High-Order Accurate Implicit-Explicit Time-Stepping Schemes for Wave Equations on Overset Grids

Core Concepts

New implicit and implicit-explicit time-stepping methods are developed for efficiently solving the wave equation in second-order form on overset grids, with applications to two and three-dimensional problems.

Abstract

The article presents new implicit and implicit-explicit time-stepping schemes for solving the wave equation in second-order form on overset grids. The key highlights are:

Implicit modified equation (IME) schemes are developed for second and fourth-order accurate time-stepping, which are single step, three levels in time, and based on the modified equation approach. These IME schemes incorporate upwind dissipation for stability on overset grids.

The fully implicit IME schemes are useful for applications like the WaveHoltz algorithm for solving Helmholtz problems, where very large time-steps are desired.

For geometrically stiff problems with localized regions of small grid cells, the implicit IME scheme is combined with an explicit scheme (EME) in a spatially partitioned implicit-explicit (SPIE) scheme. The IME scheme is used on component grids with small cells, while the EME scheme is used on other grids like background Cartesian grids. This partitioned SPIE scheme can be significantly faster than using an explicit scheme everywhere.

Stability analyses, including von Neumann and matrix stability analyses, are performed to establish the stability properties of the proposed IME and SPIE schemes. Numerical results demonstrate the accuracy and efficiency of the new schemes for various wave propagation problems on overset grids.

Stats

The wave speed is denoted by c > 0.
The time-step is denoted by Δt.
The grid spacing in the d-th coordinate direction is denoted by hd.
The CFL parameter in the d-th coordinate direction is defined as λxd = c Δt / hd.

Quotes

"New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids."
"The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids."
"The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired."

Key Insights Distilled From

High-order Accurate Implicit-Explicit Time-Stepping Schemes for Wave Equations on Overset Grids

by Allison M. C... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14592.pdf

Deeper Inquiries

How can the proposed implicit-explicit time-stepping schemes be extended to other wave propagation problems beyond the wave equation, such as Maxwell's equations or elasticity

The proposed implicit-explicit time-stepping schemes can be extended to other wave propagation problems beyond the wave equation by adapting the discretization and formulation to the specific equations governing the physical phenomena. For example, to apply the schemes to Maxwell's equations for electromagnetics, the wave operator in the schemes can be replaced with the appropriate differential operators for the electric and magnetic fields. The spatial approximations and boundary conditions would need to be tailored to the vector nature of the electromagnetic fields. Similarly, for elasticity problems, the wave equation would be replaced with the equations governing the stress and strain fields, and the spatial discretizations would be adjusted accordingly. By customizing the schemes to the specific equations and physics involved, the implicit-explicit time-stepping methods can be effectively applied to a wide range of wave propagation problems.

What are the potential limitations or challenges in applying the SPIE scheme to highly heterogeneous or anisotropic media, where the wave speeds vary significantly across the computational domain

Applying the SPIE scheme to highly heterogeneous or anisotropic media where wave speeds vary significantly across the computational domain can present challenges related to stability and accuracy. In such cases, the interpolation between explicit and implicit grids may need to be carefully managed to ensure smooth transitions and accurate representation of the varying wave speeds. The choice of interpolation methods and the handling of boundary conditions at the interfaces between different media types become crucial in maintaining stability and accuracy. Additionally, the upwind dissipation used in the schemes may need to be adjusted to account for the heterogeneous nature of the media and prevent numerical instabilities. Proper calibration and tuning of the parameters in the SPIE scheme would be essential to address these challenges effectively.

Can the stability and accuracy of the IME and SPIE schemes be further improved by incorporating adaptive mesh refinement or other advanced numerical techniques

The stability and accuracy of the IME and SPIE schemes can be further improved by incorporating adaptive mesh refinement (AMR) or other advanced numerical techniques. AMR allows for the local refinement of grids in regions where high resolution is needed, such as near complex geometrical features or areas with rapid variations in wave speeds. By dynamically adjusting the grid resolution based on the local properties of the solution, AMR can enhance the accuracy of the schemes while optimizing computational resources. Additionally, techniques like higher-order spatial discretizations, advanced boundary treatments, and optimized upwind dissipation methods can also contribute to improving the overall performance of the IME and SPIE schemes. By integrating these advanced numerical techniques, the schemes can achieve higher levels of accuracy, stability, and efficiency in simulating wave propagation in complex scenarios.