}}{24},\\ b_1& =\frac{3-2\sqrt{3}}{12},b_2=\frac{1}{2},b_3=\frac{3+2\sqrt{3}}{12},\\ a_{21}& =\frac{2-\sqrt{3}}{12},a_{31}=0,a_{32}=\frac{\sqrt{3}}{6}.\end{array}$$

Here, Ψ_Δt represents the numerical mapping $(u^n,v^n)\to (u^{n+1},v^{n+1})$ for the conservative system (3.2). Apparently, it is a standard symplectic method.

Remark 3.1 —

As aforementioned, the pseudo-spectral method has the major advantage that the computational cost can be dramatically reduced with the assistance of the FFT algorithm. For example, instead of straightly performing matrix-vector multiplications for $(D_x^2+D_z^2)\mathit{u}$, we use the two-dimensional forward ($\mathcal{F}$) and inverse (${\mathcal{F}}^{-1}$) finite Fourier transforms

$$(D_x^2+D_z^2)\mathit{u}={\mathcal{F}}^{-1}(\mathit{\sigma }\star \mathcal{F}(\mathit{u})),$$

where σ=(σ_1,1,…,σ_Nx,Nz) with ${\sigma }_{i,j}=k_{x_i}^2+k_{z_j}^2$. k_xi and k_zj are the discrete wavenumbers in the x- and z-directions, respectively. The star (⋆) denotes the dot product of vectors. By the FFT algorithm, the computational complexity decreases greatly from $\mathcal{O}(N^2)$ to $\mathcal{O}(N\log N)$ with N=N_x×N_z.

So far, we have successfully constructed the corresponding analytical solution of ψ_Δt of the subsystem (3.4), and the numerical solutions Ψ_Δt of the subsystem (3.2). The next step is to concatenate these two parts of solutions together to form the final solution for the original attenuated acoustic wave equation (2.1). We adopt the Strang splitting technique [29]

$$({\mathit{u}}^{n+1},{\mathit{v}}^{n+1})={\psi }_{\mathrm{\Delta }t/2}\circ {\mathrm{\Psi }}_{\mathrm{\Delta }t}\circ {\psi }_{\mathrm{\Delta }t/2}({\mathit{u}}^n,{\mathit{v}}^n),$$ 3.13

To prove the resulting scheme (3.13) is conformal symplectic, we first denote the solutions of every step in the Strang splitting as follows

• — Step 1: $({\mathit{u}}_{s1},{\mathit{v}}_{s1})={\psi }_{\mathrm{\Delta }t/2}({\mathit{u}}^n,{\mathit{v}}^n)$;

• — Step 2: (u_s2,v_s2)=Ψ_Δt(u_s1,v_s1) and

• — Step 3: $({\mathit{u}}^{n+1},{\mathit{v}}^{n+1})={\psi }_{\mathrm{\Delta }t/2}({\mathit{u}}_{s2},{\mathit{v}}_{s2})$.

The first and third steps are given by the analytic solutions

$$u_{s1}=\frac{1}{a}(1-\exp \left(\frac{-a\mathrm{\Delta }t}{2}\right))v^n+u^n,v_{s1}=\exp \left(\frac{-a\mathrm{\Delta }t}{2}\right)v^n$$

and

$$u^{n+1}=\frac{1}{a}(1-\exp \left(\frac{-a\mathrm{\Delta }t}{2}\right))v_{s2}+u_{s2},v^{n+1}=\exp \left(\frac{-a\mathrm{\Delta }t}{2}\right)v_{s2},$$

with the corresponding variational equations as

$$\mathrm{d}{u}_{s1}\wedge \mathrm{d}{v}_{s1}=\exp \left(\frac{-a\mathrm{\Delta }t}{2}\right)\hspace{0.17em}\mathrm{d}{u}^n\wedge \mathrm{d}{v}^n,\mathrm{d}{u}_{s2}\wedge \mathrm{d}{v}_{s2}=\exp \left(\frac{a\mathrm{\Delta }t}{2}\right)\hspace{0.17em}\mathrm{d}{u}^{n+1}\wedge \mathrm{d}{v}^{n+1},$$

where we have used the fact that dvⁿ∧dvⁿ=0 and so on.

The second step is a standard symplectic scheme such that it automatically preserves a discrete symplectic conservation law

$$\mathrm{d}{u}_{s2}\wedge \mathrm{d}{v}_{s2}=\mathrm{d}{u}_{s1}\wedge \mathrm{d}{v}_{s1}.$$

Combining the variational equations and the symplectic conservation law, we can obtain

$$\mathrm{d}{u}^{n+1}\wedge \mathrm{d}{v}^{n+1}=\exp (-a\mathrm{\Delta }t)(\mathrm{d}{u}^n\wedge \mathrm{d}{v}^n),$$

which is exactly the discrete conformal symplectic conservation law. In view of the above process, even if a is a piecewise constant function, the conformal symplectic conservation law still holds.

Remark 3.2 —

Note that although we apply a fourth-order symplectic time integration in the numerical part Ψ_Δt, the final temporal accuracy of the conformal symplectic scheme (3.13) is only of second order which is because that the Strang-splitting method is actually second-order accuracy in time. In order to obtain a higher-order scheme, one can choose alternative splitting methods. For example, let Φ_Δt denote the scheme (3.13), we can construct a fourth-order method by

$$({\mathit{u}}^{n+1},{\mathit{v}}^{n+1})={\mathrm{\Phi }}_{{\gamma }_3\mathrm{\Delta }t}\circ {\mathrm{\Phi }}_{{\gamma }_2\mathrm{\Delta }t}\circ {\mathrm{\Phi }}_{{\gamma }_1\mathrm{\Delta }t}({\mathit{u}}^n,{\mathit{v}}^n),$$ 3.14

where γ₁=γ₃=1/(2−2^1/3), γ₂=−2^1/3/(2−2^1/3). Apparently, the higher precision the splitting schemes possess, the more computational cost will be demanded. Nevertheless, a high-order scheme has more advantages not only in the accuracy, but also in the suppression of numerical dispersion as well as the stability. In this work, we just choose the second-order splitting (3.13) for illustration, which already exhibits superior numerical behaviours in suppressing dispersion and maintaining dissipation rate.

4. Stability analysis

In this section, we present the stability result for the conformal symplectic scheme (3.13), which contains the analysis for the numerical and analytical solutions in the Strang splitting. We first focus on the stability condition for Ψ_Δt based on the standard Fourier analysis [11,33].

Substituting a plane wave solution

$$\left(\begin{array}{c}\mathit{u}\\ \overline{\mathit{v}}\end{array}\right)=\left(\begin{array}{c}U\\ V\end{array}\right)\exp (\exp (\mathrm{i}(\tilde{\omega }t-k_{x}x-k_{z}z)))$$

into equations (3.12), where $\overline{\mathit{v}}=\mathit{v}\mathrm{\Delta }t$ for sake of obtaining a polynomial in ckΔt with $k=\sqrt{k_x^2+k_z^2}$. Eliminate $\exp (\mathrm{i}(\tilde{\omega }t-k_{x}x-k_{z}z))$, we have

$$\exp (\mathrm{i}\tilde{\omega }\mathrm{\Delta }t)\left(\begin{array}{c}U\\ V\end{array}\right)=\left(\begin{array}{cc}1-\frac{1}{2}{\theta }^2+\frac{1}{24}{\theta }^4-\frac{3-\sqrt{3}}{1728}{\theta }^6& 1-\frac{1}{6}{\theta }^2+\frac{1}{72}{\theta }^4-\frac{1}{1728}{\theta }^6\\ -{\theta }^2+\frac{1}{6}{\theta }^4-\frac{1}{288}{\theta }^6& 1-\frac{1}{2}{\theta }^2+\frac{1}{24}{\theta }^4-\frac{3+\sqrt{3}}{1728}{\theta }^6\end{array}\right)\left(\begin{array}{c}U\\ V\end{array}\right),$$ 4.1

where θ=ckΔt. Obviously, equation (4.1) forms a eigenvalue problem that can be easily solved using the symbolic calculation by Maple or Mathematica. The eigenvalues are

$$\lambda =(1-\frac{1}{2}{\theta }^2+\frac{1}{24}{\theta }^4-\frac{1}{576}{\theta }^6)\pm \sqrt{{(1-\frac{1}{2}{\theta }^2+\frac{1}{24}{\theta }^4-\frac{1}{576}{\theta }^6)}^2-1}.$$ 4.2

To ensure that |λ|=1, which requires

$$|1-\frac{1}{2}\theta +\frac{1}{24}{\theta }^2-\frac{1}{576}{\theta }^3|\le 1.$$ 4.3

It can be further solved as

$$\theta \le 4(2+\sqrt[3]{2}-\sqrt[3]{4}).$$ 4.4

For a uniform spacing Δx=Δz=h, we have the Nyquist frequencies k_x=k_z=π/h. Substituting this relation into θ, we get the stability condition for the time step

$$\mathrm{\Delta }t\le \frac{2\sqrt{2+\sqrt[3]{2}-\sqrt[3]{4}}}{\sqrt{2{\pi }^2(c^2/h^2)}}.$$ 4.5

While for the analytical part ψ_Δt (3.5), we rewrite it in the matrix form

$$\left(\begin{array}{c}{\mathit{u}}^{n+1}\\ {\mathit{v}}^{n+1}\end{array}\right)=Q\left(\begin{array}{c}{\mathit{u}}^n\\ {\mathit{v}}^n\end{array}\right)=\left(\begin{array}{cc}1& \frac{(1-\exp (-a\mathrm{\Delta }t))}{a}\\ 0& \exp (-a\mathrm{\Delta }t)\end{array}\right)\left(\begin{array}{c}{\mathit{u}}^n\\ {\mathit{v}}^n\end{array}\right).$$

Therefore, the stability condition for this part just relies on the spectral radius ρ(Q) of the matrix Q. If ρ(Q)≤1, then the analytical solution is stable. By calculation, the two eigenvalues are ${\lambda }_1=1,{\lambda }_2=\exp (-a\mathrm{\Delta }t)$, respectively. Because a≥0, the spectral radius ρ(Q)=1 with no constraint on Δt and a. Consequently, the stability condition for the conformal symplectic scheme (3.13) is only decided by the second numerical step, where the condition (4.5) should be satisfied.

Nevertheless, in general splitting methods, all substeps may need to satisfy an individual stability condition. One has to combine all the conditions to give a complete analysis for the final scheme. However, the splitting steps in this work contains two analytical parts which further are dissipative systems. Hence, the stability condition is automatically satisfied as we mentioned above. In the following numerical tests, all the temporal steps are chosen according to the criterion (4.5).

5. Numerical experiments

(a). Benchmark test

For simplicity, in order to verify the effectiveness of our established conformal symplectic method (3.13), we first perform three tests on benchmark models of (2.1) in a homogeneous media with a velocity of 1 km s⁻¹. The coefficient of damping term a is taken as 0.5 s⁻¹, which corresponds to the source frequency and the quality factor [34,35]. We choose the initial condition as

$$\begin{array}{rl}u(x,z,0)& =\cos (K_{1}x+K_{2}z),\\ \frac{\mathrm{\partial }u(x,z,0)}{\mathrm{\partial }t}& =-\frac{a}{2}\cos (K_{1}x+K_{2}z)+W\sin (K_{1}x+K_{2}z),\end{array}\}$$ 5.1

with $W=\sqrt{c^2(K_1^2+K_2^2)-a^2/4}$. In this case, we have the analytical solution

$$u(x,z,t)={\mathrm{e}}^{-at/2}\cos (K_{1}x+K_{2}z-Wt).$$ 5.2

First, we test the temporal accuracy to verify the correctness of the proposed conformal symplectic method (3.13) by choosing a set of parameters K₁,K₂ and a. Without loss of generality, K₁ and K₂ are taken as the same value. We can find in table 1 that for any a or K₁,K₂ the temporal accuracy is uniformly of second order which confirms the effectiveness of the conformal symplectic scheme (3.13). We also present the convergence results for the fourth-order scheme (3.14) in table 2. It is clear that higher precision does make the numerical error much smaller. Nevertheless, it also brings considerable computation cost, that is at least a triple amount of work than the second-order scheme (3.13) because it contains three stages of composition and each requires an implementation of scheme (3.13). In the following numerical experiments, we choose only the second-order scheme (3.13) for modelling of seismic attenuation, which also has excellent behaviours in dissipation preservation and dispersion suppression with lower cost.

Table 1.

Temporal accuracy test for scheme (3.13) with different damping coefficients a and K₁,K₂. The spatial steps are fixed to Δx=Δz= 2π/80 km.

		K1=K2=1		K1=K2=4		K1=K2=8
a	Δt	error	order	error	order	error	order
0.5	0.02	4.1106×10−5	—	9.8749×10−5	—	3.2261×10−4	—
	0.01	1.0276×10−5	2.0001	2.4656×10−5	2.0018	7.9075×10−5	2.0285
	0.005	2.5690×10−6	2.0000	6.1620×10−6	2.0005	1.9667×10−5	2.0074
1	0.02	7.2461×10−5	—	1.8250×10−4	—	5.5292×10−4	—
	0.01	1.8115×10−5	2.0000	4.5587×10−5	2.0012	1.3679×10−4	2.0151
	0.005	4.5286×10−6	2.0000	1.1394×10−5	2.0003	3.4106×10−5	2.0039
1.5	0.02	9.5062×10−5	—	2.5074×10−4	—	7.1089×10−4	—
	0.01	2.3765×10−5	2.0000	6.2644×10−5	2.0009	1.7642×10−4	2.0106
	0.005	5.9412×10−6	2.0000	1.5658×10−5	2.0003	4.4021×10−5	2.0028

Table 2.

Temporal accuracy test for scheme (3.14) with different damping coefficients a and K₁=K₂=8. The spatial steps are fixed to Δx=Δz= 2π/80 km.

	a=0.5		a=1		a=1.5
Δt	error	order	error	order	error	order
0.02	7.4154×10−5	—	6.4286×10−5	—	6.0617×10−5	—
0.01	4.0732×10−6	4.1863	3.5236×10−6	4.1894	3.2264×10−6	4.2317
0.005	2.3851×10−7	4.0940	2.0597×10−7	4.0965	1.8918×10−7	4.0921

To check the numerical preservation of the dissipation rate, we define a scale function of the amplitude of the wavefield

$$d(t^n)=\ln (max({\mathit{u}}^n))+\frac{a}{2}{t}^n.$$

Therefore, for the exact solution (5.2), the value of the scale function d(tⁿ)≡0. To make numerical comparisons, we consider two more numerical schemes which also respectively adopt the pseudo-spectral method for spatial discretizations and the backward or central difference method for the damping term, as

$$\frac{{\mathit{u}}^{n+1}-2{\mathit{u}}^n+{\mathit{u}}^{n-1}}{{\mathrm{\Delta }t}^2}=c^2(D_x^2{\mathit{u}}^n+D_z^2{\mathit{u}}^n)-a\frac{{\mathit{u}}^n-{\mathit{u}}^{n-1}}{\mathrm{\Delta }t}$$ 5.3

and

$$\frac{{\mathit{u}}^{n+1}-2{\mathit{u}}^n+{\mathit{u}}^{n-1}}{{\mathrm{\Delta }t}^2}=c^2(D_x^2{\mathit{u}}^n+D_z^2{\mathit{u}}^n)-a\frac{{\mathit{u}}^{n+1}-{\mathit{u}}^{n-1}}{2\mathrm{\Delta }t}.$$ 5.4

The computational domain is set on Ω=[−π,π]×[0,2π] km. The space steps are Δx=Δz=2π/80 km, and the temporal step is set to Δt=0.02 s based on the stable condition (4.5). Parameters in the initial condition (5.1) are chosen as K₁=K₂=8.

In figure 1, the corresponding dissipation rates of the above two schemes and the conformal symplectic method (3.13) are plotted with respect to time. It reveals that the central difference method and the conformal symplectic method shows no drift in the dissipation rate. While the backward difference method suffers a linear increment error apparently.

Figure 1.

Difference of numerical dissipation rates and the exact dissipation rate with respect of time. Δx=Δz=2π/80 km and the temporal step is set to Δt=0.02 s for all the three schemes. The damping factor a= 0.5 s⁻¹. (Online version in colour.)

Figure 2 also confirms this observation from the corresponding cross profiles of t=100 s at the surface (z=0). Although the central difference method (5.4) conserves the dissipation rate well, a severe phase drift occurs which is mainly, because it is not a structure-preserving method. However, the conformal symplectic scheme (3.13) exhibits a terrific capacity of preserving not only the dissipation rate, but also the wave profiles over a large time simulation, even though the amplitude of the wavefield is as extremely small as (10⁻¹¹). Note that the central difference scheme (5.4) and the conformal symplectic scheme (3.13) have both second-order accuracy in time and spectral convergence in space. From figure 2, it is clear that to precisely simulate the seismic wave propagation in attenuating media only using high-order methods is not enough, more importantly, one must consider the intrinsic property of the system and design structure-preserving schemes.

Figure 2.

(a-c) Comparison of the analytical and numerical solutions from the backward (5.3), central (5.4) difference scheme and the conformal symplectic scheme (3.13) at the surface (z=0). Δx=Δz=2π/80 km and the temporal step is set to Δt=0.02 s for all the three schemes. The damping factor a=0.5 s⁻¹. (Online version in colour.)

To illustrate that the stability condition is independent of a, we further test this example with a relative large value of a but with fixed time step Δt=0.02 s. Note that we choose different computation times in our experiments, which is mainly because that the machine precision can only reach 10⁻¹⁶ in a given computing environment and after that the wave amplitude will not decay any more, but just oscillate around the machine precision. In these three examples, we choose a=1,2,4 s⁻¹, respectively. We can observe that the wave amplitude matches well with the analytical one at magnitude 10⁻¹³ in figure 3. Furthermore, we also plot the measure function d(t) in figure 4 to illustrate the efficiency of our scheme in preserving the dissipation rate after long-term simulation for large a.

Figure 3.

Comparison of the analytical and numerical solutions of the conformal symplectic scheme at the surface (z=0) with a=1 s⁻¹ (a), a= 2 s⁻¹ (b) and a=4 s⁻¹ (c), Δx=Δz=2π/80 km, Δt= 0.02 s. (Online version in colour.)

Figure 4.

Evolutions of measure function d(t) against time with a=1 s⁻¹ (a), a=2 s⁻¹ (b) and a=4 s⁻¹ (c), Δx=Δz=2π/80 km, Δt=0.02 s. (Online version in colour.)

(b). Homogeneous model

In the second example, we use the initial conditions [10]

$$\begin{array}{rl}& u(x,z,0)=\exp (-{\alpha }^2({(x-x_0)}^2+{(z-z_0)}^2))\\ \text{and}& \frac{\mathrm{\partial }u(x,z,0)}{\mathrm{\partial }t}=0,\end{array}\}$$ 5.5

where x₀,z₀ indicate the horizontal and depth positions of the wave source. α is a constant determining the width of the pulse and generally taken from αΔx=0.5 to guarantee the reflected signal quality [10]. Here, we set x₀=0,z₀=3000 m and α=0.01.

First, we consider a perfectly elastic media (i.e. a=0), which means there is no attenuation for the wavefield, to demonstrate the large time behaviour and conservative property of our scheme (3.13). Under this assumption, the attenuated acoustic wave equation becomes

$$u_{tt}=c^2(u_{xx}+u_{zz}),$$ 5.6

which is just the Hamiltonian system [32], and admits the energy conservation law

$$\mathcal{E}(t)={\int }_{\mathrm{\Omega }}(v^2+c^2(u_x^2+u_z^2))\hspace{0.17em}\mathrm{d}x\hspace{0.17em}\mathrm{d}z=\mathrm{const}.$$ 5.7

In our experiment, we use a discrete form of energy

$$E^n=[{({\mathit{v}}^n)}^T{\mathit{v}}^n-C^2({({\mathit{u}}^n)}^T{D}_x^2{\mathit{u}}^n+{({\mathit{u}}^n)}^T{D}_z^2{\mathit{u}}^n)]\mathrm{\Delta }x\mathrm{\Delta }z.$$ 5.8

Figure 5 shows the pulse profiles at time t=1 and t=20 s, respectively. They are computed by the conformal symplectic scheme in a homogeneous media with the wave velocity c=2 km s ⁻¹. The wave source is set in the centre of the area at depth z₀.

Figure 5.

Snapshots of the wavefield at time t=1 s (a) and t=20 s (b) in non-attenuating media (a=0). The spatial and temporal steps are Δx=Δz= 50 m, Δt=0.01 s. (Online version in colour.)

It demonstrates from the snapshot at t=20 s that the pulse response is clearly simulated even after a large time of propagation. Figure 6 shows the evolution of the relative energy errors defined by

$$R{E}^n=|E^n-E^0|/E^0$$ 5.9

evolving with respect to time when a=0, confirming that the wave energy by the conformal symplectic scheme (3.13) is also well preserved.

Figure 6.

Relative energy error REⁿ (5.9) against the time step in non-attenuating media (a=0). The spatial and temporal steps are Δx=Δz= 50 m, Δt=0.01 s. (Online version in colour.)

Next we test our method in a realistic attenuating media with the damping coefficient taken as a=0.2 s⁻¹. The model settings are the same as that in the conservative case above. Nevertheless, the energy of this system is no longer conservative. Instead of the energy conservation law (5.7), it is easy to obtain that an energy dissipative law holds for the attenuated acoustic wave equation (2.1) by

$$\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathrm{\Omega }}(v^2+c^2(u_x^2+u_z^2))\hspace{0.17em}\mathrm{d}x\hspace{0.17em}\mathrm{d}z=-2{\int }_{\mathrm{\Omega }}a{(u_t)}^2\hspace{0.17em}\mathrm{d}x\hspace{0.17em}\mathrm{d}z.$$ 5.10

For the sake of checking the dissipation-preserving property of our proposed conformal symplectic scheme (3.13), we measure the relative energy decay rate in a energy dissipation law (5.10)

$$R^n=\left|\frac{E^{n+1}-E^n}{2A{({\mathit{v}}^n)}^T{\mathit{v}}^n\mathrm{\Delta }x\mathrm{\Delta }z\mathrm{\Delta }t}\right|,$$ 5.11

where Eⁿ is defined as (5.8).

Remark 5.1 —

Because, in our numerical models, the amplitude of discrete energy reaches more than 10⁵, the difference between the left and right sides of (5.10) can hardly reveal the relative errors in the energy dissipation law. Therefore, we use the relative energy decay rate (5.11) to measure the preservation of the dissipation law (5.10) such that if Rⁿ has values around 1, the dissipation law is regarded to be well preserved.

The wavefronts at the same time are presented in figure 7. Although they have identical profiles, their amplitudes are quite different which is exactly caused by the dissipation. As a consequence, we can speculate that the dissipation in a homogeneous media only affects the amplitudes of the wavefield and leaves its profiles unchanged.

Figure 7.

Snapshots of the wavefield at time t=1 (a) and t=20 s (b) with the damping coefficient a=0.2 s⁻¹. The spatial and temporal steps are Δx=Δz=50 m, Δt=0.01 s. (Online version in colour.)

Unlike the conservative case, the energy of the acoustic wave is attenuated in the attenuating media. Figure 8 demonstrates the variations of the discrete energy (5.7) as well as the relative energy decay rate in the dissipation law (5.10). A clear exponential decay of energy can be observed from the energy evolution corresponding to the dissipative subsystem (3.3), which exactly admits an exponential decay of the wavefield. Moreover, the relative energy decay rate shows that our proposed method cannot only accurately simulate the energy decay curve, but also preserve the energy dissipation rate well during a large time computation.

Figure 8.

Discrete energy Eⁿ (5.8) (a) and relative energy decay rate Rⁿ (5.11) (b) against the time step with the damping coefficient a= 0.2 s⁻¹. The spatial and temporal steps are Δx=Δz=50 m, Δt=0.01 s. (Online version in colour.)

(c). Absorbing boundary condition

In the realistic modelling of seismic waves, ABCs are inevitably adopted to avoid boundary reflections. The development of the PML boundary conditions significantly improve both the effectiveness and stability of ABCs, and thereby is widely applied on seismic propagation modellings (see [37,36] and reference therein). In this work, we follow the unsplit PML ABC proposed in [36], which requires less auxiliary variables but satisfies strong numerical stability. Because we introduce the attenuated term in the model equation, the corresponding modified PML equations are different from the standard ones in [36]. For short, we include a detailed derivation of the modified PML equations as well as the corresponding discretization in appendix.

The test model is the same as that in the above homogeneous case with the difference of perfect absorbing layers surrounding the interest domain, as demonstrated in figure 9.

Figure 9.

Numerical model with the interest domain surrounded by PMLs. The thickness of layers is uniformly set to 600 m.

The parameters in the initial condition (5.5) are set to x₀=−2000 m, z₀=1000 m. We set the wave velocity c=2 km s⁻¹ and the damping factor a=0.5 s⁻¹. The damping profile ξ_i(x_i) in (7.3) is given by

$$\begin{array}{rl}{\xi }_1(x)& =\{\begin{array}{ll}0& \text{for}\hspace{0.17em}|x|<3000,\\ 40{\left(\frac{|x|-3000}{600}\right)}^2& \text{for}\hspace{0.17em}|x|\ge 3000,\end{array}\\ {\xi }_2(z)& =\{\begin{array}{ll}0& \text{for}\hspace{0.17em}0<x<6000,\\ 40{\left(\frac{z}{600}\right)}^2& \text{for}\hspace{0.17em}z\le 0,\\ 40{\left(\frac{z-6000}{600}\right)}^2& \text{for}\hspace{0.17em}z\ge 6000.\end{array}\end{array}$$

In figure 10, we present the wavefield snapshots in the interest region [−3,3]×[0,6] km. We can hardly see any visible reflected waves in these figures, which demonstrates the efficiency of our PML formulation (A 7)–(A 9) as well as the numerical discrete method (A 10)–(A 12). Meanwhile, the attenuation of the wave amplitude is clearly observed, as shown from the colour bars. It confirms that the combination of our proposed conformal symplectic method (3.13) and the PML ABC can well simulate the attenuated acoustic wave equation in a truncated domain. Although we test only one kind of PML boundary conditions [36], other variations can be also extended to the framework of our conformal symplectic method.

Figure 10.

Snapshots of the displacement with PML boundary condition (a) t=0.3 s; (b) t=1 s; (c) t=2 s and (d) t=3 s. The spatial and temporal steps are Δx=Δz=50 m, Δt=0.01 s. (Online version in colour.)

The main purpose of numerical experiments is to verify the long-time stability and dissipation-preserving properties of the conformal symplectic method. In the following tests, we still apply the periodic boundary condition to guarantee the wavefront can propagate in the computational domain over a relative long time period.

(d). Two-layer model

To further validate the conformal symplectic scheme (3.13), we present an additional experiment in a two-layer acoustic model with a point source, which is a Ricker wavelet with a frequency of f₀=35 Hz defined as [17]:

$$f(t)=-5.76f_0^2[1-16{(0.6f_{0}t-1)}^2]\exp [-8{(0.6f_{0}t-1)}^2].$$ 5.12

In this case, we consider the initial conditions for the attenuated acoustic wave equation as

$$u(x,z,0)=0\text{and}\frac{\mathrm{\partial }u(x,z,0)}{\mathrm{\partial }t}=0.$$ 5.13

It is worth emphasizing that, with the presence of the Ricker wavelet wave source, the model equation is not conservative any more either in a attenuating media or not. Fortunately, the Ricker wavelet decays very fast, and becomes zero at the very beginning of the wave propagation. Thereafter, the system promptly comes to be conservative or purely dissipative soon depending on the media. This ensures the effectiveness of our method right after the wave source becomes zero.

The two-layer velocity model is shown in figure 11. It consists of two layers with different wave velocities, separating the whole region in half. The wave velocity is 2.4 km s⁻¹ in the top layer and 5 km s⁻¹ in the bottom layer. It brings strong interface effects between these two layers, such as the reflection, refraction and diffraction. The damping coefficients of these two layers are set a=0.4 s⁻¹ and a=0.2 s⁻¹, respectively. The wave source S is located at (x_S,z_S)=(0 km,2 km).

Figure 11.

The two-layer velocity model with a strong interface at the depth of z=3 km.

In order to illustrate clearly the propagation of the acoustic wave, we run a short-time simulation until time t=0.8 s. Figure 12 apparently shows the transmitted and reflected waves without any visible numerical dispersion even in the case of large velocity contrasts between two adjacent layers, exhibiting the significant performance of our proposed conformal symplectic scheme (3.13) in suppressing of numerical dispersions. The corresponding energy curve is plotted in figure 13a, which can be divided into three phases:

• Phase 1: the Ricker wavelet generates the acoustic wave with a sudden increasing of energy. Simultaneously, the wave propagates in the upper attenuating media which causes an energy dissipation. Both parts contribute to the variation of the energy in the first phase.

• Phase 2: the effect of the Ricker wavelet terminates and the acoustic wave spreads and decays in the upper region with constant velocity and damping rate, which is quite identical with that in the homogeneous case. The energy experiences an exponential dissipation in the second phase. The tendency looks like a linear decrement mainly because of the very short period of time.

• Phase 3: at the beginning of the third phase, the wavefront reaches the interface between the two layers. Owing to the discontinuity of the wave velocities on the interface, the energy curve has an instantaneous perturbation. Later, the transmitted and reflected waves start to dissipate in the bottom and upper layers, respectively, and so does their corresponding wave energy.

Figure 12.

Snapshots of the wavefield at time t=0.8 s in two-layer attenuating media. The spatial grid sizes and time step size are chosen as Δz=Δz= 25 m and Δt=0.002 s, respectively. (Online version in colour.)

Figure 13.

Evolutions of the discrete energy Eⁿ (5.8) against the time step in two-layer attenuating media until t=0.8 s (a) and relative energy decay rate Rⁿ (5.11) in the energy dissipation law (b). (Online version in colour.)

In addition, figure 13b presents the relative energy decay rate Rⁿ (5.11) in the energy dissipation law (5.10). It can be apparently observed that there are two oscillations in the energy residual curve. The first one is generated by the Ricker wavelet and the time window coincides exactly. The second one represents the increment of the velocity which affects the energy dissipation law. The remaining parts of the relative energy decay rate Rⁿ (5.11) almost equal to one which satisfies the dissipation law (5.10) very well. In this sense, our conformal symplectic scheme (3.13) not only can preserve the discrete dissipation law, but also indicates the effects of wave source and the wave distinction between heterogeneous mediums.

We then choose a relatively large time period of t=2 s for another test. Figure 14 shows the large time stability of our scheme. With the time evolving, the amplitude of the wavefield becomes smaller owing to the presence of the attenuation. Furthermore, it can be found that there is still hardly any evidence of numerical dispersion in either the high or low attenuating media. We also plot the variance of energy with respect to time in figure 15a, which, however, cannot be explained part by part like that in figure 13a because of the multiple collisions of the wavefront. Even so, we can also observe an oscillation of the relative energy decay rate around 10⁰ with spikes corresponding to these collisions.

Figure 14.

Snapshots of the wavefield at time t=2 s in two-layer attenuating media. The spatial grid sizes and time step size are chosen as Δz=Δz= 25 m and Δt=0.002 s, respectively. (Online version in colour.)

Figure 15.

Evolutions of the discrete energy Eⁿ (5.8) against the time step in two-layer attenuating media until t=2 s (a) and relative energy decay rate Rⁿ (5.11) in the energy dissipation law (b). (Online version in colour.)

(e). Marmousi model

We now test the conformal symplectic method in a more complex case, Marmousi model, which is popular in examining the accuracy of numerical methods for wave propagating in strongly heterogeneous media. Figure 16 displays the velocity profile of the Marmousi model. The wave velocity varies from 1500 to 5500 m s⁻¹. The point source is located at the centre of this model consisting of a Ricker wavelet with a frequency of f₀=25 Hz. For simplicity, we uniformly set the damping coefficient a=0.2 s⁻¹.

Figure 16.

Velocity profile of Marmousi mode. (Online version in colour.)

Figure 17 is the snapshot of the wavefield after 600 ms generated by the conformal symplectic method. It can be seen that there is hardly any evidence of numerical dispersion. This experiment indicates that our method also has excellent performance in strongly heterogeneous media, even though the additional damping term and the splitting strategy are introduced.

Figure 17.

Snapshot of acoustic wavefield in Marmousi model at time 600 ms. The spatial and temporal steps are Δx=Δz=25 m, Δt=0.001 s. (Online version in colour.)

6. Algorithm advantage and potential

We numerically investigate the seismic wave propagation in attenuating media through the acoustic wave equation with an additional damping term. For the conservative model without damping, structure-preserving methods, such as the symplectic and multi-symplectic methods, have been proved to have superior performance in numerical accuracy, stability and large time simulation, which is mainly because these methods are designed to preserve the intrinsic conservative structures of the original system. However, the presence of the damping term violates all the conservative properties, and thereby, the symplectic or multi-symplectic method cannot precisely capture the corresponding wave dissipation. As a consequence, the wave amplitude and phase, as well as the long time simulation will be incorrect or with high numerical dispersion. In this context, we put forward a novel conformal symplectic method for the attenuated acoustic wave equation, which can not only effectively suppress the numerical dispersion, but also, more strikingly, capture the wave amplitude, the decay rate of the wave amplitude and the discrete energy simultaneously. Meanwhile, the proposed method can be adapted to simulate the conservative systems with clear wavefield and discrete energy conservation law.

Most importantly, our proposed construction of the conformal symplectic method does not depend on specified spatial discrete methods. Other existing efficient numerical methods, such as the spectral element method, can be also taken into this framework. Therefore, the conformal symplectic method may become one of the classic methods for solving the attenuated acoustic wave equation. Similarly, such a method can further be applied on the attenuated elastic wave equations which also possesses the conformal multi-symplectic structure, and thereby we can adopt similar procedures to construct the associated conformal symplectic scheme. Although we have only discussed the two-dimensional cases, this approach can be directly generalized to the realistic three-dimensional cases.

7. Conclusion

We have proposed a conformal symplectic method which is elaborately designed to preserve the conformal symplectic structure of the attenuated acoustic wave equation. The fundamental idea is the operator splitting method. Specifically, we first divide the attenuated model equation into a conservative Hamiltonian system and a dissipative system. For the conservative system, an explicit symplectic scheme is successfully applied on the time integration which is owing to the separable property of the displacement and velocity components. While the pseudo-spectral method is adopted for the spatial discretization as it admits high-order accuracy and fast implementation. For the dissipative system, it is actually degenerated into a linear ODE system and therefore can be analytically solved. Eventually, we concatenate the two parts of solutions by a composition method to complete the conformal symplectic method which can well preserve the dissipation rate of the original attenuated model.

A benchmark test is constructed to verify the proposed conformal symplectic method and compare with other two non-conformal methods, which show an absolute advantage of our method in both numerical phase and amplitude. For the homogeneous model, we consider cases with and without damping terms to demonstrate our scheme is adaptive from a energy-preserving method to a dissipation-preserving method during a long-term simulation. By adding perfect absorbing layers on the truncated edges of the homogeneous model, we propose a modified PML equation as well as the corresponding discrete scheme. In conjunction with the conformal symplectic method, the wave reflections are well suppressed in the computation domain. Heterogeneous models including the classic Marmousi model are also tested which provide clear wavefronts without any visible numerical dispersion.

Appendix A. Formulation of the modified perfect matched layer equations

By introducing the Laplace transform of u defined as

$$\hat{u}=(\mathit{x},s)={\int }_0^{\mathrm{\infty }}{\text{e}}^{st}u(\mathit{x},t)\hspace{0.17em}\mathrm{d}t,s\in \mathbb{C},$$ A 1

we have the corresponding form of the two-dimensional damped acoustic wave equation (2.1) in the frequency domain

$$s^2\hat{u}+as\hat{u}=c^2\mathrm{\Delta }\hat{u}.$$ A 2

Following the complex coordinate stretching

$${\tilde{x}}_i:=x_i+\frac{1}{s}{\int }_0^{x_i}{\xi }_i(x)\hspace{0.17em}\mathrm{d}x,i=1,2,$$ A 3

where the damping profile ξ_i is positive inside the PML domain but vanishes inside Ω, we obtain the modified equation of (A 2) in stretched coordinates, we have

$$s^2\hat{u}+as\hat{u}=c^2\tilde{\mathrm{\Delta }}\hat{u},$$ A 4

where the Laplace operator $\tilde{\mathrm{\Delta }}$ is with respect to ${\hat{x}}_i$, i=1,2.

Next, we transform equation (A 4) back into the time domain. Given the relation that

$$\frac{\mathrm{\partial }}{\mathrm{\partial }{\hat{x}}_i}=\frac{s}{s+{\xi }_i}\frac{\mathrm{\partial }}{\mathrm{\partial }{\hat{x}}_i},$$

and multiplying (s+ξ₁)(s+ξ₂)/s² on both sides, we can replace the partial derivatives in the Laplace operator $\tilde{\mathrm{\Delta }}$ from complex coordinates into real physical coordinates as follows

$$\begin{array}{rl}& s^2\hat{u}+s({\xi }_1+{\xi }_2+a)\hat{u}+({\xi }_1{\xi }_2+a({\xi }_1+{\xi }_2))\hat{u}+\frac{a{\xi }_1{\xi }_2}{s}\hat{u}=c^2\mathrm{\Delta }\hat{u}\\ & +c^2[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}\left(\frac{{\xi }_2-{\xi }_1}{s+{\xi }_1}\frac{\mathrm{\partial }\hat{u}}{\mathrm{\partial }{x}_1}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}\left(\frac{{\xi }_1-{\xi }_2}{s+{\xi }_2}\frac{\mathrm{\partial }\hat{u}}{\mathrm{\partial }{x}_2}\right)].\end{array}$$ A 5

Introducing the auxiliary variables ν=(ν₁,ν₂)^T and η, such that

$$\begin{array}{rl}s\eta & =\hat{u},\\ (s+{\xi }_1){\hat{\nu }}_1& =({\xi }_2-{\xi }_1)\frac{\mathrm{\partial }\hat{u}}{\mathrm{\partial }{x}_1}\\ \text{and}(s+{\xi }_2){\hat{\nu }}_2& =({\xi }_1-{\xi }_2)\frac{\mathrm{\partial }\hat{u}}{\mathrm{\partial }{x}_2}.\end{array}\}$$ A 6

Finally, combining equations (A 5) and (A 6) and using the inverse Laplace transform, we derive the modified PML equations of the wavefield in the time domain as

$$\begin{array}{rl}& \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}+({\xi }_1+{\xi }_2+a)\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+({\xi }_1{\xi }_2+a({\xi }_1+{\xi }_2))u=c^2\mathrm{\Delta }u+c^2\mathrm{\nabla }\cdot \mathit{\nu }-a{\xi }_1{\xi }_2\eta ,\end{array}$$ A 7

$$\begin{array}{rl}& \frac{\mathrm{\partial }\mathit{\nu }}{\mathrm{\partial }t}={\mathrm{\Gamma }}_1\mathit{\nu }+{\mathrm{\Gamma }}_2\mathrm{\nabla }u,\end{array}$$ A 8

$$\begin{array}{rl}& \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}=u,\end{array}$$ A 9

where

$${\mathrm{\Gamma }}_1=\left(\begin{array}{cc}-{\xi }_1& 0\\ 0& -{\xi }_2\end{array}\right),{\mathrm{\Gamma }}_2=\left(\begin{array}{cc}{\xi }_2-{\xi }_1& 0\\ 0& {\xi }_1-{\xi }_2\end{array}\right).$$

It is worth noting that the final modified PML equations (A 7)–(A 9) have several differences from the standard one in [36], which not only occurs in the coefficients in equation (A 7), but also in the occurrence of the additional differential equation (A 9). Similarly, when taking other kinds of PML boundary conditions, the resulting modified PML equations also have slight differences from the conservative case such that one should pay extra attention to construct numerical schemes. In addition, the modified PML equations (A 7)–(A 9) no longer admit the conformal multi-symplectic structure. Hence, they cannot be taken into the same splitting strategy, as the corresponding numerical discretizations.

In the following, we present a separated discrete scheme to solve these equations in the PML domain. For the discretization of equation (A 7), we use

$$\begin{array}{rl}& \frac{{\mathit{u}}^{n+1}-2{\mathit{u}}^n+{\mathit{u}}^{n-1}}{\mathrm{\Delta }{t}^2}+({\mathrm{\Xi }}_1+{\mathrm{\Xi }}_2+A)\frac{{\mathit{u}}^{n+1}-{\mathit{u}}^{n-1}}{2\mathrm{\Delta }t}+({\mathrm{\Xi }}_1{\mathrm{\Xi }}_2+A({\mathrm{\Xi }}_1+{\mathrm{\Xi }}_2)){\mathit{u}}^n\\ & =C^2(D_x^2+D_z^2){\mathit{u}}^n+C^2(D_x{\mathit{\nu }}_1^n+D_z{\mathit{\nu }}_2^n)-A{\mathrm{\Xi }}_1{\mathrm{\Xi }}_2\frac{{\mathit{\eta }}^{n+1/2}+{\mathit{\eta }}^{n-1/2}}{2},\end{array}$$ A 10

where Ξ₁=I_z⊗ξ₁, Ξ₂=ξ₂⊗I_x. Here ξ₁,ξ₂ are the column vectors of damping profiles with the length of N_x and N_z, respectively. I_x,I_z are unit column vectors with the length of N_x and N_z. The associated vector forms of ν_i,η are denoted by ν_i,η, respectively. The Crank–Nicolson scheme is adopted to discretize the second equation (A 8)

$$\begin{array}{rl}\frac{{\mathit{\nu }}_1^{n+1}-{\mathit{\nu }}_1^n}{\mathrm{\Delta }t}& =-{\mathrm{\Xi }}_1{\mathit{\nu }}_1^{n+1/2}+({\mathrm{\Xi }}_2-{\mathrm{\Xi }}_1)D_x{\mathit{u}}^{n+1/2}\\ \text{and}\frac{{\mathit{\nu }}_2^{n+1}-{\mathit{\nu }}_2^n}{\mathrm{\Delta }t}& =-{\mathrm{\Xi }}_2{\mathit{\nu }}_2^{n+1/2}+({\mathrm{\Xi }}_1-{\mathrm{\Xi }}_2)D_z{\mathit{u}}^{n+1/2},\end{array}\}$$ A 11

where ${\mathit{\nu }}_i^{n+1/2}=\frac{1}{2}({\mathit{\nu }}_i^{n+1}+{\mathit{\nu }}_i^n)$ and ${\mathit{u}}^{n+1/2}=\frac{1}{2}({\mathit{u}}^{n+1}+{\mathit{u}}^n)$.

While for the approximation of η, we simply use

$$\frac{{\mathit{\eta }}^{n+1/2}-{\mathit{\eta }}^{n-1/2}}{\mathrm{\Delta }t}={\mathit{u}}^n.$$ A 12

Then the computation flow chart of the displacement from uⁿ to uⁿ⁺¹ is divided into two steps. Firstly, we compute a temporary wavefield u* at level n+1 by the conformal symplectic method (3.13), both in the interest and the PML domains. Secondly, we apply the scheme (A 10)–(A 12) to recompute the wavefield in the PML domain as u^pml. Because the spatial discretization is based on the global pseudo-spectral method, we also compute the solution in the entire domain. Thereafter, we substitute u^pml for the wavefield u* in the PML domain to complete uⁿ⁺¹.

Taking advantage of the fast Fourier transform, the low-cost computational cost is affordable. As mentioned above, the conformal symplectic method does not depend on the detailed spatial discretizations. Other methods, such as the high-order finite difference or the spectral methods, can also be used to form the conformal symplectic scheme, which is local and therefore more flexible to compute the PML equations separately. In this work, we just illustrate the derivation of modified PML equations for the attenuated acoustic wave equation (2.1), and demonstrate that the conformal symplectic scheme does not affect the application of ABCs eventually.

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This work does not have any experimental data.

Authors' contributions

W.C. designed the computational method, carried out numerical experiments and computational code writing, drafted the manuscript; H.Z provided the subject and modified the manuscript; Y.W. verified the computational method. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This research work is supported by National Natural Science Foundation of China (grant nos. 41274103, 11271195, 41504078), National Basic Research Programme of China (2014CB845906).