Richard M. Beam

An implicit finite difference algorithm for hyperbolic systems in conservation law form

Abstract An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.

An extension ofA-stability to alternating direction implicit methods

Completely implicit, noniterative, finite-difference schemes have recently been developed by several authors for nonlinear, multidimensional systems of hyperbolic and mixed hyperbolic-parabolic partial differential equations. The method of Douglas and Gunn or the method of approximate factorization can be used to reduce the computational problem to a sequence of one-dimensional or alternating direction implicit (ADI) steps. Since the eigenvalues of partial differential equations (for example, the equations of compressible fluid dynamics) are often widely distributed with large imaginary parts,A-stable integration formulas provide ideal time-differencing approximations. In this paper it is shown that if anA-stable linear multistep method is used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation, then one can always construct an ADI scheme by the method of approximate factorization which is alsoA-stable, i.e., unconditionally stable. A more restrictive result is given for three spatial dimensions. Since necessary and sufficient conditions forA-stability can easily be determined by using the theory of positive real functions, the stability analysis of the factored partial difference equations is reduced to a simple algebraic test.

Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations

Abstract Newtons method is applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations in two spatial dimensions. The finite-difference equations are written in generalized curvilinear coordinates and strong conservation-law form and a turbulence model is included. We compute the flow field about a lifting airfoil for subsonic and transonic conditions. We investigate both the requirements for an initial guess to insure convergence and the computational efficiency of freezing the Jacobian matrices (approximate Newton method). We consider the necessity for auxiliary methods to evaluate the temporal stability of the steady-state solutions. We demonstrate the ability of Newtons method in conjunction with a continuation method to find nonunique solutions of the finite-difference equations, i.e., three different solutions for the same flow conditions.

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

Toeplitz matrices occur in many mathematical as well as scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on n...

Discrete Multiresolution Analysis Using Hermite Interpolation: Biorthogonal Multiwavelets

We generalize Hartens interpolatory multiresolution representation to include Hermite interpolation. Compact Hermite interpolation with optimal order accuracy is used in both the decomposition and reconstruction algorithm. The resulting multiple basis functions (biorthogonal multiwavelets) are symmetric or skew-symmetric, compact, and analytic. Hartens approach has several advantages: the multiresolution scheme is inherently discrete, nonperiodic boundary conditions are easy to implement, and the representation can be extended to nonuniform grids in bounded domains. We demonstrate the compression features of the new multiple basis functions by application to several examples.

Multiresolution Analysis and Supercompact Multiwavelets

The Haar wavelets can represent exactly any piecewise constant function. The motivation for the present development is Alperts family of compact orthogonal multiwavelets that can represent exactly any piecewise polynomial function. We choose to derive the algorithm in the style and notation of Hartens multiresolution analysis as extended to multiwavelets by the authors. We begin with a description of the nested grid hierarchy. Next comes the decomposition, which is the heart of the algorithm, and finally the reconstruction. The basis functions (which are nonfractal) retain the spatial compactness of the Haar basis functions, which enhances the algorithm application to nonperiodic and piecewise continuous data.

Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Abstract Implicit noniterative finite-difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time-step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture has been that unconditional stability requires implicit numerical boundary conditions. An apparent counterexample was the space-time extrapolation considered by Gustafsson, Kreiss, and Sundstrom. In this paper we examine space (implicit) and space-time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals. The results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit.

Supercompact multiwavelets for flow field simulation

Abstract A supercompact multiwavelet scheme for computational fluid dynamics is presented. Beam and Warmings supercompact wavelet method is an appropriate wavelet for fluid simulation data in the sense that it can provide compact support. The compactness of the wavelets avoids unnecessary interaction with remotely located data (e.g. across a shock discontinuity or vortex) and significantly reduces computational data processing time. Thresholding for data compression with the supercompact wavelets is applied based on a covariance vector structure of multiwavelets. The extension of this scheme to three dimensions is analyzed. Numerical tests demonstrate that the analytic advantages actually result in large data compression ratios.

On the torsional static stability and response of open section tubes subjected to thermal radiation loading

Abstract The torsional static equilibrium of structural members of open circular cross section exposed to deformation-dependent thermal loading is investigated. The equilibrium and boundary condition equations are developed. Stability criteria are established for small deformations, and equilibrium shapes are obtained for large (postbuckling) deformations. Effects of initial imperfection are introduced and solutions obtained. The stability criteria are compared with those of previous investigations. It is shown that flaws in the theory of previous investigations lead to incorrect criteria. The present theory is shown to be in good agreement with experimental results.

Factored, a-stable, linear multistep methods: an alternative to the method of lines for multidimensions

Historically, the development and analysis of methods for ordinary differential equations (ODEs) have been more advanced than those for partial differential equations (PDEs). The present state of numerical methods is no exception; therefore, it behooves the numerical analyst to exploit sophisticated ODE methods for the numerical solution of PDEs.