Conservative bicharacteristic upwind schemes for hyperbolic conservation laws II

For two-dimensional hyperbolic conservation laws, the goal of constructing conservative bicharacteristic upwind (CBU) schemes has been achieved. They are direct two-dimensional upwind schemes which follow certain bicharacteristics and which are conservative. A particular explicit CBU scheme is presented; it is an extension of the authorʹs upwind scheme for one-dimensional hyperbolic conservation laws. Ideas of partial upwind scheme time stepping and conservation in the limit are discussed. The corresponding nonreflecting boundary schemes are also presented. From the numerical results of a one-dimensional Riemann problem of the Eulerʹs equations rotated in a two-dimensional region, we see that the CBU scheme is valid for calculation of discontinuities, and that the corresponding nonreflecting boundary schemes work well for discontinuities going out of the computational region. The resolution of shocks is quite satisfactory, especially for the present first order scheme, and the stability is not restrictive for suitably chosen bicharacteristics.