Fast Elliptic Solver Research Articles

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 ( r, θ, z) with free-space radiation conditions. By combining integral equation methods in the radial variable r with Fourier methods in θ and z, 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 z 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.

An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC–FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

Krylov deferred correction accelerated method of lines transpose for parabolic problems

In this paper, a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented. Unlike traditional method of lines (MoL), the new Krylov deferred correction (KDC) accelerated method of lines transpose(MoLT) first discretizes the temporal direction using Gaussian type nodes and spectral integration, and symbolically applies low-order time marching schemes to form a preconditioned elliptic system, which is then solved iteratively using Newton–Krylov techniques such as Newton–GMRES or Newton–BiCGStab method. Each function evaluation in the Newton–Krylov method is simply one low-order time-stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated MoLT technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time-stepsizes in long-time simulations.

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.

Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers

AbstractThe computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self‐adjoint second‐order elliptic partial differential eigenvalue problems in two‐ or three‐dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos algorithm that the original problem. In these experiments, the underlying Poisson and elasticity problems are solved using a standard multigrid method. Copyright © 2001 John Wiley & Sons, Ltd.

Viscous flow simulation of a two-dimensional channel flow with complex geometry using the grid-particle vortex method

To avoid a large consumption of computer time instead of the direct vortex methods with Biot-Savart interaction law, the vortex in cell method was used. The local velocity of a particle was obtained by the solution of the Poisson equation for the stream function, its differentiation, and then interpolation of velocity to the vortex particle position from the grid nodes. The Poisson equation for the stream function was solved by fast elliptic solvers. To be able to solve the Poisson equation in a region with a complex geometry, the capacitance matrix technique was used. The viscosity of the fluid was taken in a stochastic manner. A suitable stochastic differential equation was solved by the Huen method. The non-slip condition on the wall was realized by the generation of the vorticity. The program was tested by solving several flows in the channels with a different geometry and at a different Reynolds number. Here we present the testing results concerning the flow in a channel with sudden symmetric expansion.

Parallel implementation of fast elliptic solver

A fast elliptic solver for separable elliptic equations on rectangular domains is considered. The method is referred to as FASV (fast algorithm for separation of variables) and is based on the odd-even block elimination technique in combination with the method for discrete separation of variables. The algorithm is connected with solving systems of algebraic equations with sparsity whose right-hand sides have only a few nonzero block components. The method is effective and stable by construction. Only a few of the block solution components are needed and hence these problems might be solved incompletely. Parallel implementation of the method proposed using the public domain PVM software is described in terms of decomposition of the original rectangular domain into a number of strips. Numerical results for a model problem on a cluster of a few IBM workstations are reported.

On a fast direct elliptic solver by a modified Fourier method

We describe high order numerical algorithms for the solution of second order elliptic equations in rectangular domains. These algorithms are based on the Fourier method in combination with a subtraction procedure. The singularities at the corner points, arising due to non-smoothness of the boundaries, are treated explicitly using properly constructed singular corner functions. The present algorithm is a generalization of the Fast Poisson Solver developed in our previous paper.

Two-dimensional device modeling and analysis of GaInAs metal–semiconductor–metal photodiode structures

A two-dimensional self-consistent time-dependent simulation technique has been developed to investigate electron-hole transport processes in the active region of metal–semiconductor–metal (MSM) interdigitated photodiode structures and to analyze their high-speed response. The distribution of the electric field inside the MSM device is determined by numerically solving the two-dimensional Poisson’s equation by the modified fast elliptic solver method. A set of superparticles photogenerated at a particular wavelength is analyzed with a given initial distribution of the potential and given boundary conditions, and the evolution of the particles is traced in time through the active region of the MSM device. Circuit loading, electric field effects in the MSM structure with various finger separations, background doping, carrier trapping, and recombination are included in the simulation program. Owing to miniaturization of devices, the classical scaling laws lose their validity while various performance degrading effects appear. The simulations show that the main problem in MSM devices with a small contact separation is the low electric field penetration depth. This results in different electron and hole collection rates and in a poor response time. The trade-off between the high-speed response and the internal quantum efficiency is examined and ways to improve the high-speed response are indicated. Modeling results are compared with experimental data on Ga0.47In0.53As based MSM photodiodes.

A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations

Abstract In this article we discuss a fictitious domain method for the numerical solutions of three-dimensional elliptic problems with Dirichlet boundary conditions and also of the Navier-Stokes equations modeling incompressible viscous flow. The methodology for the Navier-Stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection, imbedding and incompressibility. Due to the decoupling, fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry. The resulting methodology is applied to two-dimensional unsteady external incompressible viscous flow problems and three-dimensional Stokes problems.

Novel scalar-vector potential formulation for three-dimensional, inviscid, rotational flow problems

A computational method for the calculation of steady, strongly rotational, inviscid, subsonic flowfields in three-dimensional ducts is presented. The method is based on the decomposition of the velocity vector into a potential and rotational part through the Helmholtz theorem. The computational algorithm requires the solution of elliptic-type equations for the scalar and vector potentials, where a completely novel approach for solving the vector potential equation has been adopted. The new formulation has better physical meaning than our previous one and has the great advantage of decoupling the vector potential boundary conditions. The use of very fast elliptic solvers, based on preconditioned minimization techniques, leads to very economic computations. The transport equations are handled in their Lagrangian form.

A Triangulation Algorithm for Fast Elliptic Solvers Based on Domain Imbedding

The following triangulation problem is considered. Let R be a rectangle, and $\Omega $ an open domain whose closure is contained in R. Consider a rectangular grid covering R. Perturb this grid by shifting points close to the boundary $\partial \Omega $ onto $\partial \Omega $. This results in a quadrilateral, almost rectangular grid. Divide each cell of this grid into two triangles along one of its diagonals. This results in a triangulation of R. A subset of this triangulation is a triangulation of an approximation $\Omega ^h $ to $\Omega $. The distance between $\partial \Omega ^h $ and $\partial \Omega $ is required to be $O(h^2 )$, where h denotes the maximum meshwidth of the rectangular grid. All triangles should be nondegenerate.Triangulations of this kind are needed for finite element domain imbedding methods for elliptic boundary value problems. A particularly simple example of such a method is reviewed. The convergence theory for this method motivates our definition of nondegeneracy of triangles.F...

The modified operator strongly implicit scheme, MOSI, as a fast elliptic solver

A new fast elliptic solver is introduced that is suited for systems of finite difference equations associated with five point molecules. This modified operator strongly implicit scheme, MOSI, is a further development of the modified strongly implicit scheme, MSI, which is an approximate LU decomposition sparse matrix solver. Comparisons are made between MOSI and some other commonly used equation solvers, which indicate that it is faster than these solvers, including the MSI scheme.

A fast elliptic solver for simply connected domains

We present a fast method for solving the Laplace, Poisson, and the Helmholtz equations on two-dimensional simply connected regions with smooth curved boundaries. The region is first mapped onto the unit disk, using a conformal transformation. The transformed equations are then solved on the unit disk, using mainly Rapid Elliptic Solvers (RES). The construction of the mapping itself gives rise to large linear systems of equations in whose solution RES play a major role. Our numerical experiments on the Alliant FX/8 and one CPU of the Cray X-MP/48 indicate that the scheme presented here is suitable for parallel and vector machines, respectively. Our scheme is efficient on any architecture which supports powerful fast Fourier transforms.

Fast elliptic solvers—an overview

The purpose of this paper is to present a brief survey of fast direct methods for solving elliptic boundary-value problems. The methods reviewed are based on Fourier analysis, block reduction techniques, and marching algorithms. First, the Poisson equation with Dirichlet and mixed boundary conditions are considered. Then we go to more general elliptic problems and irregular regions.

Finite difference solution to the flow between two rotating spheres

Abstract This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.

Direct solution of the discretized poisson—Neumann problem on a domain composed of rectangles

Finite-difference approximations for the two-dimensional Poisson-Neumann problem ( 1 λ) div λ grad p = q , λ > 0on R λ n · grad p = g R on ∂R result in a large system of linear equations Lp = q. The n × n matrix L is singular with rank( L) = n − 1. A solution exists if q satisfies the consistency condition v T q = 0, where L T V = 0, v ≠ 0. A direct-solution scheme is described for the case where R can be decomposed into a set of rectangular domains each having a possibly different but constant material coefficient λ. We assume that this problem has to be solved repeatedly for many vectors q. The method is efficient in that the number of operations is of order n log n. This efficiency is gained by using fast elliptic solvers for each single rectangle and a proper variant of the socalled influence- or capacitance-matrix technique. Here, on each rectangle a separate Poisson-Neumann problem has to be solved. The essential point is the manner in which the consistency condition is enforced for each rectangle. The new method has been successfully applied in a code FLUX to analyze fluid-structure interactions.

ITERATIVE NONLINEAR THERMAL ANALYSIS OF DIRECTIONAL SOLIDIFICATION USING A FAST ELLIPTIC SOLVER

The material characteristics of directionally solidified superalloy crystals require carefully controlled solidification rates, temperature field distributions, and maximum temperature gradients. Experimentally, these can be determined by vacuum casting of cylindrical rods. Corresponding analytical studies yield nonlinear equation systems due to radiative boundary conditions and latent heat effects, which in the past required lengthy finite-difference or finite-element solutions. For the quasi-steady freezing encountered in long cylindrical alloy bars, we employed an efficient iteration scheme based on a fast elliptic equation solver previously reported in the literature. This paper discusses the principal aspects of the computational scheme and gives results showing the two-dimensional temperature fields encountered in solidification processes subject to various input conditions.

Iterative Nonlinear Thermal Analysis of Directional Solidification Using a Fast Elliptic Solver

Iterative Nonlinear Thermal Analysis of Directional Solidification Using a Fast Elliptic Solver

Fast elliptic solvers and three-dimensional fluid-structure interactions in a pressurized water reactor

A numerical method is described for analysis of the response of internal structures in the vessel of a pressurized water reactor to a sudden depressurization (“blowdown”). The three-dimensional geometry and fluid-structure interactions are taken into account. Potential flow is assumed for a compressible fluid with constant speed of sound which is appropriate for subcooled water. Any linear-elastic dynamic model in terms of a finite number of degrees of freedom can be employed for the internal structure, in particular the core barrel; here the model “CYLDY2” is implemented. An implicit and stable time-differencing scheme is used. Spatially, finite differences or spectral approximations are introduced. The resultant set of linear equations is solved efficiently by means of fast elliptic solvers and the capacitance, or influence, matrix technique. The resultant code, FLUX2, is applied to predict the dynamics in case of the planned HDR experiment. A sensitivity analysis shows that truncation errors can be kept sufficiently small. By comparison of results with and without fluid-structure interactions it is found that the coupling has an essential effect on the resultant maximum stress.