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This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.
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Moving interfaces in peridynamic diffusion models and the influence of discontinuous initial conditions: Numerical stability and convergence
We derive numerical stability conditions and analyze convergence to analytical nonlocal solutions of 1D peridynamic models for transient diffusion with and without a moving interface. In heat transfer or oxidation, for example, one often encounters initial conditions that are discontinuous, as in thermal shock or sudden exposure to oxygen. We study the numerical error in these models with continuous and discontinuous initial conditions and determine that the initial discontinuities lead to lower convergence rates, but this issue is present at early times only. Except for the early times, the convergence rates of models with continuous and discontinuous initial conditions are the same. In problems with moving interfaces, we show that the numerical solution captures the exact interface location well, in time. These results can be used in simulating a variety of reaction-diffusion type problems, such as the oxidation-induced damage in zirconium carbide at high temperatures.
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- Award ID(s):
- 2206741
- PAR ID:
- 10526376
- Publisher / Repository:
- Computers & Mathematics with Applications
- Date Published:
- Journal Name:
- Computers & Mathematics with Applications
- Volume:
- 151
- Issue:
- C
- ISSN:
- 0898-1221
- Page Range / eLocation ID:
- 384 to 396
- Subject(s) / Keyword(s):
- Peridynamics Diffusion-reaction problem Numerical stability Discontinuous initial conditions Phase change Moving interface
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.camwa.2023.10.016
