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Finite Difference Discretizations (finite + difference_discretization)
Selected AbstractsNonlinear multigrid for the solution of large-scale Riccati equations in low-rank and ,-matrix formatNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2008L. Grasedyck Abstract The algebraic matrix Riccati equation AX+XAT,XFX+C=0, where matrices A, B, C, F,,,,n × n are given and a solution X,,,,n × n is sought, plays a fundamental role in optimal control problems. Large-scale systems typically appear if the constraint is described by a partial differential equation (PDE). We provide a nonlinear multigrid algorithm that computes the solution X in a data-sparse, low-rank format and has a complexity of ,,(n), subject to the condition that F and C are of low rank and A is the finite element or finite difference discretization of an elliptic PDE. We indicate how to generalize the method to ,-matrices C, F and X that are only blockwise of low rank and thus allow a broader applicability with a complexity of ,,(nlog(n)p), p being a small constant. The method can also be applied to unstructured and dense matrices C and X in order to solve the Riccati equation in ,,(n2). Copyright © 2008 John Wiley & Sons, Ltd. [source] A study into the feasibility of using two parallel sparse direct solvers for the Helmholtz equation on Linux clustersCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 7 2006G. Z. M. Berglund Abstract Two state-of-the-art parallel software packages for the direct solution of sparse linear systems based on LU-decomposition, MUMPS and SuperLU_DIST have been tested as black-box solvers on problems derived from finite difference discretizations of the Helmholtz equation. The target architecture has been Linux clusters, for which no consistent set of tests of the algorithms implemented in these packages has been published. The investigation consists of series of memory and time scalability checks and has focused on examining the applicability of the algorithms when processing very large sparse matrices on Linux cluster platforms. Special emphasis has been put on monitoring the behaviour of the packages when the equation systems need to be solved for multiple right-hand sides, which is the case, for instance, when modelling a seismic survey. The outcome of the tests points at poor efficiency of the tested algorithms during application of the LU-factors in the solution phase on this type of architecture, where the communication acts as an impasse. Copyright © 2005 John Wiley & Sons, Ltd. [source] Fast direct solver for Poisson equation in a 2D elliptical domainNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004Ming-Chih Lai Abstract In this article, we extend our previous work M.-C. Lai and W.-C. Wang, Numer Methods Partial Differential Eq 18:56,68, 2002 for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second- and fourth-order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth-order accuracy can be achieved by a three-point compact stencil which is in contrast to a five-point long stencil for the disk case. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 72,81, 2004 [source] Fast direct solvers for Poisson equation on 2D polar and spherical geometriesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2002Ming-Chih Lai Abstract A simple and efficient class of FFT-based fast direct solvers for Poisson equation on 2D polar and spherical geometries is presented. These solvers rely on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the second- and fourth-order finite difference discretizations. Using a grid by shifting half mesh away from the origin/poles, and incorporating with the symmetry constraint of Fourier coefficients, the coordinate singularities can be easily handled without pole condition. By manipulating the radial mesh width, three different boundary conditions for polar geometry including Dirichlet, Neumann, and Robin conditions can be treated equally well. The new method only needs O(MN log2N) arithmetic operations for M × N grid points. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 56,68, 2002 [source] Analysis of lattice Boltzmann boundary conditionsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003M. Junk Dr. The correct implementation of Navier-Stokes boundary conditions in the framework of lattice Boltzmann schemes is complicated by the non-availability of analytical methods to assess the consistency of such discretizations. To close this gap, we propose a simple direct asymptotic analysis which is readily applicable to finite difference discretizations of initial boundary value problems in general and to lattice Boltzmann methods in particular. Results of the analysis applied to the classical lattice Boltzmann scheme with bounce back boundary condition are reported. [source] | ||||||||||