2 research outputs found
A simplified Cauchy-Kowalewskaya procedure for the implicit solution of generalized Riemann problems of hyperbolic balance laws
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.Comment: 32 pages, 3 figure
A universal centred high-order method based on implicit Taylor series expansion with fast second order evolution of spatial derivatives
In this paper, a centred universal high-order finite volume method for
solving hyperbolic balance laws is presented. The scheme belongs to the family
of ADER methods where the Generalized Riemann Problems (GRP) is a building
block. The solution to these problems is carried through an implicit Taylor
series expansion, which allows the scheme to works very well for stiff source
terms. A von Neumann stability analysis is carried out to investigate the range
of CFL values for which stability and accuracy are balanced. The scheme
implements a centred, low dissipation approach for dealing with the advective
part of the system which profits from small CFL values. Numerical tests
demonstrate that the present scheme can solve, efficiently, hyperbolic balance
laws in both conservative and non-conservative form as well. An empirical
convergence rate assessment shows that the expected theoretical orders of
accuracy are achieved up to the fifth order