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

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.

Stabil, explicit numerikus algoritmusok diffúziós és hővezetési problémák megoldására

“…az ode45-öt használnánk. Legutóbb megjelent egy olyan angol nyelvű cikk (Saleh et al, 2022), amelyben 14 ilyen módszert mutatnak be és tesztelnek. Ennek során olyan nem triviális analitikus megoldást használtak referenciának, amely térfüggő diffúziós együtthatóra vonatkozik.…”

“…15. Az implicit Euler séma az eredeti (Saleh et al, 2022) cikkben sajnos nem szerepelt. Ebben a cikkben azért vettük bele az összehasonlításba, mert rendkívül gyakran használják és ezért alapvető fontosságú, hogy legyen információnk arról, milyen teljesítményt nyújt a többi vizsgált módszerhez képest.…”

Examining the Performance of MATLAB’s Matrix Capabilities, Testing on Euler’s Method Applied on the Diffusion Equation

“…Several novel explicit methods are proposed recently in the scientific literature, both for ODEs [1], and PDEs [ [3], [5]]. In order to develop and test new numerical methods, as well as to explore which one can be proposed to different problems under specific circumstances, one should have information on what the actual running time depends on.…”

A Review on Integration of Electric Vehicles and Optimal Power Management Strategy for Sustainable Development

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