Numerical solution of electromagnetic scattering from a large partly covered cavity

The paper focuses on the numerical study of electromagnetic scattering from two-dimensional (2D) large partly covered cavities, which is described by the Helmholtz equation with a nonlocal boundary condition on the aperture. The classical five-point finite difference method is applied for the discretization of the Helmholtz equation and a linear approximation is used for the nonlocal boundary condition. We prove the existence and uniqueness of the numerical solution when the medium in the cavity is y-direction layered or the number of the mesh points on the aperture is large enough. The fast algorithm proposed in Bao and Sun (2005) [2] for open cavity models is extended to solving the partly covered cavity problem with (vertically) layered media. A preconditioned Krylov subspace method is proposed to solve the partly covered cavity problem with a general medium, in which a layered medium model is used as a preconditioner of the general model. Numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ILU-type preconditioning algorithms. Our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems, particularly with large wave numbers.     

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u(x,y,z) and its Laplacian ∇²u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.     

A finite-element capacitance matrix method for exterior Helmholtz problems

Summary. We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation. The problem is reformulated as an equivalent one on a bounded domain using an exact non-local boundary condition on a circular artificial boundary. An FFT-based fast Helmholtz solver is then derived for a finite-element discretization on an annular domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation. Received May 2, 1995     

Wavelet-Galerkin method for solving parabolic equations in finite domains

A novel wavelet-Galerkin method tailored to solve parabolic equations in finite domains is presented. The emphasis of the paper is on the development of the discretization formulations that are specific to finite domain parabolic equations with arbitrary boundary conditions based on weak form functionals. The proposed method also deals with the development of algorithms for computing the associated connection coefficients at arbitrary points. Here the Lagrange multiplier method is used to enforce the essential boundary conditions. The numerical results on a two-dimensional transient heat conducting problem are used to validate the proposed wavelet-Galerkin algorithm as an effective numerical method to solve finite domain parabolic equations.     

Time stepping for vectorial operator splitting

We present a fully implicit finite difference method for the unsteady incompressible Navier-Stokes equations. It is based on the one-step θ-method for discretization in time and a special coordinate splitting (called vectorial operator splitting) for efficiently solving the nonlinear stationary problems for the solution at each new time level. The resulting system is solved in a fully coupled approach that does not require a boundary condition for the pressure. A staggered arrangement of velocity and pressure on a structured Cartesian grid combined with the fully implicit treatment of the boundary conditions helps us to preserve the properties of the differential operators and thus leads to excellent stability of the overall algorithm. The convergence properties of the method are confirmed via numerical experiments.     

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Diagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations.     

A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Fast transforms for tridiagonal linear equations

In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern ²×n ² in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.     

A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.     

An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter λ intrinsically, which can be chosen properly so that the semi‐implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method.     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.      

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirich-let boundary conditions in a convex polygonal domain in the plane.This new class of finite elements,which is called composite finite elements,was first introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial differential equations on domains with complicated geometry.The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving par-tial differential equations by domain discretization method.The composite finite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the fine-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the finite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L∞(L2)-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.     

求解块三对角方程组的新算法

根据块三对角矩阵的特殊分解,给出了求解块三对角方程组的新算法.该算法含有可以选择的参数矩阵,适当选择这些参数矩阵,可以使得计算精度较著名的追赶法高,甚至当追赶法失效时,由该算法仍可得到一定精度的解.     

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W_j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W_j converge to 0 exponentially quickly for large enough j. If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.     

Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.     

Numerical integration of a class of singular perturbation problems

In this paper, we discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments have been included to demonstrate the efficiency of the method.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

Boundary equations in the finite transfer method for solving differential equation systems

The finite transfer method is going to be used to solve a p system of linear ordinary differential equations. The complete problem is extended by adding the p boundary equations involved. It is chosen a fourth order scheme to obtain finite transfer expressions. A recurrence strategy is used in these equations and permits one to relate different points in the domain where boundary equations are defined. Finally a 2p algebraic system of equations is noted and solved. To show the efficiency and accuracy, the method is applied to determine the structural behavior of a bending beam with different supports and to solve a differential equation of second degree with different boundary conditions.