Convergence analysis of some tent-based schemes for linear hyperbolic systems

TL;DR

This paper analyzes the convergence and stability of tent-based finite element schemes for symmetric linear hyperbolic systems, providing error bounds and techniques for fully discrete scheme design on unstructured spacetime fronts.

Contribution

It introduces new convergence results and error bounds for mapped tent pitching schemes using discontinuous Galerkin methods, advancing numerical analysis for hyperbolic systems.

Findings

Convergence results for tent-based schemes established.

Error bounds for mapped tent discretizations derived.

Stability analysis on spacetime fronts demonstrated.

Abstract

Finite element methods for symmetric linear hyperbolic systems using unstructured advancing fronts (satisfying a causality condition) are considered in this work. Convergence results and error bounds are obtained for mapped tent pitching schemes made with standard discontinuous Galerkin discretizations for spatial approximation on mapped tents. Techniques to study semidiscretization on mapped tents, design fully discrete schemes, prove local error bounds, prove stability on spacetime fronts, and bound error propagated through unstructured layers are developed.