Error analysis of approximate solutions of non-linear Burgers’ equation

摘要

This paper deals with application of the maximum principle for differential equations to the finite difference method for determining upper and lower approximate solutions of the non-linear Burgers’ equation and their error range. In term of mathematical architecture, the paper is based on the maximum principle for parabolic differential equations to establish monotonic residual relations of the Burgers’ equation; and in terms of numerical method, it applies the finite difference method to discretize the equation, followed by use of the proposed Residual Correction Method for obtaining its optimal solutions under constraint conditions for inequalities. Derived by using this approach, the upper and lower transient approximate solutions are not just useful in analyzing the range of the maximum possible error between them and the analytic solutions correctly, and the numerical validations also indicate good accuracy in mean values of the upper and lower approximate solutions.