ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years, including both methodological aspects as well as very ambitious applications. In spite of methodological progress, the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem (GRP) remain the centre of attention in the ADER approach. In the original formulation of ADER schemes, the proposed solution procedure for the GRP was based on (i) Taylor series expansion of the solution in time right at the element interface, (ii) subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives, and (iii) solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion. For realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable. In this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER schemes. This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface. Still relying on a time Taylor series expansion of the solution at the interface, the time derivatives are then easily calculated from the time-reconstruction polynomial. The resulting schemes are called ADER-TR. A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method, thus deducing linear stability regions. Also, via careful numerical experiments, we deduce stability regions for the corresponding non-linear schemes. Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods. This paper is restricted to the one-dimensional scalar case with source term, but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.