J. Thomas Beale

Vortex methods. II. Higher order accuracy in two and three dimensions

In an earlier paper the authors introduced a new version of the vortex method for three-dimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume the consistency of a discrete approximation to the Biot-Savart Law. We prove this consistency statement here, and also derive substantially sharper results for two-dimensional flows. A complete, simplified proof of convergence in two dimensions is included.

Rates of convergence for viscous splitting of the Navier-Stokes equations

Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity v as it becomes small for either twoor three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate C^lAt, Strang-type splitting converges at the rate CP(At)2, and also that solutions of the Navier-Stokes and Euler equations differ by Cp in this case. Here C depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional NavierStokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.

Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean

To dry fragile low density materials without damage, gas assisted injection and suspension of damp powder to a low velocity stream of heated gas followed by cyclone drying assures long residence time and produces a free-flowing product.

A Method for Computing Nearly Singular Integrals

We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a standard quadrature rule. Then we add corrections for the errors due to smoothing and discretization, which are found by asymptotic analysis. We prove an error estimate for the corrected value, uniform with respect to the point of evaluation. One application is a simple method for solving the Dirichlet problem for Laplaces equation on a grid covering an irregular region in the plane, similar to an earlier method of A. Mayo [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 144--157]. This approach could also be used to compute the pressure gradient due to a force on a moving boundary in an incompressible fluid. Computational examples are given for the double layer potential and for the Dirichlet problem.

Convergence of a Boundary Integral Method for Water Waves

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269--1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration.

On the Accuracy of Vortex Methods at Large Times

Vortex methods simulate incompressible flow, without viscosity or at high Reynolds number, by a collection of computational elements of vorticity which are transported along computed particle paths. The velocity field can be computed from the vorticity in order to move the elements forward in time. Here we will survey the formulation and convergence theory of such methods, primarily for inviscid flow without boundaries in two or three dimensions. We also discuss a modification of the basic method intended to improve accuracy at later times and illustrate its performance with a simple test problem. It is found that the error is significantly reduced in this case. This modified method can be shown to converge, and details of the proof will be given elsewhere. Very similar ideas have been experimented with by Chris Anderson, and it is a pleasure to thank him for his helpful comments and suggestions.

Large-time behavior of discrete velocity boltzmann equations

We study the asymptotic behavior of equations representing one-dimensional motions in a fictitious gas with a discrete set of velocities. The solutions considered have finite mass but arbitrary amplitude. With certain assumptions, every solution approaches a state in which each component is a traveling wave without interaction. The techniques are similar to those in an earlier treatment of the Broadwell model [1].

Large-time behavior of the Broadwell model of a discrete velocity gas

We study the behavior of solutions of the one-dimensional Broadwell model of a discrete velocity gas. The particles have velocity ±1 or 0; the total mass is assumed finite. We show that at large time the interaction is negligible and the solution tends to a free state in which all the mass travels outward at speed 1. The limiting behavior is stable with respect to the initial state. No smallness assumptions are made.

A convergent boundary integral method for three-dimensional water waves

We design a boundary integral method for time-dependent, three-dimensional, doubly periodic water waves and prove that it converges with O(h 3 ) accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transported according to a computed velocity. An integral equation arising from potential theory is solved for the normal velocity. A new method is developed for the integration of singular integrals, in which the Greens function is regularized and an efficient local correction to the trapezoidal rule is computed. The sums replacing the singular integrals are treated as discrete versions of pseudodifferential operators and are shown to have mapping properties like the exact operators. The scheme is designed so that the error is governed by evolution equations which mimic the structure of the original problem, and in this way stability can be assured. The wavelike character of the exact equations of motion depends on the positivity of the operator which assigns to a function on the surface the normal derivative of its harmonic extension; similarly, the stability of the scheme depends on maintaining this property for the discrete operator. With n grid points, the scheme can be implemented with essentially O(n) operations per time step.

A Grid-Based Boundary Integral Method for Elliptic Problems in Three Dimensions

We develop a simple, efficient numerical method of boundary integral type for solving an elliptic partial differential equation in a three-dimensional region using the classical formulation of potential theory. Accurate values can be found near the boundary using special corrections to a standard quadrature. We treat the Dirichlet problem for a harmonic function with a prescribed boundary value in a bounded three-dimensional region with a smooth boundary. The solution is a double layer potential, whose strength is found by solving an integral equation of the second kind. The boundary surface is represented by rectangular grids in overlapping coordinate systems, with the boundary value known at the grid points. A discrete form of the integral equation is solved using a regularized form of the kernel. It is proved that the discrete solution converges to the exact solution with accuracy O(hp), p < 5, depending on the smoothing parameter. Once the dipole strength is found, the harmonic function can be computed from the double layer potential. For points close to the boundary, the integral is nearly singular, and accurate computation is not routine. We calculate the integral by summing over the boundary grid points and then adding corrections for the smoothing and discretization errors using formulas derived here; they are similar to those in the two-dimensional case given by [J. T. Beale and M.-C. Lai, SIAM J. Numer. Anal., 38 (2001), pp. 1902--1925]. The resulting values of the solution are uniformly of O(hp) accuracy, p < 3. With a total of N points, the calculation could be done in essentially O(N) operations if a rapid summation method is used.