Showing papers on "Finite difference published in 2000"

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

Abstract: In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Difference Equations: An Introduction with Applications

Abstract: Introduction. The Difference Calculus. Linear Difference Equations. Stability Theory. Asymptotic Methods. The Self-Adjoint Second Order Linear Equation. The Sturm-Liouville Problem. Discrete Calculus of Variations. Boundary Value Problems for Nonlinear Equations. Partial Difference Equations.

A Boundary Condition Capturing Method for Multiphase Incompressible Flow

Abstract: In l6r, the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid compressible Euler equations. In l11r, related techniques were used to develop a boundary condition capturing approach for the variable coefficient Poisson equation on domains with an embedded interface. In this paper, these new numerical techniques are extended to treat multiphase incompressible flow including the effects of viscosity, surface tension and gravity. While the most notable finite difference techniques for multiphase incompressible flow involve numerical smearing of the equations near the interface, see, e.g., l19, 17, 1r, this new approach treats the interface in a sharp fashion.

An unstructured grid, three-dimensional model based on the shallow water equations

Abstract: A semi-implicit finite difference model based on the three-dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi-implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.

A technique of treating negative weights in WENO schemes

Abstract: High-order accurate weighted essentially nonoscillatory (WENO) schemes have recently been developed for finite difference and finite volume methods both in structured and in unstructured meshes. A key idea in WENO scheme is a linear combination of lower order fluxes or reconstructions to obtain a higher order approximation. The combination coefficients, also called linear weights, are determined by local geometry of the mesh and order of accuracy and may become negative, such as in the central WENO schemes using staggered meshes, high-order finite volume WENO schemes in two space dimensions, and finite difference WENO approximations for second derivatives. WENO procedures cannot be applied directly to obtain a stable scheme if negative linear weights are present. The previous strategy for handling this difficulty is either by regrouping of stencils or by reducing the order of accuracy to get rid of the negative linear weights. In this paper we present a simple and effective technique for handling negative linear weights without a need to get rid of them. Test cases are shown to illustrate the stability and accuracy of this approach.

Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions: Efficient evaluation of energy and forces via the fast multipole method

Abstract: We report methodological and computational details of our Kohn-Sham density-functional method with Gaussian orbitals for systems with periodic boundary conditions. Our approach for the Coulomb problem is based on the direct space fast multipole method, which achieves not only linear scaling of computational time with system size but also very high accuracy in all infinite summations. The latter is pivotal for avoiding numerical instabilities that have previously plagued calculations with large bases, especially those containing diffuse functions. Our program also makes extensive use of other linear-scaling techniques recently developed for large clusters. Using these theoretical tools, we have implemented computational programs for energy and analytic energy gradients (forces) that make it possible to optimize geometries of periodic systems with great efficiency and accuracy. Vibrational frequencies are then accurately obtained from finite differences of forces. We demonstrate the capabilities of our methods with benchmark calculations on polyacetylene, polyphenylenevinylene, and a (5,0) carbon nanotube, employing basis sets of double zeta plus polarization quality, in conjunction with the generalized gradient approximation and kinetic-energy density-dependent functionals. The largest calculation reported in this paper contains 244 atoms and 1344 contracted Gaussians in the unit cell.

Applications of nonstandard finite difference schemes

Abstract: Nonstandard finite difference schemes, R.E. Mickens nonstandard methods for advection-diffusion-reaction equations, H.V. Kojouharov and B.M. Chen application of nonstandard finite differences to solve the wave equation and Maxwell's equations, J.B. Cole nonstandard discretization methods for some biological models, H. Al-Kahby et al an introduction to numerical integrators preserving physical properties, M.J. Gander and R. Meyer-Spasche.

Fast methods for the Eikonal and related Hamilton– Jacobi equations on unstructured meshes

Abstract: The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in Rn and on manifolds and connections to more general static Hamilton-Jacobi equations.

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples

Abstract: This article is the first of two articles on the adaptive multilevel finite element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well-known test problems. The PBE is first discretized with piece-wise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh-based finite difference or box-method algorithms, including multilevel methods, require substantially more time to reach a target PBE solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000

Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation

Abstract: In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown.

On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients

Abstract: The estimates presented here for parabolic Bellman's equations with variable coefficients extend the ones earlier obtained for constant coefficients.

A numerical method for simulating discontinuous shallow flow over an infiltrating surface

Abstract: SUMMARY A numerical method based on the MacCormack finite difference scheme is presented. The method was developed for simulating two-dimensional overland flow with spatially variable infiltration and microtopography using the hydrodynamic flow equations. The basic MacCormack scheme is enhanced by using the method of fractional steps to simplify application; treating the friction slope, a stiff source term, point-implicitly, plus, for numerical oscillation control and stability, upwinding the convective acceleration term. A higher-order smoothing operator is added to aid oscillation control when simulating flow over highly variable surfaces. Infiltration is simulated with the Green‐Ampt model coupled to the surface water component in a manner that allows dynamic interaction. The developed method will also be useful for simulating irrigation, tidal flat and wetland circulation, and floods. Copyright © 2000 John Wiley & Sons, Ltd.

Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches

Abstract: We propose an analysis (discretization techniques, convergence) of numerical schemes for Maxwell equations which use two meshes (not necessarily tetrahedral), dual to each other. Schemes of this class generalize Yee's "finite difference in time domain" method (FDTD). We distinguish network equations (the discrete equivalents of Faraday's law and Ampere's relation) which can be set up without any recourse to finite elements, and network constitutive laws, whose validity cannot be assessed without them. This establishes a complementarity between "finite integration techniques" (FIT) and the finite element method (FEM). As an example, a Yee-like method on a simplicial mesh and its so-called "orthogonal" dual, is described, and its convergence is proved.

Finite-difference tvd scheme for computation of dam-break problems

Abstract: A second-order hybrid type of total variation diminishing (TVD) finite-difference scheme is investigated for solving dam-break problems. The scheme is based upon the first-order upwind scheme and the second-order Lax-Wendroff scheme, together with the one-parameter limiter or two-parameter limiter. A comparative study of the scheme with different limiters applied to the Saint Venant equations for 1D dam-break waves in wet bed and dry bed cases shows some differences in numerical performance. An optimum-selected limiter is obtained. The present scheme is extended to the 2D shallow water equations by using an operator-splitting technique, which is validated by comparing the present results with the published results, and good agreement is achieved in the case of a partial dam-break simulation. Predictions of complex dam-break bores, including the reflection and interactions for 1D problems and the diffraction with a rectangular cylinder barrier for a 2D problem, are further implemented. The effects of bed s...

A posteriori error estimation and adaptivity for degenerate parabolic problems

Abstract: Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

Uzawa type algorithms for nonsymmetric saddle point problems

Abstract: In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier-Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.

The finite difference algorithm for higher order supersymmetry

Abstract: The higher order supersymmetric partners of the Schroedinger's Hamiltonians can be explicitly constructed by iterating a simple finite difference equation corresponding to the Baecklund transformation. The method can completely replace the Crum determinants. Its limiting, differential case offers some new operational advantages.

Blending Finite-Difference and Vortex Methods for Incompressible Flow Computations

Abstract: We describe and illustrate numerical procedures that combine grid and particle solvers for the solution of the incompressible Navier--Stokes equations. These procedures include vortex in cell (VIC) and domain decomposition schemes. Numerical comparisons with pure finite-difference methods demonstrate the effectiveness of these techniques for various flow geometries, bounded or unbounded.