High-accuracy quasi-variable mesh method for the system of 1D quasi-linear parabolic partial differential equations based on off-step spline in compression approximations

Abstract

Abstract In this article, we propose a new two-level implicit method of accuracy two in time and three in space based on spline in compression approximations using two off-step points and a central point on a quasi-variable mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations. The new method is derived directly from the continuity condition of the first-order derivative of the spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we solve the generalized Burgers-Fisher equation, generalized Burgers-Huxley equation, coupled Burgers-equations and heat equation in polar coordinates. We demonstrate that the proposed method enables us to obtain high accurate solution for high Reynolds number.

Keywords

• quasi-linear parabolic equations
• quasi-variable mesh
• spline in compression
• generalized Burgers-Fisher equations
• coupled Burgers equation
• Newton’s iterative method