THE SOLUTION TO THE EQUATION OF THE SHOCK WAVE BURGERS IN THE PTC MATHCAD ENVIRONMENT

Abstract

The system of equations of ordinary differential equations is integrated by means of the Euler computational scheme, and the step size is maintained small enough that the errors associated with numerical integration are negligible in comparison with the approximation error. It is found that the mean square error decreases rapidly with increasing order of the approximate solution and the trial solution converges to the exact solution. The main source of error is that it is difficult for the approximate solution to keep track of the rapid changes determined by the exact solution for the shock wave. However, this error is detected through the values of the nonlinear term included in the differential equation, which is determined according to the binomial representation through the quadratic forms of the matrix of the trial function. These trends allow us to obtain an approximate solution from a limited number of members of the trial 92 function sufficient for accurate simulation of the solution in a wide range of Reynolds numbers characterizing the ratio of nonlinear and dissipative terms in the Navier - Stokes equation. The ability of this approximate solution to identify the steep profile of the shock wave becomes obvious, since irregular oscillations arise in the solution that do not have a physical justification. The simplest way to deal with this would be to increase the number of members in the approximate solution, which leads to a slight increase in the machine time of the account. The module of numerical calculation intended for the solution of the burgers equation by means of the traditional Galerkin method in the environment of engineering calculations PTC MathCAD is given. The method of solving the burgers equation by Galerkin method applied to the problem ofpropagating shock wave is considered. The solution is based on the reduction of a parabolic partial differential equation to a system of ordinary time-dependent differential equations, taking into account the nonlinear term of the differential equation.