Paul R. Woodward

The Piecewise Parabolic Method (PPM) for gas-dynamical simulations☆

Abstract We present the piecewise parabolic method, a higher-order extension of Godunovs method. There are several new features of this method which distinguish it from other higher-order Godunov-type methods. We use a higher-order spatial interpolation than previously used, which allows for a steeper representation of discontinuities, particularly contact discontinuities. We introduce a simpler and more robust algorithm for calculating the nonlinear wave interactions used to compute fluxes. Finally, we recognize the need for additional dissipation in any higher-order Godunov method of this type, and introduce it in such a way so as not to degrade the quality of the results.

THE NUMERICAL SIMULATION OF TWO – DIMENSIONAL FLUID FLOW WITH STRONG SHOCKS

Abstract Results of an extensive comparison of numerical methods for simulating hydrodynamics are presented and discussed. This study focuses on the simulation of fluid flows with strong shocks in two dimensions. By “strong shocks,” we here refer to shocks in which there is substantial entropy production. For the case of shocks in air, we therefore refer to Mach numbers of three and greater. For flows containing such strong shocks we find that a careful treatment of flow discontinuities is of greatest importance in obtaining accurate numerical results. Three approaches to treating discontinuities in the flow are discussed—artificial viscosity, blending of low- and high-order-accurate fluxes, and the use of nonlinear solutions to Riemanns problem. The advantages and disadvantages of each approach are discussed and illustrated by computed results for three test problems. In this comparison we have focused our attention entirely upon the performance of schemes for differencing the hydrodynamic equations. We have regarded the nature of the grid upon which such differencing schemes are applied as an independent issue outside the scope of this work. Therefore we have restricted our study to the case of uniform, square computational zones in Cartesian coordinates. For simplicity we have further restricted our attention to two-dimensional difference schemes which are built out of symmetrized products of one-dimensional difference operators.

KOLMOGOROV-LIKE SPECTRA IN DECAYING THREE-DIMENSIONAL SUPERSONIC FLOWS

A numerical simulation of decaying supersonic turbulence using the piecewise parabolic method (PPM) algorithm on a computational mesh of 5123 zones indicates that, once the solenoidal part of the velocity field, representing vortical motions, is fully developed and has reached a self‐similar regime, a velocity spectrum compatible with that predicted by the classical theory of Kolmogorov develops. It is followed by a domain with a shallower spectrum. A convergence study is presented to support these assertions. The formation, structure, and evolution of slip surfaces and vortex tubes are presented in terms of perspective volume renderings of fields in physical space.

Application of the Piecewise Parabolic Method (PPM) to Meteorological Modeling

Abstract The Piecewise Parabolic Method (PPM), a numerical technique developed in astrophysics for modeling fluid flows with strong shocks and discontinuities is adapted for treating sharp gradients in small-scale meteorological flows. PPM differs substantially from conventional gridpoint techniques in three ways. First, PPM is a finite volume scheme, and thus represents physical variables as averages over a grid zone rather than single values at discrete points. Second, a unique, monotonic parabola is fit to the zone average of each dependent variable using information from neighboring zone averages. As shown in a series of one- and two-dimensional linear advection experiments, the use of parabolas provides for extremely accurate advection, particularly of sharp gradients. Furthermore, the monotonicity constraint renders PPMs solutions free from Gibbs oscillations. PPMs third attribute is that each zone boundary is treated as a discontinuity. Using the method of characteristic the nonlinear flux of qua...

On the Divergence-free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows

An approach to maintain exactly the eight conservation laws and the divergence-free condition of magnetic fields is proposed for numerical simulations of multidimensional magnetohdyrodynamic (MHD) equations. The approach is simple and may be easily applied to both dimensionally split and unsplit Godunov schemes for supersonic MHD flows. The numerical schemes based on the approach are second-order accurate in both space and time if the original Godunov schemes are. As an example of such schemes, a scheme based on the approach and an approximate MHD Riemann solver is presented. The Riemann solver is simple and is used to approximately calculate the time-averaged flux. The correctness, accuracy, and robustness of the scheme are shown through numerical examples. A comparison in numerical solutions between the proposed scheme and a Godunov scheme without the divergence-free constraint implemented is presented.

Inertial range structures in decaying compressible turbulent flows

Simulations of decaying compressible turbulent flows have been performed using the PPM algorithm on grids of 5123 and 10243 computational cells. Although the run on the finer grid has not yet been carried out to a time large enough for the spectra to relax fully, it adds significantly to the results on the coarser grid by lengthening the range of wave numbers in which the flow exhibits a self-similar character. There is an inertial range of scales in the decaying flow on the finer mesh that is free from direct effects of dissipation, forcing, boundary conditions, or initial conditions. Favre averaging of the high resolution data is performed on different scales from which the vorticity structures in the inertial range may be visualized and characterized without confusion from the smaller-scale features of the near dissipation range. We find that the vorticity structures of the inertial range are filamentary as well, but qualitatively different—shorter and more curved—than those of the dissipation range. Q...

CONVECTIVE-REACTIVE PROTON-12C COMBUSTION IN SAKURAI'S OBJECT (V4334 SAGITTARII) AND IMPLICATIONS FOR THE EVOLUTION AND YIELDS FROM THE FIRST GENERATIONS OF STARS

Depending on mass and metallicity as well as evolutionary phase, stars occasionally experience convectivereactive nucleosynthesis episodes. We specifically investigate the situation when nucleosynthetically unprocessed, H-rich material is convectively mixed with a He-burning zone, for example in convectively unstable shell on top of electron-degenerate cores in AGB stars, young white dwarfs or X-ray bursting neutron stars. Such episodes are frequently encountered in stellar evolution models of stars of extremely low or zero metal content, such as the first stars. We have carried out detailed nucleosynthesis simulations based on stellar evolution models and informed by hydrodynamic simulations. We focus on the convective-reactive episode in the very-late thermal pulse star Sakurai’s object (V4334 Sagittarii). Asplund et al. (1999) determined the abundances of 28 elements, many of which are highly non-solar, ranging from H, He and Li all the way to Ba and La, plus the C isotopic ratio. Our simulations show that the mixing evolution according to standard, one-dimensional stellar evolution models implies neutron densities in the He intershell (. few 10 11 cm -3 ) that are too low to obtain a significant neutron capture nucleosynthesis on the heavy elements. We have carried out 3D hydrodynamic He-shell flash convection simulations in 4 geometry to study the entrainment of H-rich material. Guided by these simulations we assume that the ingestion process of H into the He-shell convection zone leads only after some delay time to a sufficient entropy barrier that splits the convection zone into the original one driven by He-burning and a new one driven by the rapid burning of ingested H. By making such mixing assumptions that are motivated by our hydrodynamic simulations we obtain significantly higher neutron densities ( few 10 15 cm -3 ) and reproduce the key observed abundance trends found in Sakurai’s object. These include an overproduction of Rb, Sr and Y by about 2 orders of magnitude higher than the overproduction of Ba and La. Such a peculiar nucleosynthesis signature is impossible to obtain with the mixing predictions in our one-dimensional stellar evolution models. The simulated Li abundance and the isotopic ratio 12 C/ 13 C are as well in agreement with observations. Details of the observed heavy element abundances can be used as a sensitive diagnostic tool for the neutron density, for the neutron exposure and, in general, for the physics of the convective-reactive phases in stellar evolution. For example, the high elemental ratio Sc/Ca and the high Sc production indicate high neutron densities. The diagnostic value of such abundance markers depends on uncertain nuclear physics input. We determine how our results depend on uncertainties of nuclear reaction rates, for example for the 13 C(; n) 16 O reaction. Subject headings: stars: AGB and post-AGB — stars: abundances — stars: evolution — stars: interior — stars: individual (V4334 Sagittarii) — physical data and processes: hydrodynamics — physical data and processes: nuclear reactions, nucleosynthesis, abundances

Synthesized spectra of turbulent clouds

A comparison has been made of the velocity structure of non-star-forming dense molecular clouds and the output of a 512(exp 3) hydrodynamic code for compressible gas displaying features of turbulence. A remarkable similarity exists between the interstellar line profiles and the velocity profiles found by integrating, with an optically thin assumption, through subsections of the simulation. This similarity is in terms of both the structure of individual spectra, showing skewness, multiple peaks, and wings, and also the variations of the spectra from point to point. It is possible to see the origin of the various velocity substructures and how they correlate with regions of high density and velocity divergence, or regions of high vorticity, by deprojecting individual spectra of the simulation. It is found that the interstellar medium spectra are best represented by the epoch of the simulation in which the energy spectrum is essentially Kolmogorov-like and initially developed shocks have been largely converted into vorticity structures. Further, it is found that the vorticity peaks ascribed to intermittency contribute to the non-Gaussian features and wings of the line profile, but at an intensity level that is weak compared to the contribution of the bulk of the flow.

Three-dimensional simulation of a Richtmyer-Meshkov instability with a two-scale initial perturbation

Three-dimensional high-resolution simulations (up to 8 billion zones) have been performed for a Richtmyer–Meshkov instability produced by passing a shock through a contact discontinuity with a two-scale initial perturbation. The setup approximates shock-tube experiments with a membrane pushed through a wire mesh. The simulation produces mixing-layer widths similar to those observed experimentally. Comparison of runs at various resolutions suggests a mixing transition from unstable to turbulent flow as the numerical Reynolds number is increased. At the highest resolutions, the spectrum exhibits a region of power-law decay, in which the spectral flux is approximately constant, suggestive of an inertial range, but with weaker wave number dependence than Kolmogorov scaling, about k−6/5. Analysis of structure functions at the end of the simulation indicates the persistence of structures with velocities largest in the stream-wise direction. Comparison of three-dimensional and two-dimensional runs illustrates th...

A Numerical Laboratory

In a series of talks in 1946, John von Neumann envisioned the use of high‐speed computers to generate solutions to nonlinear problems, particularly in fluid dynamics. He pointed out that scientists were conducting expensive and difficult experiments to observe physical behavior even when the underlying principles and governing equations were known. “The purpose of the experiment is not to verify a proposed theory but to replace a computation from an unquestioned theory by direct measurements,” he wrote. “Thus wind tunnels are used at present, at least in large part, as computing devices of the so‐called analogy type to integrate the nonlinear partial differential equations of fluid dynamics.”