A monolithic multi-time-step computational framework for first-order transient systems with disparate scales
Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time-st...
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| Published in: | Computer methods in applied mechanics and engineering Vol. 283; pp. 419 - 453 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
01.01.2015
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| Subjects: | |
| ISSN: | 0045-7825, 1879-2138 |
| Online Access: | Get full text |
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| Summary: | Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time-steps (multi-time-step coupling), and different time integrators and different numerical formulations (mixed methods) in different regions of the computational domain. To this end, we present two new monolithic multi-time-step mixed coupling methods for first-order transient systems. We shall employ unsteady advection–diffusion–reaction equation with linear decay as the model problem, which offers several unique challenges in terms of non-self-adjoint spatial operator and rich features in the solutions. We shall employ the dual Schur domain decomposition technique to split the computational domain into an arbitrary number of subdomains. It will be shown that the governing equations of the decomposed problem, after spatial discretization, will be differential/algebraic equations. This is a crucial observation to obtain stable numerical results. Two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting. A systematic theoretical analysis (which includes numerical stability, influence of perturbations, bounds on drift along the subdomain interface) will be performed. The first coupling method ensures that there is no drift along the subdomain interface, but does not facilitate explicit/implicit coupling. The second coupling method allows explicit/implicit coupling with controlled (but non-zero) drift in the solution along the subdomain interface. Several canonical problems will be solved to numerically verify the theoretical predictions, and to illustrate the overall performance of the proposed coupling methods. Finally, we shall illustrate the robustness of the proposed coupling methods using a multi-time-step transient simulation of a fast bimolecular advective–diffusive–reactive system.
•We propose two new effective monolithic multi-time-step coupling methods for first-order transient systems.•We provide a detailed study on the stability, influence of perturbation, and drifts in compatibility constraints of the proposed methods.•Additional criteria, compared to second-order transient systems, have to be satisfied to ensure stability.•Implicit/explicit time integration is not always possible in the case of first-order transient systems.•We demonstrate the theoretical findings, as well as accuracy and stability of the new methods using several numerical experiments. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0045-7825 1879-2138 |
| DOI: | 10.1016/j.cma.2014.10.003 |