Home About us Contact | |||

## Poisson Equation (poisson + equation)
Kinds of Poisson Equation
## Selected Abstracts## General Gyrokinetic Equations for Edge Plasmas CONTRIBUTIONS TO PLASMA PHYSICS, Issue 7-9 2006H. QinAbstract During the pedestal cycle of H-mode edge plasmas in tokamak experiments, large-amplitude pedestal build-up and destruction coexist with small-amplitude drift wave turbulence. The pedestal dynamics simultaneously includes fast time-scale electromagnetic instabilities, long time-scale turbulence-induced transport processes, and more interestingly the interaction between them. To numerically simulate the pedestal dynamics from first principles, it is desirable to develop an effective algorithm based on the gyrokinetic theory. However, existing gyrokinetic theories cannot treat fully nonlinear electromagnetic perturbations with multi-scale-length structures in spacetime, and therefore do not apply to edge plasmas. A set of generalized gyrokinetic equations valid for the edge plasmas has been derived. This formalism allows large-amplitude, time-dependent background electromagnetic fields to be developed fully nonlinearly in addition to small-amplitude, short-wavelength electromagnetic perturbations. It turns out that the most general gyrokinetic theory can be geometrically formulated. The Poincaré-Cartan-Einstein 1-form on the 7D phase space determines particles' worldlines in the phase space, and realizes the momentum integrals in kinetic theory as fiber integrals. The infinitesimal generator of the gyro-symmetry is then asymptotically constructed as the base for the gyrophase coordinate of the gyrocenter coordinate system. This is accomplished by applying the Lie coordinate perturbation method to the Poincaré-Cartan-Einstein 1-form. General gyrokinetic Vlasov-Maxwell equations are then developed as the Vlasov-Maxwell equations in the gyrocenter coordinate system, rather than a set of new equations. Because the general gyrokinetic system developed is geometrically the same as the Vlasov-Maxwell equations, all the coordinate-independent properties of the Vlasov-Maxwell equations, such as energy conservation, momentum conservation, and phase space volume conservation, are automatically carried over to the general gyrokinetic system. The pullback transformation associated with the coordinate transformation is shown to be an indispensable part of the general gyrokinetic Vlasov-Maxwell equations. As an example, the pullback transformation in the gyrokinetic Poisson equation is explicitly expressed in terms of moments of the gyrocenter distribution function, with the important gyro-orbit squeezing effect due to the large electric field shearing in the edge and the full finite Larmour radius effect for short wavelength fluctuations. The familiar "polarization drift density" in the gyrocenter Poisson equation is replaced by a more general expression. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Numerical stability of unsteady stream-function vorticity calculations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2003E. SousaAbstract The stability of a numerical solution of the Navier,Stokes equations is usually approached by con- sidering the numerical stability of a discretized advection,diffusion equation for either a velocity component, or in the case of two-dimensional flow, the vorticity. Stability restrictions for discretized advection,diffusion equations are a very serious constraint, particularly when a mesh is refined in an explicit scheme, so an accurate understanding of the numerical stability of a discretization procedure is often of equal or greater practical importance than concerns with accuracy. The stream-function vorticity formulation provides two equations, one an advection,diffusion equation for vorticity and the other a Poisson equation between the vorticity and the stream-function. These two equations are usually not coupled when considering numerical stability. The relation between the stream-function and the vorticity is linear and so has, in principle, an exact inverse. This allows an algebraic method to link the interior and the boundary vorticity into a single iteration scheme. In this work, we derive a global time-iteration matrix for the combined system. When applied to a model problem, this matrix formulation shows differences between the numerical stability of the full system equations and that of the discretized advection,diffusion equation alone. It also gives an indication of how the wall vorticity discretization affects stability. Despite the added algebraic complexity, it is straightforward to use MATLAB to carry out all the matrix operations. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005Daisuke IshiharaAbstract This paper describes a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve an interaction problem of incompressible viscous fluid and an elastic body. The PPE is derived so as to be consistent with the coupled equation system for the fluid-structure interaction (FSI). Based on this approach, we develop two kinds of efficient monolithic methods. In both methods, the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint, and all other unknown variables are derived fully explicitly or partially explicitly. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties. The arbitrary Lagrangian,Eulerian (ALE) method is employed for the fluid part in order to take into account the deformable fluid-structure interface. To demonstrate fundamental performances of the proposed approach, the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003N. SukumarArticle first published online: 11 MAR 200Abstract Voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two-fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well-known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive-definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Bounds on outputs of the exact weak solution of the three-dimensional Stokes problem INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2009Zhong ChengAbstract A method for obtaining rigorous upper and lower bounds on an output of the exact weak solution of the three-dimensional Stokes problem is described. Recently bounds for the exact outputs of interest have been obtained for both the Poisson equation and the advection-diffusion-reaction equation. In this work, we extend this approach to the Stokes problem where a novel formulation of the method also leads to a simpler flux calculation based on the directly equilibrated flux method. To illustrate this technique, bounds on the flowrate are calculated for an incompressible creeping flow driven by a pressure gradient in an endless square channel with an array of rectangular obstacles in the center. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A collocated, iterative fractional-step method for incompressible large eddy simulation INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Giridhar JothiprasadAbstract Fractional-step methods are commonly used for the time-accurate solution of incompressible Navier,Stokes (NS) equations. In this paper, a popular fractional-step method that uses pressure corrections in the projection step and its iterative variants are investigated using block-matrix analysis and an improved algorithm with reduced computational cost is developed. Since the governing equations for large eddy simulation (LES) using linear eddy-viscosity-based sub-grid models are similar in form to the incompressible NS equations, the improved algorithm is implemented in a parallel LES solver. A collocated grid layout is preferred for ease of extension to curvilinear grids. The analyzed fractional-step methods are viewed as an iterative approximation to a temporally second-order discretization. At each iteration, a linear system that has an easier block-LU decomposition compared with the original system is inverted. In order to improve the numerical efficiency and parallel performance, modified ADI sub-iterations are used in the velocity step of each iteration. Block-matrix analysis is first used to determine the number of iterations required to reduce the iterative error to the discretization error of. Next, the computational cost is reduced through the use of a reduced stencil for the pressure Poisson equation (PPE). Energy-conserving, spatially fourth-order discretizations result in a 7-point stencil in each direction for the PPE. A smaller 5-point stencil is achieved by using a second-order spatial discretization for the pressure gradient operator correcting the volume fluxes. This is shown not to reduce the spatial accuracy of the scheme, and a fourth-order continuity equation is still satisfied to machine precision. The above results are verified in three flow problems including LES of a temporal mixing layer. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Pressure segregation methods based on a discrete pressure Poisson equation. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008An algebraic approachAbstract In this paper, we introduce some pressure segregation methods obtained from a non-standard version of the discrete monolithic system, where the continuity equation has been replaced by a pressure Poisson equation obtained at the discrete level. In these methods it is the velocity instead of the pressure the extrapolated unknown. Moreover, predictor,corrector schemes are suggested, again motivated by the new monolithic system. Key implementation aspects are discussed, and a complete stability analysis is performed. We end with a set of numerical examples in order to compare these methods with classical pressure-correction schemes. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A 3-D non-hydrostatic pressure model for small amplitude free surface flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2006J. W. LeeAbstract A three-dimensional, non-hydrostatic pressure, numerical model with k,, equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non-hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure,Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Pressure boundary condition for the time-dependent incompressible Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2006R. L. SaniAbstract In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111,1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a so-called equivalence theorem (assertion) that stated/asserted that if the Navier,Stokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) would be exactly the same as if the ,primitive' equations, in which the PPE plus Neumann BC is replaced by the usual divergence-free constraint (, · u = 0), were solved instead. This issue is explored in sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the original/primitive equations that require no BC for the pressure, the new results establish the same thing when the PPE approach is employed. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Further experiences with computing non-hydrostatic free-surface flows involving water waves INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005Marcel ZijlemaAbstract A semi-implicit, staggered finite volume technique for non-hydrostatic, free-surface flow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non-hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary-fitted co-ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller-box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An efficient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and efficiency of the proposed method are presented. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Coupled lattice-Boltzmann and finite-difference simulation of electroosmosis in microfluidic channels INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2004Dzmitry HlushkouAbstract In this article we are concerned with an extension of the lattice-Boltzmann method for the numerical simulation of three-dimensional electroosmotic flow problems in porous media. Our description is evaluated using simple geometries as those encountered in open-channel microfluidic devices. In particular, we consider electroosmosis in straight cylindrical capillaries with a (non)uniform zeta-potential distribution for ratios of the capillary inner radius to the thickness of the electrical double layer from 10 to 100. The general case of heterogeneous zeta-potential distributions at the surface of a capillary requires solution of the following coupled equations in three dimensions: Navier,Stokes equation for liquid flow, Poisson equation for electrical potential distribution, and the Nernst,Planck equation for distribution of ionic species. The hydrodynamic problem has been treated with high efficiency by code parallelization through the lattice-Boltzmann method. For validation velocity fields were simulated in several microcapillary systems and good agreement with results predicted either theoretically or obtained by alternative numerical methods could be established. Results are also discussed with respect to the use of a slip boundary condition for the velocity field at the surface. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Numerical methods for large-eddy simulation in general co-ordinates INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2004Gefeng TangAbstract Large scale unsteady motions in many practical engineering flows play a very important role and it is very unlikely that these unsteady flow features can be captured within the framework of Reynolds averaged Navier,Stokes approach. Large-eddy simulation (LES) has become, arguably, the only practical numerical tool for predicting those flows more accurately since it is still not realistic to apply DNS to practical engineering flows with the current and near future available computing power. Numerical methods for the LES of turbulent flows in complex geometry have been developed and applied to predict practical engineering flows successfully. The method is based on body-fitted curvilinear co-ordinates with the contravariant velocity components of the general Navier,Stokes equations discretized on a staggered orthogonal mesh. For incompressible flow simulations the main source of computational expense is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. A multigrid 3D pressure solver is developed to speed up the solution. In addition, the Poisson equation for pressure takes a simpler form with no cross-derivatives when orthogonal mesh is used and hence resulting in increased convergence rate and producing more accurate solutions. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## A high-order finite difference method for incompressible fluid turbulence simulations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2003Eric VedyAbstract A Hermitian,Fourier numerical method for solving the Navier,Stokes equations with one non-homogeneous direction had been presented by Schiestel and Viazzo (Internat. J. Comput. Fluids 1995; 24(6):739). In the present paper, an extension of the method is devised for solving problems with two non-homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian finite differences in the non-periodic directions whereas Fourier pseudo-spectral developments are used in the remaining periodic direction. Pressure,velocity coupling is solved by a simplified Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent flow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional flows. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Computation of unsteady viscous incompressible flows in generalized non-inertial co-ordinate system using Godunov-projection method and overlapping meshes INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2002H. PanAbstract Time-dependent incompressible Navier,Stokes equations are formulated in generalized non-inertial co-ordinate system and numerically solved by using a modified second-order Godunov-projection method on a system of overlapped body-fitted structured grids. The projection method uses a second-order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence-free vector fields. The second-order Godunov method is applied for numerically approximating the non-linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving-boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two- and three-dimensional flow problems formulated in the non-inertial co-ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## An approximate projection method for incompressible flow INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2002David E. StevensThis paper presents an approximate projection method for incompressible flows. This method is derived from Galerkin orthogonality conditions using equal-order piecewise linear elements for both velocity and pressure, hereafter Q1Q1. By combining an approximate projection for the velocities with a variational discretization of the continuum pressure Poisson equation, one eliminates the need to filter either the velocity or pressure fields as is often needed with equal-order element formulations. This variational approach extends to multiple types of elements; examples and results for triangular and quadrilateral elements are provided. This method is related to the method of Almgren et al. (SIAM J. Sci. Comput. 2000; 22: 1139,1159) and the PISO method of Issa (J. Comput. Phys. 1985; 62: 40,65). These methods use a combination of two elliptic solves, one to reduce the divergence of the velocities and another to approximate the pressure Poisson equation. Both Q1Q1 and the method of Almgren et al. solve the second Poisson equation with a weak error tolerance to achieve more computational efficiency. A Fourier analysis of Q1Q1 shows that a consistent mass matrix has a positive effect on both accuracy and mass conservation. A numerical comparison with the widely used Q1Q0 (piecewise linear velocities, piecewise constant pressures) on a periodic test case with an analytic solution verifies this analysis. Q1Q1 is shown to have comparable accuracy as Q1Q0 and good agreement with experiment for flow over an isolated cubic obstacle and dispersion of a point source in its wake. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Three-dimensional numerical modelling of free surface flows with non-hydrostatic pressure INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2002Musteyde B. KoēyigitAbstract A three-dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds-averaged Navier,Stokes equations with a non-hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co-ordinate system, with a semi-implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five-diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non-hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind-induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## GENSMAC3D: a numerical method for solving unsteady three-dimensional free surface flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001M.F. ToméAbstract A numerical method for solving three-dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier,Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user-friendly input data file is employed to allow almost any arbitrary three-dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Numerical simulation of high-Reynolds number flow around circular cylinders by a three-step FEM,BEM model INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2001D. L. YoungAbstract An innovative computational model, developed to simulate high-Reynolds number flow past circular cylinders in two-dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier,Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three-step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high-Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Practical techniques for a three-dimensional FEM analysis of incompressible fluid flow contained with slip walls and a downstream tube bundle INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2001Yuzuru EguchiAbstract Two practical techniques are proposed in this paper to simulate a flow contained in a plenum with a downstream tube bundle under a PC environment. First, a technique to impose slip wall conditions on smooth-faced planes and sharp edges is proposed to compensate for the mesh coarseness relative to boundary layer thickness. In particular, a new type of Poisson equation is formulated to simultaneously satisfy both such velocity boundary conditions on walls and the incompressibility constraint. Second, a numerical model for a downstream tube bundle is proposed, where hydraulic resistance in a tube is imposed as a traction boundary condition on a fluid surface contacting the tube bundle end. The effectiveness of the techniques is numerically demonstrated in the application to a flow in a condenser water box. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## An adaptive remeshing technique based on hierarchical error estimates for simulation of semiconductor devices INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 1 2004Geng YangAbstract We discuss first hierarchical error estimates and a criterion for mesh refinement. Then we describe briefly the hydrodynamic model of semiconductor model. Based on artificial viscosity technique about electron velocity, we propose to solve a Poisson equation to obtain a correction about mesh optimization. Finally, we simulate a GaAs MESFET's device with a gate of 0.3 µm length and give some discussions about numerical results. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Thinking inside the box: Novel linear scaling algorithm for Coulomb potential evaluation INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 4 2006David C. ThompsonAbstract Beginning with the Poisson equation, and expanding the electronic potential in terms of sine functions, the natural orbitals for describing the particle-in-a-box problem, we find that simple analytic forms can be found for the evaluation of the Coulomb energy for both the interacting and non-interacting system of N -electrons in a box. This method is reminiscent of fast-Fourier transform and scales linearly. To improve the usefulness of this result, we generalize the idea by considering a molecular system, embedded in a box, within which we determine the electrostatic potential, in the same manner as that described for our model systems. Within this general formalism, we consider both periodic and aperiodic recipes with specific application to systems described using Gaussian orbitals; although in principle the method is seen to be completely general. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] ## Linear augmented Slater-type orbital method for free standing clusters JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 8 2009K. S. KangAbstract We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schrödinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yields the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009 [source] ## Single dopant diffusion in semiconductor technology MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2004A. GlitzkyAbstract The paper deals with the analysis of pair diffusion models in semiconductor technology. The underlying model contains reaction-drift-diffusion equations for the mobile point defects and dopant-defect pairs as well as reaction equations for immobile dopants which are coupled with a non-linear Poisson equation for the chemical potential of the electrons. For homogeneous structures we present an existence and uniqueness result for strong solutions. Starting with energy estimates we derive further a priori estimates such that fixed point arguments due to Leray,Schauder guarantee the solvability of the model equations. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains MATHEMATISCHE NACHRICHTEN, Issue 12 2008Annegret GlitzkyAbstract We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain ,0 of the domain of definition , of the energy balance equation and of the Poisson equation. Here ,0 corresponds to the region of semiconducting material, , \ ,0 represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W1,p -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Iterated Neumann problem for the higher order Poisson equation MATHEMATISCHE NACHRICHTEN, Issue 1-2 2006H. BegehrAbstract Rewriting the higher order Poisson equation ,nu = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher-order Neumann (Neumann- n ) problems for ,nu = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann- n problem for ,nu = f . The representation formula provides the tool to treat more general partial differential equations with leading term ,nu in reducing them into some singular integral equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Azimuthally symmetric theory of gravitation , I. On the perihelion precession of planetary orbits MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 3 2010G. G. NyambuyaABSTRACT From a purely non-general relativistic standpoint, we solve the empty space Poisson equation (,2,= 0) for an azimuthally symmetric setting (i.e. for a spinning gravitational system like the Sun). We seek the general solution of the form ,=,(r, ,). This general solution is constrained such that in the zeroth-order approximation it reduces to Newton's well-known inverse square law of gravitation. For this general solution, it is seen that it has implications on the orbits of test bodies in the gravitational field of this spinning body. We show that to second-order approximation, this azimuthally symmetric gravitational field is capable of explaining at least two things: (i) the observed perihelion shift of solar planets; (ii) the fact that the mean Earth,Sun distance must be increasing (this resonates with the observations of two independent groups of astronomers, who have measured that the mean Earth,Sun distance must be increasing at a rate between about 7.0 ± 0.2 m century,1 and 15.0 ± 0.3 m cy,1). In principle, we are able to explain this result as a consequence of the loss of orbital angular momentum; this loss of orbital angular momentum is a direct prediction of the theory. Further, we show that the theory is able to explain at a satisfactory level the observed secular increase in the Earth year (1.70 ± 0.05 ms yr,1). Furthermore, we show that the theory makes a significant and testable prediction to the effect that the period of the solar spin must be decreasing at a rate of at least 8.00 ± 2.00 s cy,1. [source] ## Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010B. SeiboldAbstract Large linear systems with sparse, non-symmetric matrices are known to arise in the modeling of Markov chains or in the discretization of convection,diffusion problems. Due to their potential of solving sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an algebraic multilevel iteration (AMLI) type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## A time-independent approach for computing wave functions of the Schrödinger,Poisson system NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008C.-S. ChienAbstract We describe a two-grid finite element discretization scheme for computing wave functions of the Schrödinger,Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrödinger,Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h4). Copyright © 2007 John Wiley & Sons, Ltd. [source] ## A direct Schur,Fourier decomposition for the efficient solution of high-order Poisson equations on loosely coupled parallel computers NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2006F. X. TriasAbstract In this paper a parallel direct Schur,Fourier decomposition (DSFD) algorithm for the direct solution of arbitrary order discrete Poisson equations on parallel computers is proposed. It is based on a combination of a Direct Schur method and a Fourier decomposition and allows to solve each Poisson equation almost to machine accuracy using only one communication episode. Thus, it is well suited for loosely coupled parallel computers, that have a high network latency compared with the CPU performance. Several three-dimensional direct numerical simulations (DNS) of wall-bounded turbulent incompressible flows have been carried out using the DSFD algorithm. Numerical examples illustrating the robustness and scalability of the method on a PC cluster with a conventional 100 Mbits/s network are also presented. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Fast direct solver for Poisson equation in a 2D elliptical domain NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004Ming-Chih LaiAbstract 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] |