Fourth-Order Numerical Scheme Based on Half-Step Non-Polynomial Spline Approximations for 1D Quasi-Linear Parabolic Equations

摘要

In this article, we discuss a fourth-order accurate scheme based on non-polynomial splines in tension approximations for solving quasi-linear parabolic partial differential equations (PDEs). The proposed numerical method requires only two half-step points and a central point on a uniform mesh in spatial direction. This method is derived directly from the continuity condition for the first-order derivative of the non-polynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is applicable for solving singular parabolic problems in polar systems. The proposed method is tested on the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, and Burgers' equations in polar coordinates.