Christopher R. Anderson

On Vortex Methods

We give error estimates for fully discretized two- and three-dimensional vortex methods and introduce a new wy of evaluating the stretching of vorticity in three-dimensional vortex methods. The convergence theory of Beale and Majda is discussed and a simple proof of Cottet’s consistency result is presented. We also describe how to obtain accurate two-dimensional vortex methods in which the initial computational points are distributed on the nodes of nonrectangular grids, and compare several three-dimensional vortex methods.

An implementation of the fast multipole method without multipoles

An implementation is presented of the fast multipole method, which uses approximations based on Poisson’s formula. Details for the implementation in both two and three dimensions are given. Also discussed is how the multigrid aspect of the fast multipole method can be exploited to yield efficient programming procedures. The issue of the selection of an appropriate refinement level for the method is addressed. Computational results are given that show the importance of good level selection. An efficient technique that can be used to determine an optimal level to choose for the method is presented.

A method of local corrections for computing the velocity field due to a distribution of vortex blobs

Abstract A computationally efficient method for computing the velocity field due to a distribution of vortex blobs is presented. The method requires fewer calculations than the straightforward vortex method velocity procedure and does not sacrifice the higher-order accuracy which can be achieved using higher-order vortex core functions.

A VORTEX METHOD FOR FLOWS WITH SLIGHT DENSITY VARIATIONS

Abstract We present a grid-free numerical method for solving two-dimensional, inviscid, incompressible flow problems with small density variations. The method, an extension of the vortex method, is based on a discretization of the equations written in the vorticity-stream formulation. The method is tested on an exact solution and is found to be both stable and accurate. An application to the motion of a two-dimensional line thermal is also presented.

Rapid computation of the discrete Fourier transform

Algorithms for the rapid computation of the forward and inverse discrete Fourier transform for points which are nonequispaced or whose number is unrestricted are presented. The computational procedure is based on approximation using a local Taylor series expansion and the fast Fourier transform (FFT). The forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse Fourier transform. This latter system is solved using the iterative method GMRES with preconditioning. Numerical results are given to confirm the efficiency of the algorithms.

Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows

Abstract In this paper we present boundary conditions appropriate for the vorticity formulation of the two-dimensional incompressible viscous Navier-Stokes equations. These boundary conditions are incorporated into a finite difference scheme and the resulting method is of the “vorticity creation” type; i.e., vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied. The results of computations with this finite difference method are presented for flow past a circular cylinder. A difference scheme and computational results for a model problem, the Prandtl boundary layer equations describing flow over a semi-infinite flat plate are also presented.

On the accurate calculation of vortex shedding

The issue of accuracy and convergence for numerical calculations in situations in which the exact solution of the Navier–Stokes equations (under the imposed initial and boundary conditions) is unstable is discussed. If one relies on roundoff error or truncation error (or the randomness of a stochastic scheme) in order to perturb the computed solution away from the unstable one, then the computed result cannot be accurate. Instead, one must explicitly provide perturbations in order to compute accurately flows of physical interest. Computational results for flow past an impulsively started cylinder are presented.

A Rayleigh-Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices

A procedure is presented for finding a number of the smallest eigenvalues and their associated eigenvectors of large sparse Hermitian matrices. The procedure, a modification of an inverse subspace iteration procedure, uses adaptively determined Chebyshev polynomials to approximate the required application of the inverse operator on the subspace. The method is robust, converges with acceptable rapidity, and can easily handle operators with eigenvalues of multiplicity greater than one. Numerical results are shown that demonstrate the utility of the procedure.

Efficient solution of the Schroedinger-Poisson equations in layered semiconductor devices

In this paper we present several mathematical models that can be used to create approximate solutions of the three-dimensional Schroedinger-Poisson equation in layered semiconductor devices. A general algorithmic strategy that can be used to create efficient solution procedures for each of these models is described. Computational results demonstrating the accuracy and efficiency that can be obtained with the use of these models is presented.

Alternative to Ritt's pseudodivision for finding the input-output equations of multi-output models.

Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritts pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the characteristic set, of which a subset, the input-output equations, is used for identifiability analysis. A simpler algorithm is proposed for this step, using Gröbner Bases, along with a proof of the method that includes a reduced upper bound on derivative requirements. Efficacy of the new algorithm is illustrated with several biosystem model examples.