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Unsplit algorithms for multidimensional systems of hyperbolic conservation laws with source terms. (English) Zbl 0999.65089
Summary: This work describes an unsplit, second-order accurate algorithm for multidimensional systems of hyperbolic conservation laws with source terms, such as the compressible Euler equations for reacting flows. It is a MUSCL-type, shock-capturing scheme that integrates all terms of the governing equations simultaneously, in a single time-step, thus avoiding dimensional or time-splitting. Appropriate families of space-time manifolds are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding 1-D homogeneous system. The local geometry, of these manifolds depends on the source terms and the spatial derivatives of the flow variables.
Numerical integration of the characteristic equations is performed along these manifolds in the upwinding part of the algorithm. Numerical simulations of two-dimensional detonations with simplified kinetics are performed to test the accuracy and robustness of the algorithm. These flows are unstable for a wide range of parameters and may exhibit chaotic behavior. Grid-convergence studies and comparisons with earlier results, obtained with traditional schemes, are presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics, general
76L05 Shock waves and blast waves in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
80A25 Combustion
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
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