Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation

Abstract

The computer modeling system for numerical solution of the nonlinear one-dimensional non-stationary Burgers’ equation is described. The numerical solution of the Burgers’ equation is obtained by a meshless scheme using the method of partial solutions and radial basis functions. Time discretization of the one-dimensional Burgers’ equation is obtained by the generalized trapezoidal method (θ-scheme). The inverse multiquadric function is used as radial basis functions in the computer modeling system. The computer modeling system allows setting the initial conditions and boundary conditions as well as setting the source function as a coordinate- and time-dependent function for solving partial differential equation. A computer modeling system allows setting such parameters as the domain of the boundary-value problem, number of interpolation nodes, the time interval of non-stationary boundary-value problem, the time step size, the shape parameter of the radial basis function, and coefficients in the Burgers’ equation. The solution of the nonlinear one-dimensional non-stationary Burgers’ equation is visualized as a three-dimensional surface plot in the computer modeling system. The computer modeling system allows visualizing the solution of the boundary-value problem at chosen time steps as three-dimensional plots. The computational effectiveness of the computer modeling system is demonstrated by solving two benchmark problems. For solved benchmark problems, the average relative error, the average absolute error, and the maximum error have been calculated.