A modified fractional step method for the accurate approximation of detonation waves (Q2706446)

A modified fractional step method that overcomes numerical difficulties, which are caused by different time scales of the transport part and the reactive part of the model equations for combustion processes is considered. The method allows the use of a mesh width and time step determined by the nonreactive part, without precisely resolving the very small reaction zone. NEWLINENEWLINENEWLINEThe modified fractional step scheme needs information about the structure of the Riemann solution in order to determine the mesh cells over which the leading shock of a detonation wave is smeared. Furthermore, the distance between the shock and contact discontinuity in these mesh cells is required. This information will automatically be provided by an exact Riemann solver. The distance between the shock and contact discontinuity determines the part of a mesh cell where the source term will be applied in a stiff calculation. NEWLINENEWLINENEWLINEFor 2D calculations this area depends on all Riemann problems which have an influence to the mesh cell, including the Riemann problems which are solved in order to obtain the fluxes in the transverse direction. However, this further information can be obtained with less effort than the calculation of the change of the cell average due to the solution of the homogeneous problem. Moreover, the modified treatment of the source term is only necessary in those mesh cells which approximate the smeared-out leading shock, e.g., in about three mesh cells for the calculation of the 1D example. Therefore, the modification requires only slightly more effort than the classical fractional step scheme and permits the use of much coarser underresolved grids. NEWLINENEWLINENEWLINEHigh-resolution Godunov methods are employed and the structure of the Riemann solution is used to determine where burning should occur in each time step. Numerical results for 1D and 2D detonation waves are shown, including a detonation wave diffracting around a corner. The modification is implemented in the software package CLAWPACK. NEWLINENEWLINENEWLINEThis approach could also give more insight into the numerical problems occurring when solving conservation laws with stiff source terms.