Summary.
Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h 1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.
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Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15
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Dedner, A., Rohde, C. Numerical approximation of entropy solutions for hyperbolic integro-differential equations. Numer. Math. 97, 441–471 (2004). https://doi.org/10.1007/s00211-003-0502-9
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DOI: https://doi.org/10.1007/s00211-003-0502-9