Problems In Computational Physics Research Articles

New insights into large eddy simulation

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

Monte Carlo methods for the solution of nonlinear partial differential equations

Stochastic models for the solution of nonlinear partial differential equations are discussed. They consist of a discretized version of these equations and Monte Carlo techniques. The Markov transitions are based on a priori estimates of the solution. To improve the efficiency of stochastic smoothers a Monte Carlo multigrid method is presented. The numerical results presented show the convergence of these methods. Some directions for the parallelization of the Monte Carlo algorithms presented are outlined. The techniques introduced make possible the extension of Monte Carlo methods to nonlinear problems, offering a new approach with an analytic potential for a wide range of problems in computational physics.

Large-scale urbulence simulation on the Navier-Stokes computer

There exists a wide range of problems in computational physics where no analytical and few experimental results exist. Of particular interest at the Navier-Stokes Supercomputer Laboratory of Princeton University is a range of problems dealing with fluid flow. Most flows of interest are complex in nature, involving non-simple boundaries and initial conditions. A typical case involves turbulent flow over arbitrary air and water vehicles. Of particular interest is the active control of these flows. To simulate such flows directly, with boundary and initial conditions directed towards active control schemes, one must numerically solve the Navier-Stokes equations of fluid motion. These equations form a set of nonlinear, coupled, partial differential equations that account for the conservation of mass, momentum and energy in continuum flow. Because of the vast amount of computer memory and associated processing speed required to tackle even simple numerical simulations, a typical general-purpose supercomputer (e.g. Cray-2) would require unreasonably large computer time to tackle most flows involving turbulence. To overcome this difficulty, a multi-purpose parallel supercomputer has been designed and is being fabricated. Called the Navier-Stokes Computer (NSC), its primary function is the direct simulation of complex flows. The NSC is comprised of a multi-dimensional array of processing nodes with local memory. At 20 MHz, each node has a 640 MFLOP parallel-pipelined reconfigurable arithmetic unit coupled to two Gbytes of local memory. Early hardware and software benchmarks indicate that a single node is roughly comparable to the measured performance of a Cray-2. An overview of the NSC architecture along with a discussion of a turbulence simulation of this architecture is presented.

Molecular dynamics and Monte Carlo simulations in solid-state and elementary particle physics

We address the task of doing computational physics problems on a computer which has a highly parallel architecture. Our experience has been gained on the ICL DAP, the first commercially available parallel processor with over a thousand processing elements, but the results have relevance to all computers of this generic class. The DAP itself is described, along with some aspects of the extension to the Fortran language that have been made in order for the user to exploit the DAP's parallelism (or concurrency). Again although herein specific, these language constructs are necessary in one form or another on any such highly parallel computer. The physics we cover starts with molecular dynamics as the concepts here are intuitively obvious--given a governing function it is possible to follow the time development of a system initially randomized, this motion being then a simulation of the motion in a real-life system. The first system models the approach to melting of a molecular solid, naphthalene. It is shown that molecules enjoy increasing reorientational freedom in a narrow temperature interval, and show behavior as suggested from experimental work. The second system modeled is SF 6 , which undergoes a phase transition from plastic to partially plastic crystalline, and then another transition to a truly crystalline phase. The simulation allows us to watch the organizational behavior of the molecules while the system is in transition, giving a very realistic model at the atomic level of solid-state phase transitions. We continue with lattice gauge theory (or quantum chromodynamics) which governs the behavior of quarks and gluons. The method used here is Monte Carlo, a method where statistical configurations are chosen by the use of pseudo-random numbers from the appropriate distribution. A lattice is overlaid on a region of space-time, and the function represented on this lattice leads to many results which test the basic theory, such as the mass spectrum for baryons and mesons. This is an area of physics of fundamental importance, and may yet need an increase of computational resource of about a factor of 100 before its full potential is realized. For this reason we treat the problem in some detail. In conclusion we look at the three-dimensional Ising model and its study through real-space renormalization. Because this is a very simply defined problem it is ideal for gaining insight into the methods of using the renormalization group, methods which will have to be fully understood before the QCD results can be interpreted in full.

Improper problems in computational physics

Several examples of improper problems in computational physics are considered. The examples are from astrophysics, atomic physics, biophysics, geophysics, high-energy physics, hydrodynamics, nuclear physics, plasma physics, solid state physics and statistical mechanics. The basic methods for solving improper problems - the method of regularization and the method of selection - are briefly discussed.