Space–time domain decomposition for advection–diffusion problems in mixed formulations

Hoang, Thi-Thao-Phuong; Japhet, Caroline; Kern, Michel; Roberts, Jean E.

doi:10.1016/j.matcom.2016.11.002

Cited by 22 publications

(19 citation statements)

References 57 publications

(163 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The aforementioned literatures applied space-time domain decomposition method to mostly mechanics problems. On the other hand, prior work regarding flow mainly focused on linear single phase flow and transport problems where flow is naturally decoupled from the advection-diffusion component transport [17,18]. [31] first validates the space-time approach for non-linear coupled multiphase flow and transport problems on static grid.…”

Section: Introductionmentioning

confidence: 87%

Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems

Leung

Wheeler

2020

Journal of Computational Physics

View full text Add to dashboard Cite

Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and unconditionally stable fully implicit schemes. The price that comes with restricting time steps to small scales is the enormous computational load, especially in large-scale models. To address this problem, we introduce a sequential local mesh refinement framework to optimize convergence rate and prevent convergence failure, while not restricting the whole system to small time steps, thus improving computational efficiency. We test the algorithm with the non-linear two-phase flow model. Starting by solving the problem on the coarsest space-time mesh, the algorithm refines the domain in time at water saturation front to prevent convergence failure. It then deploys fine spatial grid in regions with large saturation variation to assure solution accuracy. After each refinement, the solution from the previous mesh is used to estimate the initial guess of unknowns on the current mesh for faster convergence. Numerical results are presented to confirm accuracy of our algorithm as compared to the uniformly fine time step and fine spatial discretization solution. We observe approximately 25 times speedup in the solution time by using our algorithm.

show abstract

Section: Introductionmentioning

confidence: 87%

Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems

Leung

Wheeler

2020

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…Both simplifications were made in order to follow the set-up of [48], and because this paper is the first application of the OSWR method to this type of problems. We emphasize that both restrictions can be lifted: the multidomain coupled problem (without domain decomposition) has been treated in [30], while the OSWR method has been extended to the diffusion-advection case in [66], See also [2] for related work, and [83] for a different domain decomposition method applied to the full two-phase flow model. For simplicity, we consider only Dirichlet boundary conditions on ∂Ω.…”

Section: Presentation Of the Problemmentioning

confidence: 99%

“…Robin-Robin methods have also been used to improve the transmission condition in coupled problems, see for instance [39,45,91] for the Darcy-Stokes system, or [27,88,77,87] for oceanatmosphere models. The use of a global-in-time DD method together with discontinuous Galerkin (DG) for the time discretization provides flexibility in using different time steps in different parts of the domain, adapted to the physical properties of each subdomain, see [62,64,66] for diffusion and advection-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

Space–time domain decomposition for two-phase flow between different rock types

Ahmed

Japhet

Kern

2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

In [3] a space-time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error. In this paper we provide a mathematical and numerical analysis of this space-time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space-time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method.

show abstract

“…In this paper we have not pursued this idea, but have used the same time steps in the fracture as in the matrix. However, we refer the interested reader to the series of articles [27,29,28], where a method for implementing this idea is described. h .…”

Section: Time Steppingmentioning

confidence: 99%

A reduced fracture model for two-phase flow with different rock types

Ahmed

Jaffr

Roberts

2017

Mathematics and Computers in Simulation

Self Cite

View full text Add to dashboard Cite

This article is concerned with the numerical discretization of a model for incompressible twophase flow in a porous medium with fractures. The model is a discrete fracture model in which the fractures are treated as interfaces of dimension 2 in a 3-dimensional simulation, with fluid exchange between the 2-dimensional fracture flow and the 3-dimensional flow in the surrounding rock matrix. The model takes into account the change in the relative permeabilities and in the capillary pressure curves which occurs at the interface between the fracture and the rock matrix. The model allows for barriers which are fractures with low permeability. Mixed finite elements and advective upstream weighting are used to discretize the problem and numerical experiments are shown.

show abstract

Space–time domain decomposition for advection–diffusion problems in mixed formulations

Cited by 22 publications

References 57 publications

Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems

Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems

Space–time domain decomposition for two-phase flow between different rock types

A reduced fracture model for two-phase flow with different rock types

Contact Info

Product

Resources

About