Optimal convergence rate of the explicit Euler method for convection–diffusion equations

Abstract

Revisiting the explicit Euler method of the classical diffusion equation, a new difference scheme with the optimal convergence rate four is achieved under the condition of the specific step-ratio r=1/6. Applying the corrected idea to the convection–diffusion equation, a new corrected numerical scheme is obtained which owns a similar fourth-order optimal convergence rate. Rigorous numerical analysis is carried out by the maximum principle. Compared with the standard difference schemes, the new proposed difference schemes have obvious advantage in accuracy. Extensive numerical examples with and without exact solutions confirm our theoretical results. Moreover, extending our technique to nonlinear problems such as the Fisher equation and viscous Burgers’ equation is available.