Finite volume schemes on Lorentzian manifolds. (English) Zbl 1179.35027

Amorim, Paulo; LeFloch, Philippe G.; Okutmustur, Baver

Commun. Math. Sci. 6, No. 4, 1059-1086 (2008).

The goal of a present paper is to show strong convergence of a class of monotone first-order finite volume schemes to the weak entropy solution of the multidimensional scalar conservation laws posed on globally hyperbolic Lorentzian manifolds. The class of spacetime triangulations is quite general and the elements may become degenerate in the spatial direction. The authors follow the strategy originaly developed by B. Cockburn, F. Coquel and P. G. LeFloch [SIAM J. Numer. Anal. 32, No. 3, 687–705 (1995; Zbl 0845.65051)] for conservation laws posed on a fixed (time-independent) Euclidean background. In the present paper, however, new estimates are required to take into account the geometric effects, especially those arising from time evolution of the scheme. The proof is based on the local entropy estimates and global-in-space entropy inequality. Further, the authors establish the \(L^\infty\) stability of the scheme and global entropy inequality. Strong convergence towards an entropy solution follows from DiPerna’s uniqueness theorem.

Reviewer: Mária Lukáčová (Hamburg)

MSC:

35A35	Theoretical approximation in context of PDEs
35L65	Hyperbolic conservation laws
76L05	Shock waves and blast waves in fluid mechanics
65M08	Finite volume methods for initial value and initial-boundary value problems involving PDEs

Keywords:

entropy condition; measure-valued solutions; convergence analysis; geometric effects; local entropy estimates

Citations:

Zbl 0845.65051