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

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.

45 pages, 12 figures, in revision for Journal of Scientific Computing

References in corpus (3)

• Review of Summation-by-parts schemes for initial-boundary-value problems
• Stable Coupling of Nonconforming, High-Order Finite Difference Methods
• High order finite difference methods for the wave equation with non-conforming grid interfaces

Cited by in corpus (4)

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• Difference Potentials Method for Models with Dynamic Boundary Conditions and Bulk-Surface Problems
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• Local-basis Difference Potentials Method for elliptic PDEs in complex geometry