Abstract
In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.
Highlights
Academic Editor: Ali Cemal BenimReceived: 31 July 2021Accepted: 13 August 2021Published: 20 August 2021This paper can be considered to be a continuation of our previous work [1,2,3,4], in which we developed novel numerical algorithms to solve the heat or diffusion equation.The phenomenon of diffusion is described by the heat or diffusion equation, which, in its simplest form, is the following linear parabolic partial differential equation (PDE): ∂u = α ∇2 u ∂tPublisher’s Note: MDPI stays neutral (1)with regard to jurisdictional claims in iations
We investigate the analytical solutions of the diffusion equation where the diffusion constant has a power-law time dependence:
The present paper is the natural continuation of our previous work on explicit algorithms for solving the time-dependent diffusion equation
Summary
This paper can be considered to be a continuation of our previous work [1,2,3,4], in which we developed novel numerical algorithms to solve the heat or diffusion equation. Most consider only systems with relatively simple geometries, such as a cylindrical inclusion in an infinite medium [5] Parameters such as the diffusion coefficient or the heat conductivity are typically taken as constants, or a fixed and relatively simple function of the space variables [6] at best. This means that, for intricate geometries, and in the case of space- and time-dependent coefficients in general, numerical calculations cannot be avoided.
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