High‐accuracy difference schemes for the nonlinear transfer equation

Abstract

In the present paper, for the initial boundary value problem for the non‐homogeneous nonlinear transport equation

the basic principles for constructing difference schemes of any order of accuracy O(#GT^M), M ≥ 1, on characteristic grids with the minimal stencil were introduced. To construct a difference scheme the Steklov averaging idea for the right‐hand side

was used. The case of f(u) = λu² was investigated in detail. A strict analysis of the order of approximation, stability, and convergence in nonlinear case was made. The performed numerical experiments justify theoretical results.

Key words:

• High accuracy difference scheme
• nonlinear transport equation
• method of characteristics