[Paper Review] Fourth order finite difference methods for the wave equation with mesh refinement interfaces

Siyang Wang, N. Anders Petersson|arXiv (Cornell University)|Sep 12, 2018

Numerical methods in engineering31 references3 citations

TL;DR

This paper presents a fourth-order accurate finite difference method for the wave equation with mesh refinement interfaces using summation-by-parts (SBP) operators. By exploiting a novel relationship between SBP operators with and without ghost points, the authors develop a new SBP-GP method that uses ghost points on only one side of the interface, reducing system size and improving conditioning while maintaining fourth-order accuracy and energy stability. The method allows for larger time steps compared to penalty-based approaches.

ABSTRACT

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

Motivation & Objective

• To develop a high-order accurate, energy-stable finite difference method for the wave equation on non-uniform Cartesian grids with hanging nodes.
• To reduce computational cost in ghost point updates by using ghost points from only one side of the mesh refinement interface.
• To establish a theoretical connection between SBP operators with and without ghost points for improved discretization design.
• To achieve fourth-order convergence for both smooth and discontinuous material properties.
• To compare the efficiency and stability of the new SBP-GP method with the traditional SBP-SAT (penalty) method.

Proposed method

• The method uses a fourth-order accurate SBP finite difference discretization with summation-by-parts properties to ensure energy stability.
• A new SBP-GP formulation is developed that uses ghost points only on the coarse side of the refinement interface, reducing the number of unknowns in the linear system.
• The approach combines SBP operators with and without ghost points via a previously unexplored mathematical relation, enabling one-sided ghost point treatment.
• Interface conditions are enforced strongly using interpolation and restriction stencils that maintain fourth-order accuracy.
• The method is extended to two dimensions and validated using a second-order wave equation in terms of acoustic pressure.
• A discrete energy estimate is derived to prove stability, and the time-step restriction is shown to be close to the von Neumann limit for periodic problems.

Experimental results

Research questions

• RQ1 Can a fourth-order accurate SBP finite difference method be constructed for wave equations with mesh refinement interfaces using only one-sided ghost points?
• RQ2 What is the relationship between SBP operators that use ghost points and those that do not, and how can it be exploited for improved accuracy and efficiency?
• RQ3 How does the stability and convergence rate of the new SBP-GP method compare to the SBP-SAT (penalty) method for both smooth and discontinuous material properties?
• RQ4 Does the new method maintain a large time-step restriction, close to the von Neumann limit, for non-periodic problems with hanging nodes?
• RQ5 Can the proposed method be extended to higher-order accuracy and three-dimensional problems with realistic topography?

Key findings

• The proposed SBP-GP method achieves fourth-order convergence for both smooth and discontinuous material properties, as confirmed by numerical experiments.
• The method leads to a smaller and better-conditioned linear system for ghost point updates compared to two-sided ghost point approaches.
• The time-step restriction for the new SBP-GP method is very close to the von Neumann limit for the corresponding periodic problem, enabling efficient time integration.
• The SBP-SAT method, in contrast, requires a smaller time step that depends on penalty parameters and exhibits only third-order convergence unless modified.
• The improved SBP-SAT method, with two proposed remedies, achieves fourth-order convergence, but still shows larger solution errors than the SBP-GP method for the same grid and time step.
• The new SBP-GP method is energy stable and maintains high-order accuracy even with discontinuous material interfaces when subdomains are aligned with discontinuities.