A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Endre Kovács, Ádám Nagy and Mahmoud Saleh
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Endre Kovács: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Ádám Nagy: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Mahmoud Saleh: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary

Mathematics, 2021, vol. 9, issue 18, 1-22

Abstract: This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

Keywords: heat equation; explicit time-integration; stiff equations; parabolic partial differential equations; unconditional stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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