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.

中文翻译：

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双曲平衡律广义黎曼问题局部隐式解的简化 Cauchy-Kowalewskaya 程序

Cauchy-Kowalewskaya (CK) 过程是设计基于泰勒级数时间展开式的广义黎曼问题 (GRP) 求解器的关键构建模块。CK 程序允许我们用纯空间导数来表达时间导数。这是一个非常繁琐的过程，往往需要使用软件操纵器。在本文中，在 [Journal of Computational Physics 303 (2015) 146-172] 的框架内，在 GRP 的隐式泰勒级数展开的背景下，针对双曲平衡定律，提出了 CK 程序的简化。获得了 CK 过程的递归公式，该公式直接在计算代码中实现。建议的 GRP 求解器用于 ADER 方法的上下文中，并且解决了几个一维问题以证明本方案的适用性和效率。获得了效率方面的提高。此外，达到了预期的理论精度等级，兼顾了精度和稳定性。