A generalized Riemann problem-based compact reconstruction method for finite volume schemes

TL;DR

This paper introduces a high-order finite volume reconstruction method based on generalized Riemann problems, achieving compact stencils and stable CFL conditions, with promising efficiency demonstrated through tests on hyperbolic PDEs.

Contribution

The paper proposes a novel GRP-based reconstruction method (GRPrec) that combines compactness with stability and efficiency, outperforming traditional schemes in certain metrics.

Findings

GRPrec achieves high accuracy for smooth and discontinuous solutions.

The method maintains stability under a CFL condition independent of order.

Numerical results show competitive error and CPU cost compared to existing schemes.

Abstract

We present a Generalized Riemann Problem-based reconstruction method (GRPrec) for high-order finite volume schemes applied to hyperbolic partial differential equations. The method constructs spatial polynomials using cell averages at the current time level and GRP solution data from the previous time level. The resulting GRPrec stencil is as compact as that of discontinuous Galerkin (DG) schemes but unlike DG, our finite volume schemes obey a generous CFL stability condition that is independent of the order of accuracy. We assess the method's performance through test problems for smooth and discontinuous solutions of the linear advection equation and the Euler equations of gas dynamics in one space dimension. Results are compared against exact solutions and against numerical results from well-known spatial reconstruction finite volume and DG schemes, with all methods implemented in the fully discrete ADER framework. The performance of GRPrec is very promising, especially in terms of efficiency, that is error against CPU cost.