MONOLITHIC APPROACH FOR SOLVING STOKES-DARCY SYSTEMS

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MONOLITHIC APPROACH FOR SOLVING STOKES-DARCY SYSTEMS

PID (if applicable): arxiv:2108.13229

Problem Statement

Instationary, coupled Stokes-Darcy two-domain problem: coupled systems of free flow adjacent to permeable (porous) media

Object of Research and Objective

The motivation for this is to avoid problems when using simple direct solvers (sparse direct solvers for the (linearized) subproblems): bad parallel scaling, untrustworthy solution with bad conditioning.

Procedure

Assemble and solve the above system via block-preconditioning (implemented entirely in DuMux)

File:Https://i.imgur.com/3K2chRh.jpg

Involved Disciplines

Environmental Systems, Mathematics

Data Streams

Model

Stokes flow in the free-flow domain Darcy’s law for the porous domain

Discretization

  • Time: first-order backward Euler scheme
  • Space: finite volumes
    • Porous Domain (Darcy): mit two-point flux approximation for pressure
    • Free Flow domain (Stokes): staggered grid for pressure and velocity, upwind scheme for approximation of fluxes

Variables

Name Unit Symbol
Pressure (Dirichlet pressure) - p
Velocity (Neumann velocity) - v

Process Informationen

Process Steps

Name Description Input Output Method Parameter Environment Mathematical Area
Solving Solving with Preconditioner+Solver LGS Vector Solver: PD-GMRES, Preconditioner: AMG, Uzawa, ILU(0), Block-Jacobi, Block-Gauss-Seidel DuMux Numerical Mathematics

Applied Methods

ID Name Process Step Parameter implemented by
wikidata:Q1069090 block-Gauss-Seidel preconditioner Preconditioning
wikidata:Q1481893 Block Jacobi Preconditioner Preconditioning
- Two-domain block-Gauss-Seidel preconditioner Two-Domain Preconditioning
- Two-domain Block Jacobi Preconditioner Two-Domain Preconditioning
wikidata:Q2467290 Umfpack Solver
wikidata: Q56564057 PD-GMRES Solver k (subiteration parameter, determined automatically), tolerance: relative residual...
wikidata:Q56560244 Bi-CGSTAB Solver tolerance: relative residual...
wikidata:Q1471828 AMG method Preconditioner
wikidata: Q17144437 Uzawa-iterations Preconditioner
wikidata:Q1654069 ILU(0) factorization Preconditioner

Software used

ID Name Description Version Programming Language Dependencies versioned published documented
sw:14293 DuMux DUNE for Multi-{Phase, Component, Scale, Physics, …} flow and transport in porous media C++, python-bindings, utility-skripts in python Linux, DuNE (C++ Framework), cmake (module chains), package-config, compiler, build-essentials, dpg https://git.iws.uni-stuttgart.de/dumux-repositories/dumux https://zenodo.org/record/5152939#.YQva944zY2w https://dumux.org/docs/
sw:18749 ISTL Iterative Solver Template Library” (ISTL) which is part of the “Distributed and Unified Numerics Environment” (DUNE). C++ Linux, DuNE (C++ Rahmenwerk) https://gitlab.dune-project.org/core/dune-istl https://doi.org/10.1007/978-3-540-75755-9_82 https://www.dune-project.org/modules/dune-istl/

Hardware

ID Name Processor Compiler #Nodes #Cores
AMD EPYC 7551P CPU 1 1

Input Data

ID Name Size Data Structure Format Representation Format Exchange binary/text proprietary to publish to archive
LGS O(2*106) Matrix size Data structure in DUNE/DuMux numbers open ? ?

Output Data

ID Name Size Data Structure Format Representation Format Exchange binary/text proprietary to publish to archive
Solution Vector Data Structure in DUNE/DuMux numbers open ? ?
Runtime Behavior Data Structure in DUNE/DuMux numbers open ? ?
wikidata:Q18812775 vtk-image files, from the daten structure ? ASCII or binary open ? ?

Reproducibility

Mathematical Reproducibility

yes, by all parameters

Runtime Reproducibility

yes, for same input samples

Reproducibility of Results

Due to floating point arithmetic, no bitwise reproducibility

Reproducibility on original Hardware

Reproducibility on other Hardware

a) Serial Computation

b) Parallel Computation

Transferability to

a) similar model parameters (other initial and boundary values)

b) other models

Legend

The following abbreviations are used in the document to indicate/resolve IDs:

doi: DOI / https://dx.doi.org/

sw: swMATH / https://swmath.org/software/

wikidata: https://www.wikidata.org/wiki/