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Stable and high-order accurate finite difference methods for the diffusive viscous wave equation

Siyang Wang

TL;DR

This work addresses stable, high-order numerical solution of the diffusive viscous wave equation in diffusive and viscous media. It develops an SBP-SAT finite difference framework using summation-by-parts operators to prove discrete energy stability and provides a priori error estimates for constant and variable coefficients, as well as a multidimensional extension. The main contributions are the construction of Dirichlet and Neumann SBP-SAT discretizations with provable stability, a detailed error analysis showing high-order convergence constrained by boundary truncation, and comprehensive numerical experiments validating the theory. The approach enables accurate simulations of seismic and biomedical wave propagation and can be extended to curvilinear grids and hybrid discretizations for complex geometries.

Abstract

The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finite difference method for solving the governing equations. By designing the spatial discretization with the summation-by-parts property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and for problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.

Stable and high-order accurate finite difference methods for the diffusive viscous wave equation

TL;DR

This work addresses stable, high-order numerical solution of the diffusive viscous wave equation in diffusive and viscous media. It develops an SBP-SAT finite difference framework using summation-by-parts operators to prove discrete energy stability and provides a priori error estimates for constant and variable coefficients, as well as a multidimensional extension. The main contributions are the construction of Dirichlet and Neumann SBP-SAT discretizations with provable stability, a detailed error analysis showing high-order convergence constrained by boundary truncation, and comprehensive numerical experiments validating the theory. The approach enables accurate simulations of seismic and biomedical wave propagation and can be extended to curvilinear grids and hybrid discretizations for complex geometries.

Abstract

The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finite difference method for solving the governing equations. By designing the spatial discretization with the summation-by-parts property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and for problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.
Paper Structure (17 sections, 4 theorems, 80 equations, 4 figures)

This paper contains 17 sections, 4 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The governing equation
  3. The SBP finite difference method
  4. The SBP finite difference operators
  5. An SBP-SAT discretization for the Dirichlet problem with viscosity
  6. An SBP-SAT discretization for the Neumann problem
  7. Multidimensional problems
  8. Error estimates
  9. Case 1: $\alpha=\beta=0$
  10. Case 2: $\alpha\neq 0, \beta=0$
  11. Case 3: $\alpha= 0, \beta\neq 0$
  12. Case 4: $\alpha\neq 0, \beta\neq 0$
  13. Numerical experiments
  14. Constant coefficients in one dimension
  15. Variable coefficients in one dimension
  16. ...and 2 more sections

Key Result

Theorem 1

The semidiscretization semiD1D satisfies the energy estimate EA1D_s6 if the penalty parameters are chosen as in tau1, tau2 and tau34.

Figures (4)

  • Figure 1: Error for the Dirichlet problem (left) and Neumann problem (right).
  • Figure 2: Error plots for Case 1 (top left), Case 2 (top right), Case 3 (bottom left) and Case 4 (bottom right).
  • Figure 3: Error plots for the case with variable coefficients: the fourth order method (left) and the sixth order method (right).
  • Figure 4: Solution plots for the case with Ricker wavelet computed with $41^2$ grid points at time $t=0.1$ (top left), $t=0.5$ (top right), and $t=2$ (bottom left); error plot for the solutions at $t=0.5$ (bottom right).

Theorems & Definitions (10)

  • Definition 1: first derivative SBP identity
  • Definition 2: second derivative SBP identity
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3