A high-order compact difference method and its Richardson extrapolation for semi-linear reaction–diffusion equations with piecewise continuous argument in diffusion term

Abstract

In this paper, for the initial–boundary value problems of semi-linear reaction–diffusion equations with piecewise continuous argument in spatial derivative, we suggest Crank–Nicolson method, high-order compact difference (HOCD) method and HOCD-based Richardson extrapolation (RHOCD) method. Under the appropriate conditions, it is proved that HOCD (resp. RHOCD) method has the computational accuracy O(τ2+h4) (resp. O(τ4+h4)). This shows that RHOCD method improves the calculation accuracy of HOCD method in temporal direction. Moreover, we also analyze the stability of HOCD method and thus derive a global stability criterion of this method. Finally, with a series of numerical experiments, we further confirm the computational effectiveness and theoretical accuracy of the concerned methods.