859 research outputs found
A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method
A numerical method for solving elliptic PDEs with variable coefficients on
two-dimensional domains is presented. The method is based on high-order
composite spectral approximations and is designed for problems with smooth
solutions. The resulting system of linear equations is solved using a direct
(as opposed to iterative) solver that has optimal O(N) complexity for all
stages of the computation when applied to problems with non-oscillatory
solutions such as the Laplace and the Stokes equations. Numerical examples
demonstrate that the scheme is capable of computing solutions with relative
accuracy of or better, even for challenging problems such as highly
oscillatory Helmholtz problems and convection-dominated convection diffusion
equations. In terms of speed, it is demonstrated that a problem with a
non-oscillatory solution that was discretized using nodes was solved
in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since
the solver is direct, and the "solution operator" fits in RAM, any solves
beyond the first are very fast. In the example with unknowns, solves
require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599
The automatic solution of partial differential equations using a global spectral method
A spectral method for solving linear partial differential equations (PDEs)
with variable coefficients and general boundary conditions defined on
rectangular domains is described, based on separable representations of partial
differential operators and the one-dimensional ultraspherical spectral method.
If a partial differential operator is of splitting rank , such as the
operator associated with Poisson or Helmholtz, the corresponding PDE is solved
via a generalized Sylvester matrix equation, and a bivariate polynomial
approximation of the solution of degree is computed in
operations. Partial differential operators of
splitting rank are solved via a linear system involving a block-banded
matrix in operations. Numerical
examples demonstrate the applicability of our 2D spectral method to a broad
class of PDEs, which includes elliptic and dispersive time-evolution equations.
The resulting PDE solver is written in MATLAB and is publicly available as part
of CHEBFUN. It can resolve solutions requiring over a million degrees of
freedom in under seconds. An experimental implementation in the Julia
language can currently perform the same solve in seconds.Comment: 22 page
PoisFFT - A Free Parallel Fast Poisson Solver
A fast Poisson solver software package PoisFFT is presented. It is available
as a free software licensed under the GNU GPL license version 3. The package
uses the fast Fourier transform to directly solve the Poisson equation on a
uniform orthogonal grid. It can solve the pseudo-spectral approximation and the
second order finite difference approximation of the continuous solution. The
paper reviews the mathematical methods for the fast Poisson solver and
discusses the software implementation and parallelization. The use of PoisFFT
in an incompressible flow solver is also demonstrated
Totally parallel multilevel algorithms
Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance
SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs
The paper describes a sparse direct solver for the linear systems that arise
from the discretization of an elliptic PDE on a two dimensional domain. The
solver is designed to reduce communication costs and perform well on GPUs; it
uses a two-level framework, which is easier to implement and optimize than
traditional multi-frontal schemes based on hierarchical nested dissection
orderings. The scheme decomposes the domain into thin subdomains, or "slabs".
Within each slab, a local factorization is executed that exploits the geometry
of the local domain. A global factorization is then obtained through the LU
factorization of a block-tridiagonal reduced coefficient matrix. The solver has
complexity for the factorization step, and for each
solve once the factorization is completed.
The solver described is compatible with a range of different local
discretizations, and numerical experiments demonstrate its performance for
regular discretizations of rectangular and curved geometries. The technique
becomes particularly efficient when combined with very high-order convergent
multi-domain spectral collocation schemes. With this discretization, a
Helmholtz problem on a domain of size (for
which N=100 \mbox{M}) is solved in 15 minutes to 6 correct digits on a
high-powered desktop with GPU acceleration
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