474 research outputs found
A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries
We present a new multigrid scheme for solving the Poisson equation with
Dirichlet boundary conditions on a Cartesian grid with irregular domain
boundaries. This scheme was developed in the context of the Adaptive Mesh
Refinement (AMR) schemes based on a graded-octree data structure. The Poisson
equation is solved on a level-by-level basis, using a "one-way interface"
scheme in which boundary conditions are interpolated from the previous coarser
level solution. Such a scheme is particularly well suited for self-gravitating
astrophysical flows requiring an adaptive time stepping strategy. By
constructing a multigrid hierarchy covering the active cells of each AMR level,
we have designed a memory-efficient algorithm that can benefit fully from the
multigrid acceleration. We present a simple method for capturing the boundary
conditions across the multigrid hierarchy, based on a second-order accurate
reconstruction of the boundaries of the multigrid levels. In case of very
complex boundaries, small scale features become smaller than the discretization
cell size of coarse multigrid levels and convergence problems arise. We propose
a simple solution to address these issues. Using our scheme, the convergence
rate usually depends on the grid size for complex grids, but good linear
convergence is maintained. The proposed method was successfully implemented on
distributed memory architectures in the RAMSES code, for which we present and
discuss convergence and accuracy properties as well as timing performances.Comment: 33 pages, 15 figures, accepted for publication in Journal of
Computational Physic
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method
This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed
A self-gravity module for the PLUTO code
We present a novel implementation of an iterative solver for the solution of
the Poisson equation in the PLUTO code for astrophysical fluid dynamics. Our
solver relies on a relaxation method in which convergence is sought as the
steady-state solution of a parabolic equation, whose time-discretization is
governed by the \textit{Runge-Kutta-Legendre} (RKL) method. Our findings
indicate that the RKL-based Poisson solver, which is both fully parallel and
rapidly convergent, has the potential to serve as a practical alternative to
conventional iterative solvers such as the \textit{Gauss-Seidel} (GS) and
\textit{successive over-relaxation} (SOR) methods. Additionally, it can
mitigate some of the drawbacks of these traditional techniques. We incorporate
our algorithm into a multigrid solver to provide a simple and efficient gravity
solver that can be used to obtain the gravitational potentials in
self-gravitational hydrodynamics. We test our implementation against a broad
range of standard self-gravitating astrophysical problems designed to examine
different aspects of the code. We demonstrate that the results match
excellently with the analytical predictions (when available), and the findings
of similar previous studies.Comment: Submitted to ApJS. Comments are welcom
A Parallel Geometric Multigrid Method for Adaptive Finite Elements
Applications in a variety of scientific disciplines use systems of Partial Differential Equations (PDEs) to model physical phenomena. Numerical solutions to these models are often found using the Finite Element Method (FEM), where the problem is discretized and the solution of a large linear system is required, containing millions or even billions of unknowns. Often times, the domain of these solves will contain localized features that require very high resolution of the underlying finite element mesh to accurately solve, while a mesh with uniform resolution would require far too much computational time and memory overhead to be feasible on a modern machine. Therefore, techniques like adaptive mesh refinement, where one increases the resolution of the mesh only where it is necessary, must be used. Even with adaptive mesh refinement, these systems can still be on the order of much more than a million unknowns (large mantle convection applications like the ones in [90] show simulations on over 600 billion unknowns), and attempting to solve on a single processing unit is infeasible due to limited computational time and memory required. For this reason, any application code aimed at solving large problems must be built using a parallel framework, allowing the concurrent use of multiple processing units to solve a single problem, and the code must exhibit efficient scaling to large amounts of processing units.
Multigrid methods are currently the only known optimal solvers for linear systems arising from discretizations of elliptic boundary valued problems. These methods can be represented as an iterative scheme with contraction number less than one, independent of the resolution of the discretization [24, 54, 25, 103], with optimal complexity in the number of unknowns in the system [29]. Geometric multigrid (GMG) methods, where the hierarchy of spaces are defined by linear systems of finite element discretizations on meshes of decreasing resolution, have been shown to be robust for many different problem formulations, giving mesh independent convergence for highly adaptive meshes [26, 61, 83, 18], but these methods require specific implementations for each type of equation, boundary condition, mesh, etc., required by the specific application. The implementation in a massively parallel environment is not obvious, and research into this topic is far from exhaustive.
We present an implementation of a massively parallel, adaptive geometric multigrid (GMG) method used in the open-source finite element library deal.II [5], and perform extensive tests showing scaling of the v-cycle application on systems with up to 137 billion unknowns run on up to 65,536 processors, and demonstrating low communication overhead of the algorithms proposed. We then show the flexibility of the GMG by applying the method to four different PDE systems: the Poisson equation, linear elasticity, advection-diffusion, and the Stokes equations. For the Stokes equations, we implement a fully matrix-free, adaptive, GMG-based solver in the mantle convection code ASPECT [13], and give a comparison to the current matrix-based method used. We show improvements in robustness, parallel scaling, and memory consumption for simulations with up to 27 billion unknowns and 114,688 processors. Finally, we test the performance of IDR(s) methods compared to the FGMRES method currently used in ASPECT, showing the effects of the flexible preconditioning used for the Stokes solves in ASPECT, and the demonstrating the possible reduction in memory consumption for IDR(s) and the potential for solving large scale problems.
Parts of the work in this thesis has been submitted to peer reviewed journals in the form of two publications ([36] and [34]), and the implementations discussed have been integrated into two open-source codes, deal.II and ASPECT. From the contributions to deal.II, including a full length tutorial program, Step-63 [35], the author is listed as a contributing author to the newest deal.II release (see [5]). The implementation into ASPECT is based on work from the author and Timo Heister. The goal for the work here is to enable the community of geoscientists using ASPECT to solve larger problems than currently possible. Over the course of this thesis, the author was partially funded by the NSF Award OAC-1835452 and by the Computational Infrastructure in Geodynamics initiative (CIG), through the NSF under Award EAR-0949446 and EAR-1550901 and The University of California -- Davis
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