A class of new stable, explicit methods to solve the non-stationary heat equation

TL;DR

This paper introduces a new class of explicit, stable numerical algorithms for solving the heat equation by using neighbor-based approximations that allow analytical solutions and enhanced stability, especially for large stiff systems.

Contribution

The authors develop explicit, stable methods that avoid finite difference time derivatives, providing faster solutions for large stiff heat equations.

Findings

Methods are first and second order in time.

Significantly faster than traditional explicit or implicit methods.

Effective for large stiff systems.

Abstract

We present a class of new explicit and stable numerical algorithms to solve the spatially discretized linear heat or diffusion equation. After discretizing the space and the time variables like conventional finite difference methods, we do not approximate the time derivatives by finite differences, but use constant neighbor and linear neighbour approximations to decouple the ordinary differential equations and solve them analytically. During this process, the timestep-size appears not in polynomial, but in exponential form with negative exponents, which guarantees stability. We compare the performance of the new methods with analytical and numerical solutions. According to our results, the methods are first and second order in time and can be much faster than the commonly used explicit or implicit methods, especially in the case of extremely large stiff systems.