High-order numerical methods for 2D parabolic problems in single and composite domains
For researchers in computational PDEs, this paper provides a systematic comparison of three advanced methods for handling complex geometries and interfaces, but the results are incremental as they confirm known capabilities without introducing new techniques.
This work compares three high-order numerical methods (CutFEM, Difference Potentials, and SBP-FD) for solving 2D parabolic problems in single and composite domains, including cases with discontinuous solutions/flux at interfaces. Benchmark tests show that all methods achieve high-order accuracy and convergence, with specific error and convergence rates reported for linear parabolic problems.
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.