98 research outputs found

    Shenfun -- automating the spectral Galerkin method

    Full text link
    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

    Get PDF
    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

    Full text link
    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 22, 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 (nx,ny)(n_x,n_y) is computed in O((nxny)3/2)\mathcal{O}((n_x n_y)^{3/2}) operations. Partial differential operators of splitting rank ≄3\geq 3 are solved via a linear system involving a block-banded matrix in O(min⁥(nx3ny,nxny3))\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3})) 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 6060 seconds. An experimental implementation in the Julia language can currently perform the same solve in 1010 seconds.Comment: 22 page

    Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator

    Full text link
    In this paper, we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates (r,Ξ,z)(r,\theta,z) with free-space radiation conditions. By combining integral equation methods in the radial variable rr with Fourier methods in Ξ\theta and zz, 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 zz 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

    Get PDF
    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

    Get PDF
    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

    Full text link
    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 d’ordre Ă©levĂ©, le traitement des gĂ©omĂ©tries complexes et la rĂ©solution des Ă©quations de Navier-Stokes instationnaires

    Get PDF
    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.L’objectif 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 d’abord utilisĂ©e pour rĂ©soudre des Ă©quationsaux dĂ©rivĂ©es partielles d’ordre 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 l’approche PGD.Enfin, une sĂ©paration espace-temps est proposĂ©e pour rĂ©soudre les Ă©quations deNavier-Stokes instationnaires Ă  l’aide d’une 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 d’incompressibilitĂ©. L’adaptation d’une 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
    • 

    corecore