High-order numerical methods for 2D parabolic problems in single and composite domains

TL;DR

This paper compares three high-order numerical methods—Cut Finite Element, Difference Potentials, and summation-by-parts Finite Difference—for solving 2D parabolic diffusion problems in single and composite domains, focusing on accuracy and convergence.

Contribution

It provides a comparative analysis and benchmark tests of three advanced numerical methods for 2D parabolic problems with discontinuities at interfaces.

Findings

All methods achieve high accuracy and convergence.

Differences in handling interface discontinuities are highlighted.

Benchmark results guide method selection for specific problem types.

Abstract

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.