Parabolic Method Research Articles

Simulations of the Instability Experiments in StratifiedCylindrical Shells

A high-resolution multi-fluid parabolic piecewise method (MFPPM) ispresented to simulate the Richtmyer–Meshkov (RM) instability instratified cylindrical shells. However, the algorithm captures do nottrack the interfaces between distinct large deformation materials. Toreach verification and validation of our MFPPM code, we select twoexperimental models for simulations, one is from the Lawrence LivermoreNational Laboratory, and the other is from our Laboratory for ShockWave and Detonation Physics. The former result is in excellentagreement with the experiment and the Couple Arbitrary Lagrange–Euler(CALE) code, the latter is in general consistent with our experiment.It is found that the MFPPM code is reasonably successful in simulatingmany aspects of the gelatin-ring RM experiments.

RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code

The authors have developed a new computer code, RAM, to solve the conservative equations of special relativistic hydrodynamics (SRHD) using adaptive mesh refinement (AMR) on parallel computers. They have implemented a characteristic-wise, finite difference, weighted essentially non-oscillatory (WENO) scheme using the full characteristic decomposition of the SRHD equations to achieve fifth-order accuracy in space. For time integration they use the method of lines with a third-order total variation diminishing (TVD) Runge-Kutta scheme. They have also implemented fourth and fifth order Runge-Kutta time integration schemes for comparison. The implementation of AMR and parallelization is based on the FLASH code. RAM is modular and includes the capability to easily swap hydrodynamics solvers, reconstruction methods and physics modules. In addition to WENO they have implemented a finite volume module with the piecewise parabolic method (PPM) for reconstruction and the modified Marquina approximate Riemann solver to work with TVD Runge-Kutta time integration. They examine the difficulty of accurately simulating shear flows in numerical relativistic hydrodynamics codes. They show that under-resolved simulations of simple test problems with transverse velocity components produce incorrect results and demonstrate the ability of RAM to correctly solve these problems. RAM has been tested in one, two and three dimensions andmore » in Cartesian, cylindrical and spherical coordinates. they have demonstrated fifth-order accuracy for WENO in one and two dimensions and performed detailed comparison with other schemes for which they show significantly lower convergence rates. Extensive testing is presented demonstrating the ability of RAM to address challenging open questions in relativistic astrophysics.« less

Vanishing viscosity for hemivariational inequalities modeling dynamic problems in elasticity

In this paper we consider a mathematical model describing a dynamic linear elastic contact problem with nonmonotone skin effects. The subdifferential multivalued and multidimensional reaction–displacement law is employed. We treat an evolution hemivariational inequality of hyperbolic type which is a weak formulation of this mechanical problem. We establish a result on the existence of solutions to the Cauchy problem for the hemivariational inequality. This result is a consequence of an existence theorem for second order evolution inclusion. For the latter, using the parabolic regularization method, we obtain the solution as a limit when the viscosity term tends to zero.

Parabolic Wave Equation Method Applied to the Tropospheric Ducting Propagation Problem: A Survey

A survey is made of one of the most widely used approximation methods in the wave propagation studies—the parabolic wave equation method—applied for the specific case of microwave propagation assessment under tropospheric ducting conditions. A brief review of the methods for tropospheric refractivity profiling, the average refractivity modeling, and the applicability of the often-assumed lateral homogeneity of the refractivity is given. The parabolic wave equation derivation and the numerical methods for solving it are summarized with special emphasis on transparent boundary conditions for the upper computational window boundary. Results are reported which illustrate the application of the finite element method and split-step Fourier-based solutions of parabolic wave equations for different cases of tropospheric ducts, concerning radar and communications links performance.

The Parabolic Spline Method (PSM) for conservative transport problems

A new and efficient parabolic spline based remapping algorithm is developed and tested herein. To ensure mass conservation, the scheme solves an integral form of the transport equation rather than the differential form. The integrals are computed from reconstructed parabolic splines with mass conservation constraints. For higher dimensions, this remapping can be used within a standard directional splitting methodology or within the flow-dependent cascade splitting approach. A grid and sub-grid based monotonic filter is also incorporated into the overall scheme. A truncation error analysis of the scheme is presented and discussed in terms of results from test cases. The analysis shows that although it has a similar truncation error in the converged limit as that of the widely used Piecewise Parabolic Method (PPM) for infinitely differentiable functions, PSM is more accurate than PPM for problems with slow spectral decay. Additionally, an operation count of the scheme is given which demonstrates the computational advantage of PSM compared to PPM. © Crown copyright 2005. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. So far the impact of these schemes in the framework of inverse problem solution has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on data assimilation is the subject of this work, using the nonlinear viscous Burgers equation in 1−D. Also provided are results obtained in 2−D using a global shallow water equations model. The results obtained in this work may point out to suitability of these advection schemes for data assimilation in more complex higher dimensional models. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical dissipation and the bottleneck effect in simulations of compressible isotropic turbulence

The piece-wise parabolic method (PPM) is applied to simulations of forced isotropic turbulence with Mach numbers ∼0.1 … 1. The equation of state is dominated by the Fermi pressure of an electron-degenerate fluid. The dissipation in these simulations is of purely numerical origin. For the dimensionless mean rate of dissipation, we find values in agreement with known results from mostly incompressible turbulence simulations. The calculation of a Smagorinsky length corresponding to the rate of numerical dissipation supports the notion of the PPM supplying an implicit subgrid scale model. In the turbulence energy spectra of various flow realisations, we find the so-called bottleneck phenomenon, i.e., a flattening of the spectrum function near the wave number of maximal dissipation. The shape of the bottleneck peak in the compensated spectrum functions is comparable to what is found in turbulence simulations with hyperviscosity. Although the bottleneck effect reduces the range of nearly inertial length scales considerably, we are able to estimate the value of the Kolmogorov constant. For steady turbulence with a balance between energy injection and dissipation, it appears that C ≈ 1.7. However, a smaller value is found in the case of transonic turbulence with a large fraction of compressive components in the driving force. Moreover, we discuss length scales related to the dissipation, in particular, an effective numerical length scale Δ eff, which can be regarded as the characteristic smoothing length of the implicit filter associated with the PPM.

Level set simulations of turbulent thermonuclear deflagration in degenerate carbon and oxygen

We study the dynamics of thermonuclear flames propagating in fuel stirred by stochastic forcing. The fuel consists of carbon and oxygen in a state which is encountered in white dwarfs close to the Chandrasekhar limit. The level set method is applied to represent the flame fronts numerically. The computational domain for the numerical simulations is cubic, and periodic boundary conditions are imposed. The goal is the development of a suitable flame speed model for the small-scale dynamics of turbulent deflagration in thermonuclear supernovae. Because the burning process in a supernova explosion is transient and spatially inhomogeneous, the localized determination of subgrid scale closure parameters is essential. We formulate a semi-localized model based on the dynamical equation for the subgrid scale turbulence energy k sgs. The turbulent flame speed s t is of the order √2k sgs. In particular, the subgrid scale model features a dynamic procedure for the calculation of the turbulent energy transfer from resolved toward subgrid scales, which has been successfully applied to combustion problems in engineering. The options of either including or suppressing inverse energy transfer in the turbulence production term are compared. In combination with the piece-wise parabolic method for the hydrodynamics, our results favour the latter option. Moreover, different choices for the constant of proportionality in the asymptotic flame speed relation, s t∝√2k sgs, are investigated.

The Piecewise Parabolic Method for Multidimensional Relativistic Fluid Dynamics

We present an extension of the Piecewise Parabolic Method to special relativistic fluid dynamics in multidimensions. The scheme is conservative, dimensionally unsplit, and suitable for a general equation of state. Temporal evolution is second-order accurate and employs characteristic projection operators; spatial interpolation is piece-wise parabolic making the scheme third-order accurate in smooth regions of the flow away from discontinuities. The algorithm is written for a general system of orthogonal curvilinear coordinates and can be used for computations in non-cartesian geometries. A non-linear iterative Riemann solver based on the two-shock approximation is used in flux calculation. In this approximation, an initial discontinuity decays into a set of discontinuous waves only implying that, in particular, rarefaction waves are treated as flow discontinuities. We also present a new and simple equation of state which approximates the exact result for the relativistic perfect gas with high accuracy. The strength of the new method is demonstrated in a series of numerical tests and more complex simulations in one, two and three dimensions.

Comparing AMR and SPH Cosmological Simulations. I. Dark Matter and Adiabatic Simulations

We compare two cosmological hydrodynamic simulation codes in the context of hierarchical galaxy formation: the Lagrangian smoothed particle hydrodynamics (SPH) code GADGET, and the Eulerian adaptive mesh refinement (AMR) code Enzo. Both codes represent dark matter with the N-body method but use different gravity solvers and fundamentally different approaches for baryonic hydrodynamics. The SPH method in GADGET uses a recently developed conserving formulation of SPH, while for the mesh-based Enzo two different formulations of Eulerian hydrodynamics are employed: the piecewise parabolic method (PPM) extended with a dual energy formulation for cosmology, and the artificial viscosity-based scheme used in the magnetohydrodynamics code ZEUS. In this paper we focus on a comparison of cosmological simulations that follow either only dark matter, or also a nonradiative (adiabatic) hydrodynamic gaseous component. We perform multiple simulations using both codes with varying spatial and mass resolution with identical initial conditions. The dark matter-only runs agree generally quite well provided Enzo is run with a comparatively fine root grid and a low overdensity threshold for mesh refinement, otherwise the abundance of low-mass halos is suppressed. This can be readily understood as a consequence of the hierarchical particle-mesh algorithm used by Enzo to compute gravitational forces, which tends to deliver lower force resolution than the tree-algorithm of GADGET at early times before any adaptive mesh refinement takes place. At comparable force resolution we find that the latter offers substantially better performance and lower memory consumption than the present gravity solver in Enzo. In simulations that include adiabatic gasdynamics we find general agreement in the distribution functions of temperature, entropy, and density for gas of moderate to high overdensity, as found inside dark matter halos. However, there are also some significant differences in the same quantities for gas of lower overdensity. For example, at z = 3 the fraction of cosmic gas that has temperature log T > 0.5 is ~80% for both Enzo ZEUS and GADGET, while it is 40%-60% for Enzo PPM. We argue that these discrepancies are due to differences in the shock-capturing abilities of the different methods. In particular, we find that the ZEUS implementation of artificial viscosity in Enzo leads to some unphysical heating at early times in preshock regions. While this is apparently a significantly weaker effect in GADGET, its use of an artificial viscosity technique may also make it prone to some excess generation of entropy that should be absent in Enzo PPM. Overall, the hydrodynamical results for GADGET are bracketed by those for Enzo ZEUS and Enzo PPM but are closer to Enzo ZEUS.

An unsplit Godunov method for ideal MHD via constrained transport

We describe a single step, second-order accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, the corner transport upwind (CTU) method of Colella for multidimensional integration, and the constrained transport (CT) algorithm for preserving the divergence-free constraint on the magnetic field. We adopt the most compact form of CT, which requires the field be represented by area-averages at cell faces. We demonstrate that the fluxes of the area-averaged field used by CT can be made consistent with the fluxes of the volume-averaged field returned by a Riemann solver if they obey certain simple relationships. We use these relationships to derive new algorithms for constructing the CT fluxes at grid cell corners which reduce exactly to the equivalent one-dimensional solver for plane-parallel, grid-aligned flow. We show that the PPM reconstruction algorithm must include multidimensional terms for MHD, and we describe a number of important extensions that must be made to CTU in order for it to be used for MHD with CT. We present the results of a variety of test problems to demonstrate the method is accurate and robust.

George Herbert’s Divine Comedy: Humor in The Temple

George Herbert’s Divine Comedy: Humor in The Temple Michael C. Schoenfeldt (bio) One of the signal if unexpected pleasures of George Herbert’s poetry is its frequent recourse to affable laughter at human foibles. We tend to assume that the ascetic orientation of religion is by definition opposed to terrestrial pleasure. And Herbert in particular seems to exemplify the sober earnestness that marks a particular stereotype of Anglican worship.1 In The Temple, though, Herbert intends to reorient a range of secular pleasures toward heaven, declaring that God is at once their source and goal. Terrestrial pleasure is not negated, but it is revealed to be only a scanty appetizer for the deeply sensual joys of heaven. These joys, I want to argue, extend not only to the more carnal pleasures of food and sex – pleasures whose uneasy presence in Herbert I have tried to document elsewhere – but also to the deeply social amusements of humor.2 Herbert feels that laughter, like all pleasures, is morally and socially hazardous, and so must be carefully circumscribed, and deployed with the greatest delicacy and decorum. Attention to the myriad uses of humor throughout Herbert’s corpus discloses the anxious if indulgent opulence of a religious imagination which presumes that the “quick returns of courtesie and wit” might provide a worthy medium for the devout worship of God.3 Despite the unrelenting solemnity with which Western culture tends to drape religious devotion, Herbert was far from the first to explore potential links between religion and humor. St. Paul incisively delineated the divergent standards of wisdom that demarcate the differences between terrestrial and celestial perspectives: For the preaching of the Crosse is to them that perish, foolishnesse: but unto us which are saved, it is the power of God. For it is written, I will destroy the wisedome of the wise, and wil bring to nothing the understanding of the prudent. (I Corinthians 1:18–19; AV) [End Page 45] Building on a passage from Isaiah (29:14; “for the wisedome of their wise men shall perish, and the understanding of their prudent men shall be hid”), Paul tells the community at Corinth that Christianity entails a complete inversion of terrestrial hierarchies of power and knowledge: Hath not God made foolish the wisedome of this world? … [When] the world by wisedome knew not God, it pleased God by the foolishnesse of preaching, to save them that beleeve … The foolishnesse of God is wiser then men: and the weaknesse of God is stronger then men … God hath chosen the foolish things of the world, to confound the wise; and God hath chosen the weake things of the world, to confound the things which are mighty … that no flesh should glory in his presence. (I Corinthians 1:20–30) Paul here argues that those pursuits that the world mocks are in fact the occasion of a higher wisdom. “The wisdome of this world,” he argues, “is foolishness with God” (1 Corinthians 3:19). He deliberately aligns the Christian subversion of earthly values with the comic inversion of terrestrial hierarchies in his invocation of a decidedly theatrical metaphor: “For wee are made a spectacle unto the world, and to Angels, and to men. We are fooles for Christs sake, but ye are wise in Christ” (I Corinthians 4:9–10). This Pauline concept of holy folly is developed at length by Erasmus in the latter sections of his enormously popular work, the Encomium Moriae, or Praise of Folly. Jesus, notes Erasmus’s figure of Folly, was made something of a fool himself in order to help the folly of mankind, when he assumed the nature of man and was seen in man’s form. Nor did he wish [sinners] to be redeemed in any other way save by the folly of the cross and through his simple, ignorant apostles, to whom he unfailingly preached folly. He taught them to shun wisdom, and made his appeal through the example of children, lilies, mustard-seed and humble sparrows, all foolish, senseless things.4 Erasmus wryly suggests that Jesus’ deliberate metaphoric simplicity and parabolic method entailed an active participation in the comic mode. [End Page 46] He argues, moreover...

Turbulent Structures in a Slightly Compressible Mixing Layer

The configuration of a compressible mixing layer confined between rigid walls at high Reynolds number is studied at convective Mach number Mc=0.64, using the monotonically integrated large eddy simulation approach. The purpose is to analyze the flow configuration with attention to the turbulent mixing linked to large scales in the transition and turbulent zones. The full Navier-Stokes equations are solved with a high-order Godunov's scheme piecewise parabolic method with the approximate Riemann solver of Roe. On the basis of flow visualization, highly three-dimensional vortex lattices undergoing helical pairing of oblique structures are observed in the transition zone. A particular structure of mixing in sublayers is detected qualitatively and through probability density function analysis. Computed statistics are in good agreement with experimental measurements.

Application of the high-resolution Godunov method to the multi-fluid flow calculations

In this paper, we have numerically solved the multi-fluid problems using an operator-split two-step high-resolution Godunov PPM (parabolic piecewise method) for the flow in complex geometries. By using the front capturing method, the PPM integrator captures the interface in the solution process. The basic multi-fluid integrator is coupled to a Cartesian grid algorithm where a VOF (volume of fluid) representation of the fluid interface is also used. As an application of this method, we test the 2D interfacial advection example and simulate an experimental hypervelocity launcher model from Sandia National Laboratories. The computational design of the hypervelocity launcher is also given in the paper.

Evaluating planetesimal bow shocks as sites for chondrule formation

Abstract— We investigate the possible formation of chondrules by planetesimal bow shocks. The formation of such shocks is modeled using a piecewise parabolic method (PPM) code under a variety of conditions. The results of this modeling are used as a guide to study chondrule formation in a one‐dimensional, finite shock wave. This model considers a mixture of chondrule‐sized particles and micron‐sized dust and models the kinetic vaporization of the solids. We found that only planetesimals with a radius of ˜1000 km and moving at least ˜8 km/s with respect to the nebular gas can generate shocks that would allow chondrule‐sized particles to have peak temperatures and cooling rates that are generally consistent with what has been inferred for chondrules. Planetesimals with smaller radii tend to produce lower peak temperatures and cooling rates that are too high. However, the peak temperatures of chondrules are only matched for low values of chondrule wavelength‐averaged emissivity. Very slow cooling (<˜100s of K/hr) can only be achieved if the nebular opacity is low, which may result after a significant amount of material has been accreted into objects that are chondrule‐sized or larger, or if chondrules formed in regions of the nebula with small dust concentrations. Large shock waves of approximately the same scale as those formed by gravitational instabilities or tidal interactions between the nebula and a young Jupiter do not require this to match the inferred thermal histories of chondrules.

A convexity preserving scheme for conservative advection transport

A simple and practical scheme for advection transport equation is presented. The scheme, namely piecewise rational method (PRM), is a variant of the existing piecewise parabolic method (PPM) of Colella and Woodward (1984). Instead of the parabolic function, a rational function is used for the reconstruction. Making use of the convexity preserving nature of the rational function enables us to obtain oscillation-less numerical solutions, but avoids the adjustments of the cell-interface values to enforce the monotonicity in PPM. The PRM is very simple and computationally efficient. Our numerical results show that PRM is competitive to the PPM in many aspects, such as numerical accuracy and shape-preserving property.

Oxidation behavior of TiC-containing Ti3AlC2 based material at 500–900 °C in air

The isothermal oxidation behavior of Ti3AlC2 based material containing 5 vol% TiC inclusion in air had been investigated at 500–900 °C by means of TGA, XRD, Raman spectroscopy and SEM/EDS. It was demonstrated that, although Ti3AlC2 based material exhibited good oxidation resistance at temperatures above 700 °C, anomalous oxidation with higher oxidation rate occurred at lower temperatures of 500 and 600 °C. This interesting phenomenon was due to the formation of microcracks associated with the stress developed within the scales, mainly consisting of anatase, and the volume expansion as Ti3AlC2 based material was directly exposed to air at those temperatures. Its oxidation, at temperatures investigated with the exception of 600 °C, generally obeyed a parabolic rate law. The weight gain data for the remaining temperatures were analyzed with an instantaneous parabolic rate constant method by assuming a parabolic rate law. The variations of instantaneous parabolic rate constant with time reflected the complexity of the oxidation behavior of Ti3AlC2 based material. These variations were discussed from the viewpoint of the formation of microcracks at 500 °C, and preferred oxidation of TiC inclusion in the initial oxidation and its subsequent depletion at 800 and 900 °C on the basis of X-ray diffraction, Raman spectroscopy, SEM scale morphology observation and composition analysis using EDS. In addition, the deleterious effect of TiC inclusion on the oxidation resistance of Ti3AlC2 based material was also investigated and discussed with comparison to monolithic Ti3AlC2, which was helpful to understand the discrepancies in reports on the oxidation of Ti3AlC2.

Vortex-accelerated secondary baroclinic vorticity deposition and late-intermediate time dynamics of a two-dimensional Richtmyer–Meshkov interface

We study the vortex-accelerated secondary baroclinic vorticity deposition (VAVD) at late-intermediate times, and dynamics of sinusoidal single-mode Richtmyer–Meshkov interfaces in two dimensions. Euler simulations using a piecewise parabolic method are conducted for three post-shock Atwood numbers (A*), 0.2, 0.635, and 0.9, with Mach number (M) of 1.3. We initialize the sinusoidal interface with a slightly “diffuse” or small-but-finite thickness interfacial transition layer to facilitate comparison with experiment and avoid ill-posed phenomena associated with evolutions of an inviscid vortex sheet. The thickness of the interface is chosen so that there are no secondary structures along the interface prior to the multivalue time tM, which is defined as the time when the extracted medial axis of an interfacial layer first becomes multivalued. For an interval of 11tM beyond tM, the simulations reveal nearly monotonic strong growth of both positive and negative baroclinic circulation in a vortex bilayer pattern inside the complex roll-up region. The circulations grow and secondary baroclinic circulation dominates at intermediate times, especially for higher A*. This vorticity deposition is due to misalignment of density gradient across the interface and vortex-centripetal acceleration (secondary baroclinic), and enhanced by the intensification of interfacial density gradient arising from the vortex-induced strain. Our simulation results for A*=0.635 agree with the recent air–sulfur hexafluoride (SF6) experiment of Jacobs and Krivets [Proceedings of the 23rd International Symposium on Shock Waves, Fort Worth, Texas, (2001)], including several large-scale features of the evolving mushroom structure: The usual interface spike-bubble amplitude growth rate ȧ and the dimensions of the spike roll-up cavity. VAVD plays an important role in the intermediate time dynamics of the interfaces. Our amplitude growth rate ȧ disagrees with the O(t−1) result of Sadot et al. [Phys. Rev. Lett. 80, 1654 (1998)]. Instead, it approaches a constant which increases with A*(⩽0.9). An adjusting periodic single point vortex model which uses the calculated net circulation magnitude and its location, gives excellent results for the amplitude growth rates to late-intermediate times at low Atwood numbers (A*=0.2,0.635). The evolution of enstrophy, vorticity skewness, and flatness are quantified for the entire run duration, and one-dimensional averaged kinetic-energy spectra are presented at several times.

A version of PPM for multidimensional relativistic hydrodynamics

We present an extension of the piecewise parabolic method (PPM) to multidimensional relativistic hydrodynamics. As in the original formulation, the algorithm combines a nonlinear Riemann solver, based on the ‘two-shock’ approximation, with second-order accurate temporal evolution based on characteristic projection operators and piecewise parabolic reconstruction, which makes the scheme third-order accurate in space, away from discontinuities. We present an application of the code to the propagation of a relativistic jet.

A PARALLEL CELL-BY-CELL AMR METHOD FOR THE PPM HYDRODYNAMICS CODE

The Piecewise Parabolic Method (PPM) hydrodynamics code and other codes based on the PPM technique have been used extensively for the simulation of astrophysical phenomena. A new version of the PPM hydrodynamics code under development at the Laboratory for Computational Science and Engineering (LCSE) at the University of Minnesota is described. This new code incorporates a version of dynamic local adaptive mesh refinement (AMR) targeted specifically at improving the treatment of shocks and contact discontinuities. This AMR technique is not intended to increase grid resolution in entire regions of the problem for which standard techniques of nonuniform grids and simple grid motion are adequate. Because the AMR is targeted at surfaces within a flow that can develop complex shapes, a cell-by-cell approach to the grid refinement is adopted in order to minimize the number of refined grid cells, with the hope of controlling the computational cost. As a result of this approach, the number of refined cells needed across a given shock or captured contact discontinuity is modest (< 10). Because the PPM method is very complex, ordinarily its wide difference stencil demands that a relatively large number of extra grid cells need to be added to each end of a grid strip, even when adapting to a thin feature of the flow. To avoid the work associated in handling these extra cells, only to conveniently produce valid data in the thin strip of actual interest, we have used intermediate results of the coarse grid calculation in order to eliminate the need for all but two of these extra cells at each end of the refined grid strip. The benefits of this approach will be described through sample results in 2D, and the method for handling the boundaries of the refined grid strips efficiently will be discussed. The data structures in this method are being carefully designed to permit efficient implementation of the algorithm on clusters of shared memory machines, with automatic dynamic balancing of computational loads over the cluster members.