Approximate evolution operators for the Active Flux method

TL;DR

This paper develops approximate evolution operators for the Active Flux method, enabling structure-preserving solutions of nonlinear hyperbolic conservation laws where exact operators are unavailable.

Contribution

It introduces strategies to design accurate approximate evolution operators, extending the Active Flux method to nonlinear PDEs and balance laws.

Findings

Approximate evolution operators can preserve structure in nonlinear problems.

Active Flux method extended to nonlinear PDEs with approximate operators.

The approach maintains well-balanced properties for complex conservation laws.

Abstract

This work focuses on the numerical solution of hyperbolic conservations laws (possibly endowed with a source term) using the Active Flux method. This method is an extension of the finite volume method. Instead of solving a Riemann Problem, the Active Flux method uses actively evolved point values along the cell boundary in order to compute the numerical flux. Early applications of the method were linear equations with an available exact solution operator, and Active Flux was shown to be structure preserving in such cases. For nonlinear PDEs or balance laws, exact evolution operators generally are unavailable. Here, strategies are shown how sufficiently accurate approximate evolution operators can be designed which allow to make Active Flux structure preserving / well-balanced for nonlinear problems.