Stable finite difference methods for Kirchhoff-Love plates on overlapping grids

TL;DR

This paper develops stable, high-order finite difference algorithms with hyper-dissipation for solving Kirchhoff-Love plate equations on complex overlapping grids, ensuring accuracy and stability in challenging geometries.

Contribution

It introduces four novel algorithms incorporating hyper-dissipation for stable time integration on curvilinear overlapping grids, with stability criteria derived via Fourier analysis.

Findings

Algorithms achieve stable, accurate solutions on complex geometries.

Hyper-dissipation effectively stabilizes explicit and implicit schemes.

Numerical experiments confirm improved stability and efficiency.

Abstract

In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the resulted semi-discrete system. To address the instability, we propose to add a fourth-order hyper-dissipation to the spatially discretized system to stabilize its time integration; this additional dissipation term captures the essential upwinding effect of the original upwind scheme. The investigation of strategies for incorporating the upwind dissipation term into several time-stepping schemes (both explicit and implicit) leads to the development of four novel algorithms. For each algorithm, formulas for determining a stable time step and a sufficient dissipation coefficient on curvilinear grids are derived by performing a local Fourier analysis. Quadratic eigenvalue problems for a simplified model plate in 1D domain are considered to reveal the weak instability due to the presence of interpolating equations in the spatial discretization. This model problem is further investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments are carried out to validate the accuracy and stability of the proposed algorithms, followed by two benchmark problems to demonstrate the capability and efficiency of our approach for solving realistic applications. Results that concern the performance of the proposed algorithms are also presented.