# A simplified Cauchy-Kowalewskaya procedure for the implicit solution of generalized Riemann problems of hyperbolic balance laws

## TL;DR

This paper introduces a simplified, recursive Cauchy-Kowalewskaya procedure for implicit Taylor series solutions of hyperbolic balance laws, improving computational efficiency and maintaining accuracy in generalized Riemann problem solvers.

## Contribution

A new recursive formula for the CK procedure simplifies implementation and enhances efficiency in solving hyperbolic balance laws within the GRP framework.

## Key findings

- The proposed method achieves expected theoretical orders of accuracy.
- It improves computational efficiency in solving GRPs.
- The scheme maintains stability while enhancing performance.

## Abstract

The Cauchy-Kowalewskaya (CK) procedure is a key building block in the design of solvers for the Generalised Rieman Problem (GRP) based on Taylor series expansions in time. The CK procedure allows us to express time derivatives in terms of purely space derivatives. This is a very cumbersome procedure, which often requires the use of software manipulators. In this paper, a simplification of the CK procedure is proposed in the context of implicit Taylor series expansion for GRP, for hyperbolic balance laws in the framework of [Journal of Computational Physics 303 (2015) 146-172]. A recursive formula for the CK procedure, which is straightforwardly implemented in computational codes, is obtained. The proposed GRP solver is used in the context of the ADER approach and several one-dimensional problems are solved to demonstrate the applicability and efficiency of the present scheme. An enhancement in terms of efficiency, is obtained. Furthermore, the expected theoretical orders of accuracy are achieved, conciliating accuracy and stability.