Utilizing a third order difference scheme in numerical method of lines for solving the heat equation

Obtaining the numerical solution for partial differential equations (PDEs) is a fundamental task in computational mathematics and scientific computing. In this work, we propose a third order numerical method of lines (MOL) based on difference schemes for the efficient and accurate approximation of heat equation. The method combines the flexibility of MOL with the superior accuracy of higher order difference schemes. The proposed approach begins by discretizing the spatial domain, and then applies higher order difference schemes to approximate the spatial derivatives in the PDE. The MOL framework is utilized to transform the resultant set of ordinary differential equations to a system of algebraic equations, facilitating their solution through an appropriate numerical method of choice. We have utilized the Adams method for this purpose. We analyze the stability of the proposed method and establish conditions under which it yields accurate and reliable solutions. We present the theoretical foundations of the method and discuss its implementation. We also present numerical results that demonstrate the procedure's precision and effectiveness. The results show that the method is able to achieve high accuracy with relatively few grid points, and the results are compared with those obtained by some existing numerical schemes accessible in the literature.