Numerical-analytical method for the Burgers equation with a periodic boundary condition

We construct an efficient numerical-analytical method for solving the initial-boundary value problem for the Burgers equation on a segment with a periodic boundary condition. The method includes the reduction to a linear problem based on an explicit-implicit time discretization scheme and an analytical solution of an auxiliary linear problem at each time step using the explicit form of the corresponding Green's function. The efficiency of the constructed method is due to the fact that the algorithm for solving the auxiliary problem has only linear complexity in terms of the number of spatial discretization nodes used, without involving difference approximations of the derivatives of the desired function. On the basis of the Cole—Hopf substitution, we obtain an explicit periodic solution of the problem on the interval and compare the results of the numerical implementation of the constructed algorithm with this explicit solution. The developed method demonstrated a combination of high computational efficiency and accuracy of the result.