  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.
