MONOLITHIC APPROACH FOR SOLVING STOKES-DARCY SYSTEMS

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

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($2*10^6$) 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 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/