数学物理学报(英文版)

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JAMES GLIMM: MATHEMATICIAN, FRIEND, MENTO

Arthur Jaffe, Thomas Spencer

数学物理学报(英文版). 2010 (2): 369-376. DOI: 10.1016/S0252-9602(10)60054-9

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A description of Jim's contributions to mathematics would take us far beyond our expertise. Jim was never content to spend too long in any one area of mathematics. Operator algebras, hyperbolic systems of conservation laws, fluid dynamics, quantum field theory and statistical mechanics and the computational aspects of applied mathematics are just a few of the areas to which Jim has contributed. Here we relate a few impressionistic recollections of personal interactions, together with some ideas about his work in quantum field theory and statistical mechanics. In spite of the non-technical character of these observations, we hope nevertheless that they will provide pleasure for our dear friend and for other readers as well.

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STUDY OF CRYSTAL GROWTH AND SOLUTE PRECIPITATION THROUGH |FRONT TRACKING METHOD

Xiaolin Li, James Glimm, Xiangmin Jiao, Charles Peyser, Yanhong Zhao

数学物理学报(英文版). 2010 (2): 377-390. DOI: 10.1016/S0252-9602(10)60055-0

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Crystal growth and solute precipitation is a Stefan problem. It is a free boundary problem for a parabolic partial
differential equation with a time-dependent phase interface. The velocity of the moving interface between solute and crystal is a local function. The dendritic structure of the crystal interface, which develops dynamically, requires high resolution of the interface geometry. These facts make the Lagrangian front tracking method well suited for the problem. In this paper, we introduce an upgraded version of the front tracking code and its associated algorithms for the numerical study of crystal formation. We compare our results with the smoothed particle hydrodynamics method (SPH) in terms of the crystal fractal dimension with its dependence on the Damkohler number and density ratio.

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A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT XISYMMETRIC VESSELS

Gui-Qiang G. Chen, Weihua Ruan

数学物理学报(英文版). 2010 (2): 391-427. DOI: 10.1016/S0252-9602(10)60056-2

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We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisymmetric vessels. Early models derived are nonconservative and/or nonhomogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The
Riemann solutions may consist of four waves for some cases. The system can also be written as a 3×3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.

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AN ADAPTIVE VERSION OF GLIMM'S SCHEME

H. Kim, M. Laforest, D. Yoon

数学物理学报(英文版). 2010 (2): 428-446. DOI: 10.1016/S0252-9602(10)60057-4

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This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L¹(R) for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad-hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.

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EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

Blake Barker, Olivier Lafitte, Kevin Zumbrun

数学物理学报(英文版). 2010 (2): 447-498. DOI: 10.1016/S0252-9602(10)60058-6

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For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with γ-law gas equation of state, γ≥1, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes.
In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. Considered as three-dimensional solutions,
undercompressive shocks are Lax-type (Alfven) waves. For the monatomic and diatomic cases γ=5/3 and γ= 7/5,
with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well, both with respect to two-dimensional and (in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.

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AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

Shuqiang Wang, Roman Samulyak, Tongfei Guo

数学物理学报(英文版). 2010 (2): 499-521. DOI: 10.1016/S0252-9602(10)60059-8

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The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella (1998, J. Comput. Phys. 147, 60) has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.

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PARALLEL ADAPTIVE SIMULATION OF A PLUNGING LIQUID JET

Azat Yu. Galimov, Onkar Sahni, Richard T. Lahey Jr., Donald A. Drew, Kenneth E. Jansen

数学物理学报(英文版). 2010 (2): 522-538. DOI: 10.1016/S0252-9602(10)60060-4

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This paper is concerned with three-dimensional numerical simulation of a plunging liquid jet. The transient processes of forming an air cavity around the jet, capturing an initially large air bubble, and the break-up of this large toroidal-shaped bubble into smaller bubbles were analyzed. A stabilized finite element method (FEM) was employed under parallel numerical simulations based on adaptive, unstructured grid and coupled with a level-set method to track the interface between air and liquid. These simulations show that the inertia of the liquid jet initially depresses the pool's surface, forming an annular air cavity which surrounds the liquid jet. A toroidal liquid eddy which is subsequently formed in the liquid pool results in air cavity collapse, and in turn entrains air into the liquid pool from the unstable annular air gap region around the liquid jet.

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BOUNDARY PROCEDURES FOR THE TIME - DEPENDENT BURGERS’EQUATION UNDER UNCERTAINTY

Per Pettersson, Jan Nordstrom, Gianluca Iaccarino

数学物理学报(英文版). 2010 (2): 539-550. DOI: 10.1016/S0252-9602(10)60061-6

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The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to influence the solution; we will
discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.

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MULTISCALE ISSUES IN DNS OF MULTIPHASE FLOWS

G. Tryggvason, S. Thomas, J. Lu, B. Aboulhasanzadeh

数学物理学报(英文版). 2010 (2): 551-562. DOI: 10.1016/S0252-9602(10)60062-8

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Direct numerical simulations (DNS) have now become a well established tool to examine complex multiphase flows. Such flows typically exhibit a large range of scales and it is generally necessary to use different descriptions of the flow depending on the scale that we are examining. Here we discuss multiphase flows from a multiscale perspective. Those include both how DNS are providing insight and understanding for modeling of scales much larger than the “dominant scale”(defined where surface tension, viscous forces or inertia are important), as well as how DNS are often limited by the need to resolve processes taking place on much smaller scales. Both problems can be cast into a language introduced for general classes of multiscale problems and reveal that while the classification may be new, the issues are not.

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PRESSURE-VELOCITY EQUILIBRIUM HYDRODYNAMIC MODELS

John W. Grove

数学物理学报(英文版). 2010 (2): 563-594. DOI: 10.1016/S0252-9602(10)60063-X

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This article describes mathematical models for phase separated mixtures of materials that are in pressure and velocity equilibrium but not necessarily temperature equilibrium. General conditions for constitutive models for such mixtures that exhibit a single mixture sound speed
are discussed and specific examples are described.

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HIGH-ORDER WENO SIMULATIONS OF THREE-DIMENSIONAL RESHOCKED RICHTMYER--MESHKOV INSTABILITY TO LATE TIMES: DYNAMICS, DEPENDENCE ON INITIAL CONDITIONS, AND COMPARISONS TO EXPERIMENTAL DATA

Oleg Schilling, Marco Latini

数学物理学报(英文版). 2010 (2): 595-620. DOI: 10.1016/S0252-9602(10)60064-1

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The dynamics of the reshocked multi-mode Richtmyer--Meshkov instability is investigated using 513×257² three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations. A two-mode initial perturbation with superposed random noise is used to model the Mach 1.5 air/SF₆ Vetter--Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces, and density cross-sections are utilized to show the detailed flow structure before, during, and after reshock. It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock. The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model. A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented. Finally, the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra, as well as the evolution of the baroclinic enstrophy production, buoyancy production, and shear production terms in the enstrophy and turbulent kinetic transport equations.

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RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS -A REVIEW

Christian Klingenberg, Knut Waagan

数学物理学报(英文版). 2010 (2): 621-632. DOI: 10.1016/S0252-9602(10)60065-3

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We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4], [5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We describe the extension to higher order and multiple space dimensions. Finally we show our implementation of all this into the astrophysics code FLASH.

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THEORETICAL ANALYSIS OF A HIGH-MACH-NUMBER DILUTE INELASTIC GAS IMPINGING ON A SURFACE

Yu Hui Deng, Jonathan J. Wylie, Qiang Zhang

数学物理学报(英文版). 2010 (2): 633-644. DOI: 10.1016/S0252-9602(10)60066-5

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We consider a problem of a 2-dimensional high-Mach-number dilute inelastic gas originating from infinity impinging on an oblique surface of infinite length. By analyzing probabilities of particle-particle collisions, we derive an analytical formula for the mean force experienced by the surface when the system is dilute. We also derive conditions for the validity of our theory.

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