Dynamically adapted grids for interacting discontinuous solutions

Abstract

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is consid- ered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approx- imation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuni- form grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adapta- tion method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid con- sisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the dif- fusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking. The dynamic adaptation methods (13-21) are less complicated and make it possible to generate grids with a constant (19) or variable (18) number of grid points. In these methods, the control of the distribution of the grid points is achieved by using information about the dynamic behavior of the solution to be found. This makes it possible to concentrate a large number of grid points in the regions where the solution under- goes sharp changes (13). A close relationship between the dynamic behavior of the solution and the location of the grid points requires that the coordinates of the grid points be recalculated at each time layer. This fact places high requirements upon the consistency of the solution dynamics with the grid points motion. For this reason, the algorithms that do not use any fitting parameters have certain advantages (15). These fea- tures are especially pronounced in the generation of adaptive grids for unsteady gas dynamics problems (see (16-22)), which describe rapidly changing processes. The solution of hyperbolic differential equations,