100 research outputs found
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation of the two-phase flow system. After
discretizing in time with diagonally implicit Runge-Kutta schemes the resulting
systems of nonlinear algebraic equations are solved with Newton's method and
the arising systems of linear equations are solved efficiently and in parallel
with an algebraic multigrid method. The new scheme is investigated for various
test problems from the literature and is also compared to a cell-centered
finite volume scheme in terms of accuracy and time to solution. We find that
the method is accurate, robust and efficient. In particular no post-processing
of the DG velocity field is necessary in contrast to results reported by
several authors for decoupled schemes. Moreover, the solver scales well in
parallel and three-dimensional problems with up to nearly 100 million degrees
of freedom per time step have been computed on 1000 processors
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
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Preparing sparse solvers for exascale computing.
Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'
Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives
Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper
Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations
Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations
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