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

TL;DR

The paper tackles efficiently solving the wave equation on overset grids by developing high-order implicit-time-stepping methods based on the modified equation (ME) approach. It introduces second- and fourth-order IME schemes and two upwind-dissipation strategies, including a monolithic and a predictor-corrector formulation, and then couples them in a Spatially Partitioned Implicit-Explicit (SPIE) framework to handle geometrically stiff regions. The authors establish unconditional stability conditions via von Neumann and GKS analyses, provide a detailed implementation for curvilinear overset grids, and demonstrate substantial accuracy, stability, and computational speedups through extensive 2D/3D numerical experiments, including disk, cylinder, and knife-edge geometries, as well as WaveHoltz-related Helmholtz considerations. The results indicate that SPIE schemes can outperform fully explicit approaches by orders of magnitude in stiff regions while maintaining comparable accuracy, making them practical for complex wave-propagation problems in acoustics and electromagnetics. Overall, the work delivers a robust, scalable framework for high-order wave propagation on overset grids with broad applicability to physics-based simulations.

Abstract

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. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations.

Figures (23)

• Figure 1: Geometrically stiff problem: scattering of a modulated Gaussian plane wave from a knife edge. Left: overset grid for the geometry showing magnified views of the tip grid which has very small grid cells. Right: contours of $|u|$ computed with the new SPIE scheme; the tip grid was advanced implicitly while other grids were advanced explicitly resulting in a time-step that was about $20$ times larger than using an explicit scheme on all grids.
• Figure 2: One-dimensional overset grid used to assess the stability of the SPIE scheme. The explicit scheme is used on the left grid and the implicit scheme is used on the right. Interpolation points are marked as circles.
• Figure 3: Left: composite of a background grid ($G_1$, blue) and a boundary-fitted grid ($G_2$, green) in physical space for the domain defined by the interior of the red boundary. The grid points on $G_1$ with green dots interpolate from $G_2$ and the grid points on $G_2$ with blue dots interpolate from $G_1.$ Middle: Plot of $G_1$ showing interpolation points, ghost points (grid points which exist outside the physical boundary), and unused points (grid points which do not affect the computation). Right: The green boundary fitted grid, $G_2,$ is mapped to a unit square. The plot shows interpolation points and ghost points.
• Figure 4: One-dimensional overset grid used for the matrix stability analyses. Interpolation points are marked as circles, ghost points are marked as squares.
• Figure 5: Example stable and unstable cases for the SPIE2-UW-PC scheme with $\gamma=0.3$. Middle left: amplification factors $a$ for the stable case corresponding to the grid on the top, grid-ratio $\delta=.5$. Middle right: amplification factors $a$ for the unstable case corresponding to the grid on the bottom, grid-ratio $\delta=1.55$.

Theorems & Definitions (16)

• Definition 1: Stability
• Theorem 1: IME2 Stability
• Theorem 2: IME4 Stability
• Theorem 3
• Theorem 4
• Lemma 5.1
• Theorem 5
• proof
• Theorem : IME2 Stability
• proof