Search

In this paper we apply the generalized Taylor­Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rig- orously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic dis- continuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also con- sidered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver. [ABSTRACT FROM AUTHOR]

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small- to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann's alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware. [ABSTRACT FROM AUTHOR]

In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94-111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577-606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of sup erconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions. [ABSTRACT FROM AUTHOR]

This paper is about the construction of a BGK Navier-Stokes (BGK-NS) solver in the discontinuous Galerkin (DG) framework. Since in the DG formulation the conservative variables and their slopes can be updated simultaneously, the flow evolution in each element involves only the flow variables in the nearest neighboring cells. Instead of using the semidiscrete approach in the Runge-Kutta discontinuous Galerkin (RKDG) method, the current DG-BGK method integrates the governing equations in time as well. Due to the coupling of advection and dissipative terms in the gas-kinetic formulation, the DG-BGK method solves the viscous governing equations directly. Numerical examples for the one-dimensional compressible Navier-Stokes solutions will be presented. [ABSTRACT FROM AUTHOR]

In this paper, we develop a numerical scheme for the Smoluchowski coagulation equation, which relies on a conservative formulation and a finite volume approach. Several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model. In particular the gelation phenomenon and the long time behavior of the solution are numerically studied. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient branch-following procedure that can be used not only to compute branches of periodic solutions of periodically forced dynamical systems but also to determine the stability of the periodic solutions. The procedure combines Broyden's method with a subspace iteration method to determine the dominant eigenvalues. The method has connections with the hybrid Newton-Picard methods developed by Lust et al. in [SIAM 1. Sci. Comput., 19 (1998) pp. 1188-12091. A convergence analysis of the procedure is presented. The method is applied to the computation of periodic states of a reverse flow reactor, and its performance is compared with two variants of the hybrid Newton-Picard method. [ABSTRACT FROM AUTHOR]

This paper studies the numerical solution of two-dimensional nonlinear degenerate reaction-diffusion differential equations with singular forcing terms over rectangular domains. The equations considered may generate strong quenching singularities. This investigation focuses on a variable time step Peaceman­Rachford splitting method for the aforementioned problem. The time adaptation is implemented based on arc-length estimations of the first time derivative of the solution. The two-dimensional problem is split into several one-dimensional problems so that the computational cost is significantly reduced. The monotonicity and localized linear stability of the variable step scheme are investigated. We give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem. It is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient transformation technique for accurate numerical evaluation of weakly singular integrals with interior singularities. The present transformation technique does not require any division of the integration interval. It is composed of two parts of a sigmoidal transformation whose tails coincide with a singular point smoothly up to the order of the sigmoidal transformation employed. Therefore, in using the standard Gauss quadrature rule, the present transformation has a feature in which the integration points are clustered into the interior singular point very closely, as in the case of endpoint singular integrals. The results of some numerical examples show the superiority of the present method over the existing methods. [ABSTRACT FROM AUTHOR]

In this paper we describe a simple algorithm for the approximation of an invariant curve T of a planar map f. We are particularly interested in the case where T is attracting but not smooth. Results for the delayed logistic map illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth-order elliptic boundary value problems in a two-dimensional domain. The boundary value problem is discretized by the finite difference method, and two iterative processes, called block Jacobi and block Gauss­Seidel monotone iterations, are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration. It is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi-monotone nondecreasing. A sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system. Similar results are obtained for quasi-monotone nonincreasing functions. An analytical comparison relation between the block Jacobi and block Gauss­Seidel monotone iterations is obtained. It is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero. Numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy. Also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations. [ABSTRACT FROM AUTHOR]

We introduce spectral element-based algebraic multigrid (ρAMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically “smooth” vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of “strong” connections (relatively large coefficients in the operator matrix). One aim of ρAMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. ρAMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for ρAMGe along with numerical experiments demonstrating its robustness. [ABSTRACT FROM AUTHOR]

In this paper, a discrete singular convolution (DSC) algorithm is introduced to solve Fisher’s equation, to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem. Two novel numerical treatments, a moving frame scheme and a new asymptotic scheme, are designed to overcome the subtle difficulties involved in the numerical solution of Fisher’s equation. The moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem. The new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long-time integrations. The resulting DSC algorithm is able to correctly predict long-time traveling wave behavior. The performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches, i.e., methods of the Fourier pseudospectral, the accurate spatial derivatives, and the Crank­Nicolson schemes. The spatial and temporal accuracies of all four methods are analyzed and numerically verified. [ABSTRACT FROM AUTHOR]

The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approx- imated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier­Stokes equations, N ≤ 4. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small. [ABSTRACT FROM AUTHOR]

This paper presents a new Lagrangian approach for the two-dimensional (2-D) time-dependent Euler equations. It may be considered as a sequel to the authors' previous Lagrangian approaches for steady supersonic flow computations [C. Y. Loh and W. H. Hui, J. Comput. Phys., 89 (1990), pp. 207­240; W. H. Hui and C. Y. Loh, J. Comput. Phys., 103 (1992), pp. 450­464; W. H. Hui and C. Y. Loh, J. Comput. Phys., 103 (1992), pp. 465­471; C. Y. Loh and M. S. Liou, J. Comput. Phys., 104 (1993), pp. 150­161; C. Y. Loh and M. S. Liou, SIAM J. Sci. Comput., 15 (1994), pp. 1038­1058; C. Y. Loh and M. S. Liou, J. Comput. Phys., 113 (1994), pp. 224­248]. The theoretical background and the intrinsic flow coordinates as well as the Lagrangian conservation form are introduced based on the concept of material functions (or path functions). A TVD scheme of the Godunov type is chosen to describe the numerical procedure. Several examples are then given to justify the claimed advantages of the new methodology, namely, (a) any contact discontinuities are crisply solved and (b) grids are automatically and accurately generated following pathlines. [ABSTRACT FROM AUTHOR]

Multilevel algorithms are a successful class of optimization techniques which addresses the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimization method which refines the partition at each graph level. In this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions. We also present a formulation of the Kernighan­Lin partition optimization algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-of-the-art partitioner and shown to provide improved results. [ABSTRACT FROM AUTHOR]

Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme. [ABSTRACT FROM AUTHOR]

In this paper we introduce a moving mesh method for solving PDEs in two dimensions. It can be viewed as a higher-dimensional generalization of the moving mesh PDE (MMPDE) strategy developed in our previous work for one-dimensional problems [W. Huang, Y. Ren, and R. D. Russell, SIAM J. Numer. Anal., 31 (1994), pp. 709–730]. The MMPDE is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps. Geometrical interpretations are given for the gradient equation and functional, and basic properties of this MMPDE are discussed. Numerical examples are presented where the method is used both for mesh generation and for solving time-dependent PDEs. The results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality. [ABSTRACT FROM AUTHOR]