Method of dynamical adaption for evolution-type problems with high gradients

The method of the dynamic adaption of computational grids intended for evolution type problems of the mathematical physics, which are difficultly solved by means of traditional methods, is considered. The basic of the method is the idea of transition to arbitrary moving coordinate system. It is shown, that the free parameters of the transformation function can be determined from the condition of processes being quasistationary in the new coordinate system. The application of the method to the solution of the nonlinear. Burgers equation is considered. The large gradients of the solution appear when small value of physical viscosity $\mu=10^{-4}$ is used, leading to the unremovable oscillations. As it is shown by means of mathematical modelling, for this problem the fix grid solution has oscillations even when the grid nodes number is $N=1000\div1500$. The application of the dynamically adapting grids make possible to get the solution wihout osculation using grid with overall nodes number $N=15\div20$.