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.
Highlights
It is generally known that conductive heat transfer is described by a second-order linear parabolic partial differential equation (PDE), the so-called heat equation: Received: 31 July 2021 Accepted: 11 September 2021 Published: 16 September 2021
To obtain a reference solution, we used the implicit ode15s solver of MATLAB, which is a variable-step, variable-order solver based on the numerical differentiation formulas (NDFs) of orders 1 to 5, where the letter s indicates that the codes were suggested to use in the case of stiff systems
The purpose of this paper is to introduce and examine a set of new explicit one-step numerical schemes. We propose these methods to solve the ordinary differential equation (ODE) system which is obtained after spatially discretizing diffusion term
Summary
It is generally known that conductive heat transfer is described by a second-order linear parabolic partial differential equation (PDE), the so-called heat equation: Received: 31 July 2021 Accepted: 11 September 2021 Published: 16 September 2021. It is well known that conventional explicit methods (either Runge-Kutta or multistep Adams-Bashforth types) are only conditionally stable; the time steps must be reduced to very small values regardless of the actual accuracy requirements This is why the implicit methods with much better stability properties are widely considered to be superior [7,8] and almost exclusively used to solve these kinds of problems [9]. We have to note that, according to our knowledge, these methods haven’t even been listed together in any single document apart from our work, let alone being systematically tested against one another and against popular methods These methods have weak points, for example, some of them are only conditionally convergent or consistent, they are usually less accurate, they can be complicated to code or hardly be applied for irregular grids [28,29,30,31,32]. We mention that the method we introduce here can be considered as a generalization of our previous CN2 method [37], which was only first order because it used only integer time steps
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