98 research outputs found
Shenfun -- automating the spectral Galerkin method
With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is
made towards automating the implementation of the spectral Galerkin method for
simple tensor product domains, consisting of (currently) one non-periodic and
any number of periodic directions. The user interface to shenfun is
intentionally made very similar to FEniCS (fenicsproject.org). Partial
Differential Equations are represented through weak variational forms and
solved using efficient direct solvers where available. MPI decomposition is
achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and
all developed solver may, with no additional effort, be run on supercomputers
using thousands of processors. Complete solvers are shown for the linear
Poisson and biharmonic problems, as well as the nonlinear and time-dependent
Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational
Mechanic
A second-order continuity domain-decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems
This paper presents a second-order continuity
non-overlapping domain decomposition (DD) technique for
numerically solving second-order elliptic problems in
two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C2 (C squared) function
at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials
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
Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator
In this paper, we describe a new class of fast solvers for separable elliptic
partial differential equations in cylindrical coordinates with
free-space radiation conditions. By combining integral equation methods in the
radial variable with Fourier methods in and , we show that
high-order accuracy can be achieved in both the governing potential and its
derivatives. A weak singularity arises in the Fourier transform with respect to
that is handled with special purpose quadratures. We show how these solvers
can be applied to the evaluation of the Coulomb collision operator in kinetic
models of ionized gases.Comment: 20 pages, 5 figure
On efficient direct methods for conforming spectral domain decomposition techniques
AbstractA conforming spectral domain decomposition technique is described for the solution of Stokes flow in rectangularly decomposable domains. The matrices arising from such a spectral discretization procedure possess a block tridiagonal structure where these blocks are full submatrices. Efficient direct solution procedures are proposed to take advantage of the matrix structure. A comparison of the methods in terms of computational efficiency is made. Numerical results are presented for the flow through an abruptly contracting channel
Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations
In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate
Extreme Learning Machine-Assisted Solution of Biharmonic Equations via Its Coupled Schemes
Obtaining the solutions of partial differential equations based on various
machine learning methods has drawn more and more attention in the fields of
scientific computation and engineering applications. In this work, we first
propose a coupled Extreme Learning Machine (called CELM) method incorporated
with the physical laws to solve a class of fourth-order biharmonic equations by
reformulating it into two well-posed Poisson problems. In addition, some
activation functions including tangent, gauss, sine, and trigonometric
(sin+cos) functions are introduced to assess our CELM method. Notably, the sine
and trigonometric functions demonstrate a remarkable ability to effectively
minimize the approximation error of the CELM model. In the end, several
numerical experiments are performed to study the initializing approaches for
both the weights and biases of the hidden units in our CELM model and explore
the required number of hidden units. Numerical results show the proposed CELM
algorithm is high-precision and efficient to address the biharmonic equation in
both regular and irregular domains
Des avancĂ©es dans la rĂ©duction de modĂšle de type PGD pour les EDPs dâordre Ă©levĂ©, le traitement des gĂ©omĂ©tries complexes et la rĂ©solution des Ă©quations de Navier-Stokes instationnaires
The main purpose of this work is to describe a simulation method for the use of aPGD-based Model reduction Method (MOR) for solving high order partial differentialequations. First, the PGD method is used for solving fourth order PDEs and thealgorithm is illustrated on a lid-driven cavity problem. Transformations of coordinatesfor changing the complex physical domain into the simple computational domain arealso studied, which lead to extend the spatial PGD method to complex geometrydomains. Some numerical examples for different kinds of domain are treated toillustrate the potentialities of this methodology.Finally, a PGD-based space-time separation is introduced to solve the unsteadyStokes or Navier-Stokes equations. This decomposition makes use of common tem-poral modes for both velocity and pressure, which lead to velocity spatial modessatisfying individually the incompressibility condition. The adaptation and imple-mentation of a PGD approach into a general purpose finite volume framework isdescribed and illustrated on several analytic and academic flow examples. A largereduction of the computational cost is observed on most of the treated examples.Lâobjectif principal de ce travail est de proposer une nouvelle approche de simulationbasĂ©e sur une MĂ©thode de rĂ©duction du modĂšle (MOR) utilisant une dĂ©compositionPGD. Dans ce travail, cette approche est dâabord utilisĂ©e pour rĂ©soudre des Ă©quationsaux dĂ©rivĂ©es partielles dâordre Ă©levĂ© avec un exemple numĂ©rique pour les Ă©quations auxdĂ©rivĂ©es partielles du quatriĂšme ordre sur le problĂšme de la cavitĂ© entraĂźnĂ©e. Ensuiteun changement de coordonnĂ©es pour transformer le domaine physique complexe enun domaine de calcul simple est Ă©tudiĂ©, ce qui conduit Ă Ă©tendre la mĂ©thode PGDau traitement de certaines gĂ©omĂ©tries complexes. Divers exemples numĂ©riques pourdiffĂ©rents types de domaines gĂ©omĂ©triques sont ainsi traitĂ©s avec lâapproche PGD.Enfin, une sĂ©paration espace-temps est proposĂ©e pour rĂ©soudre les Ă©quations deNavier-Stokes instationnaires Ă lâaide dâune approche PGD. Cette dĂ©compositionest basĂ©e sur le choix de modes temporels communs pour la vitesse et la pression,ce qui conduit Ă une dĂ©composition basĂ©e sur des modes spatiaux satisfaisant in-dividuellement la condition dâincompressibilitĂ©. Lâadaptation dâune formulationvolumes finis Ă cette dĂ©composition PGD est prĂ©sentĂ©e et validĂ©e sur de premiersexemples analytiques ou acadĂ©miques pour les Ă©quations de Stokes ou Navier-Stokesinstationnaires. Une importante rĂ©duction des temps calculs est observĂ©e sur lespremiers exemples traitĂ©s
- âŠ