The comparison of accuracy and convergence for the CSPH—TVD method and some Eulerian schemes for solving gas-dynamic equations

In this article the results of the study of accuracy and convergence methods CSPH—TVD, MUSCL, PPM and WENO for solving equations of ideal gas dynamics was present.
Many physical processes are described by the equations of gas dynamics. Because of their non-linearity, exact or approximate analytical solutions can be obtained only for a limited number of special cases. Therefore, numerical methods are used for gas-dynamic problems usually.
One of these methods is the numerical scheme CSPH—TVD (Combined Smooth Particle Hydrodynamics—Total Variation Diminishing). This method is based on the consistent application of the Lagrange (SPH) and Euler (TVD) approaches.
Our results show that the relative error and orders the convergence of all the above numerical schemes are quite close. This can be due to the presence of a weak discontinuity in the structure of the solution. There are many solutions in gas dynamics problems with such discontinuities. The presence of a weak discontinuity allows us to understand the properties of numerical schemes in a real (not test) computations.
Our results shows that CSPH—TVD method has comparable to popular Euler numerical schemes accuracy and convergence. A distinctive feature of this method is the possibility of computations at the boundary with vacuum (or dry bed for shallow water case). The additional regularization is not needed. The method is successfully used for solving various gas-dynamic problems in a variety of subject areas: the dynamics of surface water, aspiration flows, astrophysical jets, accretion flows, the transfer of contaminants and others.