[Paper Review] Space-time domain decomposition for advection-diffusion problems in mixed formulations

Thi‐Thao‐Phuong Hoang, Caroline Japhet+2 more|arXiv (Cornell University)|May 19, 2016

Advanced Numerical Methods in Computational Mathematics52 references21 citations

TL;DR

This paper proposes space-time domain decomposition methods for advection-diffusion problems in mixed formulations, enabling nonconforming time grids and different time steps for advection and diffusion via operator splitting. It introduces two methods—GTP-Schur with a generalized Neumann-Neumann preconditioner and GTO-Schwarz with optimized Robin transmission conditions—demonstrating that GTO-Schwarz converges 2–2.5× faster than GTP-Schur and maintains robustness across advection- and diffusion-dominated regimes.

ABSTRACT

This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time steps can be used in different parts of the domain. Global-in-time, non-overlapping domain-decomposition methods are coupled with operator splitting making possible the different treatment of the advection and diffusion terms. Two domain-decomposition methods are considered: one uses the time-dependent Steklov--Poincar{\\'e} operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains. A generalized Neumann-Neumann preconditioner is proposed for the first method. To illustrate the two methods numerical results for two-dimensional problems with strong heterogeneities are presented. These include both academic problems and more realistic prototypes for simulations for the underground storage of nuclear waste.

Motivation & Objective

• Address the challenge of simulating heterogeneous advection-diffusion problems in porous media with widely varying time scales, particularly in nuclear waste storage applications.
• Overcome the limitations of uniform time stepping by enabling different time steps for advection and diffusion processes within subdomains.
• Develop space-time domain decomposition methods that support nonconforming time grids and preserve temporal accuracy in the presence of heterogeneous coefficients.
• Ensure mass conservation and robust convergence for both advection-dominated and diffusion-dominated regimes using mixed finite elements and optimized transmission conditions.
• Validate the methods on realistic 2D test cases, including nuclear waste repository prototypes, to demonstrate scalability and efficiency with complex geometries and material properties.

Proposed method

• Apply global-in-time, non-overlapping domain decomposition with operator splitting to treat advection and diffusion separately, allowing independent time stepping in subdomains.
• Formulate two space-time interface problems: one using a time-dependent Steklov–Poincaré operator (GTP-Schur) and another using Robin transmission conditions (GTO-Schwarz) for optimized Schwarz waveform relaxation.
• Introduce new discrete unknowns for advection in the interface problems to handle Dirichlet transmission conditions, ensuring consistency with the monodomain formulation.
• Use an optimal time projection algorithm to handle nonconforming time grids between subdomains without additional grid refinement.
• Implement a generalized Neumann-Neumann preconditioner for the GTP-Schur method to improve convergence, particularly in diffusion-dominated cases with high coefficient contrasts.
• Employ upwind finite volume for advection (explicit Euler in time) and mixed finite elements with implicit Euler for diffusion, ensuring mass conservation and stability.

Experimental results

Research questions

• RQ1 Can space-time domain decomposition methods be extended to advection-diffusion problems in mixed formulations while allowing different time steps for advection and diffusion?
• RQ2 How do the GTP-Schur and GTO-Schwarz methods perform in terms of convergence rate and robustness when applied to advection-diffusion problems with strong heterogeneities?
• RQ3 To what extent does the use of nonconforming time grids affect the temporal accuracy of the solution, and can this be mitigated effectively?
• RQ4 How does the generalized Neumann-Neumann preconditioner impact convergence in the GTP-Schur method, especially in advection-dominated regimes?
• RQ5 Does the optimized Robin parameter in the GTO-Schwarz method provide robustness and improved convergence compared to the GTP-Schur method across varying physical regimes?

Key findings

• The GTO-Schwarz method outperforms the GTP-Schur method by a factor of 2 to 2.5 in terms of subdomain solves required to achieve a fixed error reduction in the solution.
• The GTO-Schwarz method exhibits weak dependence on discretization parameters due to optimized Robin transmission conditions, making it robust across advection- and diffusion-dominated regimes.
• The GTP-Schur method with the generalized Neumann-Neumann preconditioner shows mesh-size-independent convergence in the diffusion-dominated case, while convergence without preconditioning strongly depends on mesh size.
• In advection-dominated cases, the Neumann-Neumann preconditioner slows down convergence compared to the non-preconditioned GTP-Schur, but asymptotic convergence remains weakly dependent on mesh size.
• The use of nonconforming time grids introduces an error close to that of a conforming fine grid, preserving temporal accuracy for both two-subdomain and multiple-subdomain configurations.
• With an adapted initial guess from the previous time window, only a few iterations per time window are needed to reach the scheme error, indicating efficient time-marching performance.