An asymptotic preserving method for linear systems of balance laws based on Galerkin's method

TL;DR

This paper introduces an Asymptotic Preserving (AP) finite element method for linear systems of balance laws, demonstrating its advantages over traditional methods through theoretical analysis and numerical experiments.

Contribution

The paper develops a novel AP scheme based on Galerkin's method for linear balance laws, replacing finite differences with finite elements, and provides a comprehensive analysis and comparison.

Findings

The AP method is consistent and stable.

Numerical results show the AP method outperforms Implicit Euler.

The approach effectively handles the asymptotic limit.

Abstract

We apply the concept of Asymptotic Preserving (AP) schemes to the linearized p-system and discretize the resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. The fully discrete method is analyzed with respect to consistency, and we compare it numerically with more traditional methods such as Implicit Euler's method. Numerical results indicate that the AP method is indeed superior to more traditional methods.