Number Of Mesh Points Research Articles

Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity

AbstractA preconditioning iterative algorithm is proposed for solving electromagnetic scattering from an open cavity embedded in an infinite ground plane. In this iterative algorithm, a physical model with a vertically layered medium is employed as a preconditioner of the model of general media. A fast algorithm developed in (SIAM J. Sci. Comput. 2005; 27:553–574) is applied for solving the model of layered media and classical Krylov subspace methods, restarted GMRES, COCG, and BiCGstab are employed for solving the preconditioned system. Our numerical experiments on cavity models with large numbers of mesh points and large wave numbers show that the algorithm is efficient and the number of iterations is independent of the number of mesh points and dependent upon the wave number. Copyright © 2008 John Wiley & Sons, Ltd.

Electromagnetic transitions of the hydrogen atom in a magnetic field by the Lagrange-mesh method

Very accurate wavefunctions obtained with the Lagrange-mesh method are employed to calculate dipole strengths and the dominant quadrupole strength in the high-field domain (γ ⩾ 1) within the infinite-proton-mass approximation. Calculations are performed in the semi-parabolic and spherical coordinate systems. The accuracy is tested by varying the number of mesh points and by comparison between both coordinate systems. Comparison between the position (or length) and velocity expressions of the dipole strength is found not to be a reliable test as it overestimates the accuracy. We confirm that published tables have in general a four-digit accuracy. The Lagrange-mesh technique provides a simple and fast way of obtaining similar results at arbitrary fields with γ ⩽ 1000. Using the semi-parabolic coordinate system is simpler and more robust.

DEVELOPMENT OF A THREE-POINT SIXTH-ORDER HELMHOLTZ SCHEME

A compact finite-difference scheme is presented in this paper for efficiently solving the variable Helmholtz equation. This scheme development involves relating the derivative terms, namely, uxx and uxxxx, to u at the two adjacent nodal points in order to yield a 3-point stencil implicit scheme. In addition to sixth-order accuracy, the proposed scheme can conditionally provide oscillation-free solutions using the underlying M-matrix theory. Computations have been carried out for several problems which are amenable to exact solutions. Agreement with the exact solutions is excellent even in a grid system involving fewer number of mesh points. Thus, the integrity of the compact scheme and the efficiency of the alternating direction implicit algorithm are demonstrated. The proposed scheme is applied to study the effect of variable wave number on the wave propagation characteristics for the problem, which is partly bounded by a scatter surface.

Confined hydrogen atom by the Lagrange-mesh method: Energies, mean radii, and dynamic polarizabilities

The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. The hydrogen atom confined in a sphere is studied with Lagrange-Legendre basis functions vanishing at the center and surface of the sphere. For various confinement radii, accurate energies and mean radii are obtained with small numbers of mesh points, as well as dynamic dipole polarizabilities. The wave functions satisfy the cusp condition with 11-digit accuracy.

ROBUST NUMERICAL METHOD FOR A SYSTEM OF SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS ON A RECTANGLE

A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction‐diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ϵ tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ϵ‐uniformly at the rate O ‘N−2 ln2 N + N0 −1) , where N = mins Ns, s = 1, 2, Ns + 1 and N0 + 1 are the numbers of mesh points on the axis xs and on the axis t, respectively.

Inter-comparison of the g-, f- and p-modes calculated using different oscillation codes for a given stellar model

In order to make astroseismology a powerful tool to explore stellar interiors, different numerical codes should give the same oscillation frequencies for the same input physics. This work is devoted to test, compare and, if needed, optimize the seismic codes used to calculate the eigenfrequencies to be finally compared with observations. The oscillation codes of nine research groups in the field have been used in this study. The same physics has been imposed for all the codes in order to isolate the non-physical dependence of any possible difference. Two equilibrium models with different grids, 2172 and 4042 mesh points, have been used, and the latter model includes an explicit modelling of semiconvection just outside the convective core. Comparing the results for these two models illustrates the effect of the number of mesh points and their distribution in particularly critical parts of the model, such as the steep composition gradient outside the convective core. A comprehensive study of the frequency differences found for the different codes is given as well. These differences are mainly due to the use of different numerical integration schemes. The use of a second-order integration scheme plus a Richardson extrapolation provides similar results to a fourth-order integration scheme. The proper numerical description of the Brunt-Vaisala frequency in the equilibrium model is also critical for some modes. An unexpected result of this study is the high sensitivity of the frequency differences to the inconsistent use of values of the gravitational constant (G) in the oscillation codes, within the range of the experimentally determined ones, which differ from the value used to compute the equilibrium model.

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. When ɛ vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ɛ tends to zero, a parabolic boundary layer with a characteristic width ɛ appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ɛ-uniformly at a rate of O(N −2ln2 N + N 0 −1 , where \( N = \mathop {\min }\limits_s N_s \), N s + 1 and N 0 + 1 are the numbers of mesh points on the axes x s and t, respectively.

OPTIMAL DIFFERENCE SCHEMES ON PIECEWISE‐UNIFORM MESHES FOR A SINGULARLY PERTURBED PARABOLIC CONVECTION‐DIFFUSION EQUATION

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic convection‐diffusion equation. For this problem, upwind difference schemes on the well‐known piecewise‐uniform meshes converge ϵ‐uniformly in the maximum discrete norm at the rate O(N− 1lnN + N0 −1 ), where N + 1 and N 0 + 1 are the number of mesh points in x and t respectively; the number of nodes in the x‐mesh before the transition point (the point where the step‐size changes) and after it are the same. Under the condition N Â N 0 this scheme converges at the rate O(P−1/2 ln P); here P = (N + 1)(N 0 + 1) is the total number of nodes in the piecewise‐uniform mesh. Schemes on piecewise‐uniform meshes are constructed that are optimal with respect to the convergence rate. These schemes converge ϵ‐uniformly at the rate O(P−1/2 ln1/2 P). In optimal meshes based on widths that are similar to Kolmogorov's widths, the ratio of mesh points in x and t is of O((ϵ + ln−1 P)−1). Under the condition ϵ = o( 1), most nodes in such a mesh in x are placed before the transition point.

Numerical solutions of linear and nonlinear singular perturbation problems

A new method is developed by detecting the boundary layer of the solution of a singular perturbation problem. On the non-boundary layer domain, the singular perturbation problem is dominated by the reduced equation which is solved with standard techniques for initial value problems. While on the boundary layer domain, it is controlled by the singular perturbation. Its numerical solution is provided with finite difference methods, of which up to sixth order methods are developed. The numerical error is maintained at the same level with a constant number of mesh points for a family of singular perturbation problems. Numerical experiments support the analytical results.

Performance of Multigrid in the Context of Beam Dynamics Simulations

Precise and fast 3D space charge calculations for bunches or clouds of charged particles are of growing importance in design studies for future linear accelerators and light sources. One of the possible approaches is the computation of the potential of the bunch in the rest frame by means of Poisson's equation. The software package MOEVE has been developed for space charge calculations on non-equidistant grids. It consists of several iterative Poisson solvers (MOEVE Poisson solvers), among them the state-of-the-art multigrid Poisson solver. Furthermore, the MOEVE Poisson solvers have been implemented in the tracking code Astra and GPT. In this paper the algorithms of the software package MOEVE will be described and the performance will be tested for very large linear systems of equations. The numerical results will show that the conditions for the Poisson solvers have to be chosen carefully in order to achieve optimal results for real life applications. The design of future light sources and colliders requires increasingly precise 3D beam dynamics sim- ulations. In so-called tracking simulations the particle trajectory is determined which is described by the relativistic equation of motion. The equation of motion is solved by means of an appropriate time integration scheme. In regimes of rather low energy, this implies that the space charge flelds have to be taken into account in each time step of the numerical integration. Recently, the e-cient calculation of 3D space charge flelds gained particular importance in the context of electron cloud studies for the ILC (International Linear Collider) damping rings. Based on the geometric multigrid technique fast Poisson solvers have been developed and suc- cessfully applied for 3D space charge simulations. In theory, these multigrid Poisson solvers have optimal performance, i.e., the numerical efiort depends linearly on the number of mesh points. Unfortunately, this optimal convergence rate can sometimes not be achieved in simulations of real life problems. In this paper the performance of the geometric multigrid technique is investigated in the context of space charge calculations, in particular, for huge numbers of mesh points, i.e., up to 4 million. Another problem is the behavior of the multigrid Poisson solver within the particle tracking procedure. Since space charge flelds have to be computed in each time step of the numer- ical integration, the calculated flelds of the previous time step can be used as initial guess for the iterative Poisson solver. With this approach the efiort for the new space charge calculation can be reduced. The question under investigation is how multigrid can compete with other iterative algorithms for real life tracking simulations. The iterative Poisson solvers are available as software package MOEVE 2.0 (MOEVE: Multigrid for non-equidistant grids to solve Poisson's equation) (8). Furthermore, these Poisson solvers are implemented in the tracking code Astra (DESY, Hamburg, Germany) (3) and the tracking code GPT (Pulsar Physics, Eindhoven, The Netherlands) (13). 2. MATHEMATICAL MODEL Space charge calculations for beam dynamics studies are performed within a tracking procedure. The tracking is a method to determine the trajectories of the particles which are described by the relativistic equations of motion (1). The equations of motion are solved by means of an appropriate time integration scheme. The space charge flelds have to be taken into account in each time step of the numerical integration. The space charge calculations are performed in the rest frame of the bunch by means of Poisson's equation given by i¢' = % 0 in › ‰R 3 ;

Mathematical test criteria for filtering complex systems: Plentiful observations

An important emerging scientific issue is the real time filtering through observations of noisy turbulent signals for complex systems as well as the statistical accuracy of spatio-temporal discretizations for such systems. These issues are addressed here in detail for the setting with plentiful observations for a scalar field through explicit mathematical test criteria utilizing a recent theory [A.J. Majda, M.J. Grote, Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems, Proceedings of the National Academy of Sciences 104 (4) (2007) 1124–1129]. For plentiful observations, the number of observations equals the number of mesh points. These test criteria involve much simpler decoupled complex scalar filtering test problems with explicit formulas and elementary numerical experiments which are developed here as guidelines for filter performance. The theory includes information criteria to avoid filter divergence with large model errors, asymptotic Kalman gain, filter stability, and accurate filtering with small ensemble size as well as rigorous results delineating the role of various turbulent spectra for filtering under mesh refinement. These guidelines are also applied to discrete approximations for filtering the stochastically forced dissipative advection equation with very turbulent and noisy signals with either an equipartition of energy or −5/3 turbulent spectrum with infrequent observations as severe test problems. The theory and companion simulations demonstrate accurate statistical filtering in this context with implicit schemes with large time step with very small ensemble sizes and even with unstable explicit schemes under appropriate circumstances provided the filtering strategies are guided by the off-line theoretical criteria. The surprising failure of other strongly stable filtering strategies is also explained through these off-line criteria.

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S1L. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ɛ-uniformly at a rate of O(N−1lnN + N0), where N and N0 define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N−1 + N0−1 ≪ ɛ. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S1L with respect to x and t, the convergence under the condition N−1 + N0−1 ≤ ɛ1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ɛ-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ɛ-uniformly at a rate of O(N−1lnN + N0).

Numerical analysis of the tip and root vortex position in the wake of a wind turbine

The stability of tip and root vortices are studied numerically in order to analyse the basic mechanism behind the break down of tip and root vortices. The simulations are performed using the CFD program "EllipSys3D". In the computations the so-called Actuator Line Method is used, where the blades are represented by lines of body forces representing the loading. The forces on the lines are implemented using tabulated aerodynamic aerofoil data. In this way, computer resources are used more efficiently since the number of mesh points locally around the blade is decreased, and they are instead concentrated in the wake behind the blades. We here present results of computed flow fields and evaluate the flow behaviour in the wake. In particular we compare the position of the root vortices as to the azimuthal position of the tip votices.

Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ɛi2 (i = 1, 2). The parameters ɛi take arbitrary values in the half-open interval (0, 1]. When the vector parameter ɛ = (ɛ1, ɛ2) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ɛ1 and (or) ɛ2 tend to zero, a double boundary layer with the characteristic width ɛ1 and ɛ2 appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ɛ-uniformly at the rate of O(N−2ln2N) are constructed, where N = min Ns, Ns + 1 is the number of mesh points on the axis xs.

Numerical Simulation of Subcooled Nucleate Boiling by Coupling Level-Set Method with Moving-Mesh Method

A new numerical procedure coupling the level-set method with the moving-mesh method to simulate subcooled nucleate pool boiling is proposed. Numerical test problems have validated this new method. The simulation of bubble dynamics during nucleate boiling under liquid subcooling shows that this novel adaptive method is more accurate in determining interfacial heat transfer than a computational method based on uniform grids with the same number of mesh points.

Current Status of ESTA-Task2

A report on the current status of the ESTA- Task 2 (oscillation frequencies comparison) is presented. Several types of comparisons have been made using the frequencies provided by different numerical codes. The equilibrium model is the same for all of them, and additionally some requirements such as boundary conditions, no re-meshing, etc, are also imposed. Some unexpected differences appear in the frequency and large separation comparisons. In the high frequency region these differences can be due to the order of the integration scheme used to solve the partial differential equations (or to the lack of number of mesh points in the outer layers). In the region around the fundamental radial mode and the low frequency region, important differences appear for the avoided crossing and the trapped modes respectively, the reason maybe due to the different sensitivity of each code to the profile of the Brunt-Vaisala frequency. The next step will be to test the accuracy of the codes using a model with a larger number of mesh points and with a better described Brunt-Vaisala frequency.

Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton–Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference points and order all the nodes according to their $l^p$‐metrics to those reference points. We show that these orderings satisfy the two most important properties underlying the fast sweeping method: (1) these orderings can cover all directions of information propagating efficiently; (2) any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings. We prove the convergence of the new algorithm. The computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of $O(M)$ flops for iteration steps and $O(M{\rm log}M)$ flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method, where M is the total number of mesh points. Extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size.

A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations

Dirichlet problem is considered for a singularly perturbed semilinear parabolic convection-diffusion equation on a rectangular domain. The solution of the classical finite difference scheme on a uniform mesh converges at the rate O((e + N−1)−1 N−1 + N0−) where N + 1 and N0 + 1 denote the numbers of mesh points with respect to χ and t respectively, e ∈ (0, 1] is the perturbation parameter. Using nonlinear and linearised basic classical schemes, finite difference schemes on a posteriori adaptive meshes based on uniform subgrids are constructed. The subdomains where grid refinement is required are defined by the gradients of the solutions of the intermediate discrete problems. The constructed difference schemes converge 'almost e-uniformly', namely, at the rate O((e−υN−1) + N−1/2 + N0−1) where υ is an arbitrary number from (0, 1].

Coarse-graining approximation for simulating surface reaction kinetics in particulate systems

When systems of reacting solid or liquid particles are explicitly modeled on a computational mesh, a wide distribution of initial particle sizes can lead to undesirably large numbers of mesh points. A coarse graining method is presented that avoids this problem by essentially grouping the finest particles together into a number of larger clusters. The excess surface area of these clusters is incorporated into a bias factor that can be multiplied by the intrinsic reaction rate constant to approximate the enhanced reaction rate of the clusters relative to actual particles of the same size. The bias factor can be calculated directly from knowledge of mesh point spacing and of the particle size distribution. Examples are given of the value that the bias factor can have for a model distribution of particle sizes, using computational meshes with varying resolution limits.

Lagrange‐mesh method for quantum‐mechanical problems

AbstractThe Lagrange‐mesh method is an approximate variational calculation with the simplicity of a mesh calculation because of the use of a consistent Gauss quadrature. The method can be used for quantum‐mechanical bound‐state and scattering problems with local and non‐local interactions or with a semi‐relativistic kinetic energy, and for solving the time‐dependent Schrödinger equation. No analytical evaluation of matrix elements is needed. The accuracy is exponential in the number of mesh points and is not significantly reduced with respect to the corresponding variational calculations. Lagrange functions can be based on classical or non‐classical orthogonal polynomials as well as on trigonometric and sinc functions, as described in examples. Some singularities can be handled with a regularization technique. Applications to three‐body systems are discussed for various choices of coordinate systems. The helium atom and hydrogen molecular ion are presented as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)