Piecewise parabolic method on local stencil for gasdynamic simulations

A numerical method based on piecewise parabolic difference approximations is proposed for solving hyperbolic systems of equations. The design of its numerical scheme is based on the conservation of Riemann invariants along the characteristic curves of a system of equations, which makes it possible to discard the four-point interpolation procedure used in the standard piecewise parabolic method (PPM) and to use the data from the previous time level in the reconstruction of the solution inside difference cells. As a result, discontinuous solutions can be accurately represented without adding excessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new method is compared with PPM by solving test problems for the linear advection equation and the inviscid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. A technique for solving the gas dynamics equations is described and tested for several one-and two-dimensional problems.