Fourth-order numerical scheme based on half-step nonpolynomial spline approximations for 1D quasi-linear parabolic equations

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