A universal centred high-order method based on implicit Taylor series expansion with fast second order evolution of spatial derivatives

TL;DR

This paper introduces a universal high-order finite volume method for hyperbolic balance laws, utilizing implicit Taylor series expansion within an ADER framework, effective for stiff source terms and achieving high accuracy.

Contribution

It presents a novel centred high-order scheme based on implicit Taylor series expansion, enhancing stability and accuracy for hyperbolic balance laws with stiff source terms.

Findings

Achieves expected theoretical orders of accuracy up to fifth order.

Demonstrates stability for a range of CFL values through von Neumann analysis.

Efficiently solves both conservative and non-conservative hyperbolic systems.

Abstract

In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The solution to these problems is carried through an implicit Taylor series expansion, which allows the scheme to works very well for stiff source terms. A von Neumann stability analysis is carried out to investigate the range of CFL values for which stability and accuracy are balanced. The scheme implements a centred, low dissipation approach for dealing with the advective part of the system which profits from small CFL values. Numerical tests demonstrate that the present scheme can solve, efficiently, hyperbolic balance laws in both conservative and non-conservative form as well. An empirical convergence rate assessment shows that the expected theoretical orders of accuracy are achieved up to the fifth order.