Unsplit algorithms for multidimensional systems of hyperbolic conservation laws with source terms.

*(English)*Zbl 0999.65089Summary: 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.

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 |

##### Keywords:

numerical examples; unsplit schemes; detonations; systems of hyperbolic conservation laws; compressible Euler equations; reacting flows; MUSCL-type, shock-capturing scheme; characteristic equations; algorithm
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\textit{M. V. Papalexandris} et al., Comput. Math. Appl. 44, No. 1--2, 25--49 (2002; Zbl 0999.65089)

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