A class of new stable, explicit methods to solve the non‐stationary heat equation. Issue 3 (23rd December 2020)

• Record Type: Journal Article
• Authors: Kovács, Endre
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 3(2021)
• Page Start: 2469
• Page End: 2489
• Publication Date: 2020-12-23
• Subjects: explicit time‐integration -- heat equation -- parabolic partial differential equations -- stiff equations -- unconditional stability; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22730
• Languages: English
• ISSNs: 0749-159X

Abstract

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‐neighbor approximations to decouple the ordinary differential equations and solve them analytically. During this process, the time step 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.