Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion

TL;DR

This paper introduces a spline-based discontinuous Galerkin method for wave propagation that enables explicit time-stepping, efficient mass matrix inversion, and improved stability and accuracy through optimal knot vectors.

Contribution

It develops a multi-patch discontinuous Galerkin isogeometric analysis framework with weight-adjusted mass matrix inversion and optimal knot vector selection for efficient wave simulation.

Findings

Mass matrices are efficiently invertible with high accuracy.

Spline spaces allow larger stable timesteps than traditional finite elements.

Optimal knot vectors enhance approximation and stability.

Abstract

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods.