Numerical modeling of a wave in a nonlinear medium with dissipation

The article is devoted to the numerical simulation of a nonlinear wave in a dissipative nondispersive medium described by the Burgers equation. Numerical solution of the Burgers equation at large values of viscosity runs into serious difficulties. They are mainly associated with the presence of a small parameter at the highest derivative and, as a consequence, the appearance of regions of strong spatial inhomogeneity in the solution. Then the requirements imposed on the approximation properties of numerical methods sharply increase. In this work, the indicated difficulties are overcome and it is proposed to apply the spectral-grid method. For this, in the considered interval of integration, a grid is introduced, the requirements of the continuity of the solution itself and its first derivative are imposed at the inner nodes of the grid, and the corresponding boundary conditions are satisfied at the boundary nodes of the grid. The transition from one time layer to another is carried out according to the Adams-Beshfort scheme. A computational experiment was carried out in large-scale changes in characteristic parameters such as: viscosity, the number of Chebyshev polynomials and mesh elements, for different values of time and mesh steps in time. The calculation results show the high accuracy and efficiency of the spectral-grid method in solving the initial-boundary value problem for the Burgers equation.