High-order Wave Propagation Algorithms for Hyperbolic Systems

TL;DR

This paper introduces a high-order finite volume method for hyperbolic PDEs that effectively handles nonconservative systems, spatial variations, and near-equilibrium solutions using advanced reconstruction and Riemann solvers.

Contribution

The paper develops a versatile high-order method employing wave-based reconstruction and Riemann solvers, extending applicability to complex hyperbolic systems with nonconservative terms.

Findings

Method achieves high-order accuracy and well-balanced solutions.

Numerical examples demonstrate effectiveness on complex problems.

Approach is adaptable to various hyperbolic PDEs.

Abstract

We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving Riemann problems and calculating fluctuations (not fluxes). The implementation employs weighted essentially non-oscillatory reconstruction in space and strong stability preserving Runge-Kutta integration in time. The method can be extended to arbitrarily high order of accuracy and allows a well-balanced implementation for capturing solutions of balance laws near steady state. This well-balancing is achieved through the $f$-wave Riemann solver and a novel wave-slope WENO reconstruction procedure. The wide applicability and advantageous properties of the method are demonstrated through numerical examples, including problems in nonconservative form, problems with spatially varying fluxes, and problems involving near-equilibrium solutions of balance laws.