@misc{Yang, author = {Yang, Shuli}, title = {MmB schemes on regular triangular meshes for 2-D conservation laws.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-873}, number = {SC-92-17}, abstract = {In this paper, two classes of second order accurate high resolution schemes are presented on regular triangular meshes for initial value problem of two dimensional conservation laws. The first class are called Runge-Kutta-FVM MmB (locally Maximum- minimum Bounds preserving) schemes, which are first discretized by (FVM) finite volume method in space direction and modifying numerical fluxes, and then by Runge-Kutta methods in time direction; The second class, constructed by Taylor expansion in time, and then by FVM methods and making modifications to fluxes, are called Taylor- FVM MmB schemes. MmB properties of both schemes are proved for 2-D scalar conservation law. Numerical results are given for Riemann problems of 2-D scalar conservation law and 2-D gas dynamics systems and some comparisons are made between the two classes of the schemes. Key words and phrases: MmB schemes, 2-D, conservation laws, gas dynamics systems, Runge-Kutta-FVM, Taylor-FVM.}, language = {en} } @misc{Yang, author = {Yang, Shuli}, title = {Numerical Riemann solutions in multi-pieces for 2-D gas dynamics systems I. Contact discontinuities.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-903}, number = {SC-92-20}, abstract = {The numerical solutions of Riemann problems in three, four, five and six pieces, which only contain contact discontinuities, are presented by using Taylor FVM MmB schemes on regular triangular meshes for 2-D gas dynamics systems. The 2-D Riemann initial data are as defined in [1], under the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves , or contact discontinuities. The main ends of the paper are that spirals will be shown for some configurations and the relations of the solutions between different distibutions of Riemann initial data are explained by the numerical solutions of modified Riemann problems. Key words and phrases: Riemann problem, gas dynamics systems, spiral, MmB schemes.}, language = {en} } @misc{HuamoShuli, author = {Huamo, Wu and Shuli, Yang}, title = {MmB-A New Class of Accurate High Resolution Schemes for Conservation Laws in Two Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-248}, number = {SC-89-06}, abstract = {In this paper we present the MmB schemes, which preserve the local maximum and minimum bounds of the initial data in the smallest union of mesh elements of previous time step containing the domain of dependence of the solution on the mesh element with center at point \$ P \$\ under consideration. In 1-D, the MmB schemes are almost identical with TVD schemes. As well-known, there is no second-order TVD scheme in 2-D, nevertheless, we present here two classes of 2-D second-order accurate MmB-schemes. It is proved that 1-D discrete MmB (or TVD) and 1-D semi-discrete TVD schemes may have second-order accuracy at (nonsonic) critical points, but cannot be of uniformly second-order accurate in the whole neighborhood of the critical points. New accurate high resolution flux limiters are suggested. Numerical results for 1-D and 2-D test problems are given. {\bf Keywords:} Difference scheme, TVD, MmB, flux limiter.}, language = {en} }