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

• Published in: Journal of computational physics 2021-10, Vol.443, p.110535, Article 110535
• Main Author: Montecinos, Gino I.
• Format: Article
• Language: English
• Subjects: ADER schemes; Computational physics; Empirical analysis; Evolution; Finite volume method; Generalized Riemann problems; Hyperbolic balance laws; Series expansion; Stability analysis; Taylor series; Universal scheme
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110535

•The scheme is and extension to the non-conservative case of Toro and Montecinos, JCOMP: 303:146-172, 2015.•A non-linear algebraic system is constructed, which allows an efficient fixed point iteration procedure.•The scheme works for both conservative and non-conservative problems as well.•The scheme uses the centred and low-dissipation scheme reported in JCOMP: 416:109545, 2020.•A strategy to carry out the von Neumann analysis for high order non-linear schemes is presented. 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 work very well for stiff source terms. The series expansion is high order for the state and requires the evolution in time of spatial derivatives. A Taylor expansion of second order based on a linearization around the resolved state, is proposed for evolving spatial derivatives. 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. The scheme is proven to be well-balanced, the C-property, a classical assessment of well-balancing of non-conservative schemes, is numerically demonstrated. An empirical convergence rate assessment shows that the expected theoretical orders of accuracy are achieved up to the fifth order.