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Optimized Schwarz waveform relaxation methods for wave-heat coupling in one dimensional bounded domains.
Franz Chouly1, Martin J Gander2, Véronique Martin3
1Center of Mathematics, University of the Republic, Montevideo, Uruguay.
Numerical Algorithms
|November 27, 2025
Summary
This study introduces optimized transmission conditions for heterogeneous domain decomposition methods, enhancing coupled simulations of heat and wave equations. The approach ensures faster convergence for complex physical models.
Area of Science:
- Computational mathematics
- Numerical analysis
- Scientific computing
Background:
- Heterogeneous domain decomposition methods are crucial for coupling diverse physical models (e.g., fluid-structure, ocean-atmosphere).
- These methods enable code reuse and parallel execution, accommodating different time steps and independent solvers.
- Optimized Schwarz Waveform Relaxation (OSWR) is a promising candidate due to its non-overlapping capabilities.
Purpose of the Study:
- To design and analyze novel transmission conditions for OSWR.
- To accelerate the convergence of space-time coupled partial differential equations (PDEs).
- To address the minimal model problem of coupling heat and wave equations in 1D.
Main Methods:
- Developing and evaluating two transmission strategies for OSWR.
- Strategy 1: Optimizing transmission with a single common parameter.
- Strategy 2: Utilizing wave characteristics for parameter selection and subsequent optimization.
Main Results:
- Proposed transmission conditions significantly enhance OSWR convergence rates.
- Numerical experiments validate the effectiveness of the developed strategies.
- The methods demonstrate robustness and efficiency for coupled PDE problems.
Conclusions:
- The developed transmission conditions offer a pathway to faster and more efficient heterogeneous domain decomposition.
- This work provides a foundation for tackling more complex coupled physical systems.
- Optimized coupling strategies are essential for advancing scientific simulations.
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