Stony Brook AMS - Downloadable Preprints, 1993

We consider multigrid iterative methods for the solution of electromagnetic scattering for dielectric materials. We show convergence of thetems. iteration using coarse grids which are two to four times coarser in each dimension than the fine grid. These results allow a significant increase in problem size and solution speed, for a given hardware configuration. We report in particular on the solution of scattering problems which require the solution of 31,000 equations on the fine grid, using the direct solution of 3,500 double precision equations on the coarse grid, anb project the ability to solve significantly larger systems using larger machines or an out-of-core capability.

In this paper we continue to study hyperbolic systems of conservation laws with umbilic degeneracy. With the aid of the compactness framework established in [CK1], we prove the convergence of the viscosity method, and the existence of global entropy solutions for the Cauchy problem with large initial data for canonical classes of the quadratic flux systems.

We present a numerical method for computing elasto-plastic flows in metals. The method uses a conservative Eulerian formulation of elasto-plasticity together with a higher-order Godunov finite difference method combined with tracking of material boundaries. The Eulerian approach avoids the problem of mesh distortion caused by a Lagrangian remap, and can be easily extended to the computation of flows in multiple space dimensions using operator splitting. The method is validated by a comparison of computations with experiments on one-dimensional high velocity plate impact. We obtain excellent agreement between our computations and experiment.

We consider multigrid iterative methods for the solution of electromagnetic scattering from an infinite plane with a finite cavity. We show convergence of the iteration using coarse grids which are two to four times coarser in each dimension than the fine grid. These results allow a significant increase in problem size and solution speed, for a given hardware configuration. We report in particular on the solution of scattering problems which require the solution of 31,000 equations on the fine grid, using the direct solution of 1,900 double precision equations on the coarse grid, and project the ability to solve significantly larger systems using larger machines or an out-of-core capability.

A parallel algorithm for solving the multinephron model of the renal inner medulla is developed. The intrinsic nature of this problem supplies sufficient symmetry for a high-level parallelism on distributed-memory parallel machines such as the iPSC/860, Paragon, and CM-5. Parallelization makes it feasible to study interesting models such as the rat kidney with $30,000$ nephrons. On a high-end work station, one can study systems with 100 nephrons, while on a 32-node iPSC/860 we can handle more than 1000 nephrons.

A nearly perfect speedup is achieved by even distribution of loads and minimizing the cost of communication.

Front tracking capability has been encorporated into the Partnership in Computational Science (PICS) GCT code, version 1.0. This merge adds a two dimensional, discontinuity tracking capability to the GCT 1.0 code. It supports the ability to run both on scalar platforms, specifically UNIX workstations, and on the INTEL iPSC/860 hypercube parallel architecture. Porting to the INTEL Paragon architecture will be accomplished once a stable version of the software for said architecture is available.

We explain several ideas which may, either singly or in combination, help achieve high resolution in simulations of large-deformation plasticity. Because of the large deformations, we work in the Eulerian picture. The governing equations are written in a fully conservative form, which are correct for discontinuous as well as continuous solutions. Models of shear bands are discussed. These models describe the internal dynamics of a developed shear band in terms of time-asymptotic states; in other words, the smooth internal structure is replaced by a jump discontinuity, and the shear band evolution is determined by jump relations. This analysis is useful for high resolution numerical methods, including both shock capturing and shock tracking schemes, as well as for the understanding and validation of computations, independently of the underlying method. Preliminary computations, which illustrate the feasibility of these ideas, are presented.

The acceleration of a material interface by a shock wave generates an interface instability known as the Richtmyer-Meshkov instability. Previous attempts to model the growth rate of the instability have produced values that are almost twice that of the experimental measurements. This article presents numerical simulations using front tracking that for the first time are in quantitative agreement with experiments of a shocked air-SF$_6$ interface. Moreover, the failure of the impulsive model, and the linear theory from which it is derived, to model experiments correctly is understood in terms of time limits on the validity of the linear model.

This paper concerns $2 \times 2$ systems of conservation laws with quadratic fluxes corresponding to Case II in the classification of such problems. Aided by a computer program, we have constructed the solution that satisfies the viscous profile criterion for shock admissibility. Our solution differs from that obtained using the Lax admissibility criterion, even though solutions exist and are unique for both criteria. With the viscous profile criterion, some nonlocal Lax shock waves are inadmissible; in their place, transitional waves appear in the wave patterns.

We present a parallel method for matrix multiplication on distributed-memory MIMD architectures based on Strassen's method. Our timing tests, performed on an Intel Paragon, demonstrate that our method realizes the potential of the Strassen's method with a complexity of $4.7M^{2.807}$ at the system level rather than the node level at which several earlier works have been focused. The parallel efficiency is nearly perfect when the processor number is divisible by $7$. The parallelized Strassen's method is always faster than the traditional matrix multiplication methods whose complexity is $2M^3$ coupled with the BMR method and the Ring method at system level. The speed gain depends on matrix order $M$: $20%$ for $M \approx 1000$ and more than $100%$ for $M \approx 5000$.

The hyperbolic, or conservative, description of nonlinear waves is recognized to be incomplete, and appeal to a physically more complete set of equations may be needed to resolve ambiguities, such as nonuniqueness of solutions, at the hyperbolic level. Here we consider a very simple example of a hyperbolic wave, namely a fluid interface, and show that it has potentially a very complicated structure. We survey recent progress of the authors and co-workers in understanding the structure of a mixing zone, as a diffused boundary or interface between two fluids, in several cases for which standard linear diffusion, with a growth rate O(t^{1/2}), is incorrect. The methods use a mixture of theoretical analysis, numerical simulation by the front tracking method, and comparison to experimental and field data.

We discuss the use of front tracking to simulate shock reflections and shock accelerated interfaces. Our simulations of regular Mach reflection show enhanced resolution of the primary waves in the interaction, and our computations of the growth rate of a Richtmyer-Meshkov unstable interface are the first numerical results that are in quantitative agreement with experiments on a shocked air-SF$_6$ interface. Previous computations of the growth rate of the instability produced values that were almost twice those found in experiments.

We discuss the application of front tracking to the simulation of shock reflections and shock accelerated interfaces. Some key features of the front tracking method are the elimination of numerical diffusion and the reduction of wall heating. In computations of the regular Mach reflection of a shock at an oblique ramp, we see enhanced resolution of the primary waves in the interaction. In addition, tracking allows very precise measurements to be made of the states and location of the Mach triple point. Our computations of the growth rate of a Richtmyer-Meshkov unstable interface are the first numerical results that are in quantitative agreement with experimental results of a shocked air-SF$_6$ interface. Previous attempts to model the growth rate of the instability have produced values that are almost twice that of the experimental measurements. Moreover, the failure of the impulsive model, and the linear theory from which it is derived, to model experiments correctly is understood in terms of time limits on the validity of the linear model.

We report new results on the Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Highlights include calculations of Richtmyer-Meshkov instabilities in curved geometries without grid orientation effects, improved agreement between computations and experiments in the case of Richtmyer-Meshkov instabilities at a plane interface, and a demonstration of an increase in the Rayleigh-Taylor mixing layer growth rate with increasing compressibility, along with a loss of universality of this growth rate. The principal computational tool used in obtaining these results was a code based on the front tracking method.

The method of front tracking is used to simulate the regular Mach reflection of a shock at an oblique ramp, providing enhanced resolution of the primary waves in the interaction. Some important features of the front tracking method are the elimination of numerical diffusion and the reduction of wall heating. In addition, tracking allows very precise measurements to be made of the states and location of the Mach triple point.

Mathematical models of the mechanism for making concentrated urine in the mammalian kidney compute the variables $V$ (e.g., volume flows and concentrations) as function of parameters $h$ (e.g., water and solute permeabilites). We consider the inverse problem: Given a $V$, for which $h$ is known to exist, compute $h$. We give computational evidence that $h$ can be determined well within the round-off error tolerance without any prior information about it.

Using an inner medullary shunt model of the kidney concentrating mechanism, we investigate the relationship between $z$ --- salt and urea concentrations in the {\sl Central Core}, and $h$ --- water, salt and urea permeabilities in the {\sl Henle's loop} and the {\sl Collecting Duct}. Computational results are given comparing (a) the direct problem: given $h$ compute $z$, to (b) the inverse problem: given $z$ compute $h$.