New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient

Mahmoud Saleh, Endre Kovács (), Imre Ferenc Barna and Laszlo Matyas ()
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Mahmoud Saleh: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Endre Kovács: Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary
Imre Ferenc Barna: Wigner Research Center for Physics, 1051 Budapest, Hungary

Mathematics, 2022, vol. 10, issue 15, 1-26

Abstract: We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Keywords: diffusion; heat conduction; analytical solution; explicit time-integration; unconditionally stable numerical methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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