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

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 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.