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

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.


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


Zbl 0845.65051
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