A simplified Cauchy-Kowalewskaya procedure for the local implicit solution of generalized Riemann problems of hyperbolic balance laws. (30th April 2020)

• Record Type: Journal Article
• Authors: Montecinos, Gino I.; Balsara, Dinshaw S.
• Is Part Of: Computers & fluids. Volume 202(2020)
• Publication Date: 2020-04-30
• Subjects: Finite volume schemes -- ADER schemes -- Generalized Riemann Problems -- stiff source terms; Fluid dynamics -- Data processing -- Periodicals; 532.050285
• DOI: 10.1016/j.compfluid.2020.104490
• Languages: English
• ISSNs: 0045-7930

Abstract

Highlights: GRP solutions obtained from algebraic equations derived from implicit Taylor series. A closed form for the Cauchy-Kowalewskaya functionals for hyperbolic balance laws. The Cauchy-Kowalewsky procedure requires also the derivatives of Jacobian matrices. Derivatives approximated from interpolations on a suitable set of space-time nodes. Performance of ADER schemes with implicit series, using this approach, is improved. Abstract: The Cauchy-Kowalewskaya (CK) procedure is a key building block in the design of solvers for the Generalised Riemann 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.