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SIAM J. on Numerical Analysis

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1998

Volume 35, Issue 6, pp. 2117-2485


Convex Entropies and Hyperbolicity for General Euler Equations

Ami Harten, Peter D. Lax, C. David Levermore, and William J. Morokoff

SIAM J. Numer. Anal. 35, pp. 2117-2127 (11 pages) | Cited 6 times

Online Publication Date: July 25, 2006

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The compressible Euler equations possess a family of generalized entropy densities of the form $\rho f(\sigma)$, where $\rho$ is the mass density, $\sigma$ is the specific entropy, and f is an arbitrary function. Entropy inequalities associated with convex entropy densities characterize physically admissible shocks. For polytropic gases, Harten has determined which $\rho f(\sigma)$ are strictly convex. In this paper we extend this determination to gases with an arbitrary equation of state. Moreover, we show that at every state where the sound speed is positive (i.e., where the Euler equations are hyperbolic) there exist $\rho f(\sigma)$ that are strictly convex, thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity.

Multiresolution Representation in Unstructured Meshes

Rémi Abgrall

SIAM J. Numer. Anal. 35, pp. 2128-2146 (19 pages) | Cited 5 times

Online Publication Date: July 25, 2006

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In this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components. To do so we consider a nested sequence of discretization, which corresponds to increasing levels of resolution, and we define the scales as the "difference in information" between any two successive levels. We obtain data compression by eliminating scale-coefficients which are sufficiently small. This capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale-coefficients.

High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation Laws

G. S. Jiang, D. Levy, C. T. Lin, S. Osher, and E. Tadmor

SIAM J. Numer. Anal. 35, pp. 2147-2168 (22 pages) | Cited 51 times

Online Publication Date: July 25, 2006

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We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.

Nonclassical Shocks and Kinetic Relations: Finite Difference Schemes

Brian T. Hayes and Philippe G. LeFloch

SIAM J. Numer. Anal. 35, pp. 2169-2194 (26 pages) | Cited 13 times

Online Publication Date: July 25, 2006

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We consider hyperbolic systems of conservation laws that are not genuinely nonlinear. The solutions generated by diffusive-dispersive regularizations may include nonclassical (n.c.) shock waves that do not satisfy the classical Liu entropy criterion. We investigate the numerical approximation of n.c. shocks via conservative difference schemes constrained only by a single entropy inequality.The schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms.
Limits of these schemes are characterized via the kinetic relation introduced earlier by the authors. We determine the kinetic function numerically for several examples of systems and schemes. This study demonstrates that the kinetic relation is a suitable tool for the selection of unique n.c. solutions and for the study of their sensitive dependence on the critical parameters: the ratios of diffusion/dispersion and diffusion/mesh size, the shock strength, and the order of discretization of the flux.

On the Analysis of Finite Volume Methods for Evolutionary Problems

K. W. Morton

SIAM J. Numer. Anal. 35, pp. 2195-2222 (28 pages) | Cited 16 times

Online Publication Date: July 25, 2006

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Finite volume or finite element methods now dominate the modeling of steady flows because of their natural handling of complex configurations. They should have a similar advantage for unsteady flows; however, their error analysis on nonuniform meshes has met a number of difficulties. In this paper a new approach is adopted which makes greater use of a Godunov formulation. It opens the way for a fuller comparison of characteristic-based methods with semidiscrete methods.

Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics

Frédéric Coquel and Benoît Perthame

SIAM J. Numer. Anal. 35, pp. 2223-2249 (27 pages) | Cited 16 times

Online Publication Date: July 25, 2006

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We consider the Euler equations for a compressible inviscid fluid with a general pressure law $p(\rho,\varepsilon)$, where $\rho$ represents the density of the fluid and $\varepsilon$ its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy decomposition under the form $\varepsilon= \varepsilon _1 + \varepsilon _2.$ The internal energy $\varepsilon _1$ is associated with a (simpler) pressure law $p_1(\rho,\varepsilon_1)$; the energy $\varepsilon _2$ is advected by the flow. These two energies are also subject to a relaxation process and in the limit of an infinite relaxation rate, we recover the initial pressure law p. We show that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, we can find a global convex and uniform entropy for the relaxation system.
From our construction, we also deduce the extension to general pressure laws of classical approximate Riemann solvers for polytropic gases, which only use a single call to the pressure law (per mesh point and time step). For the Godunov scheme, we show that this extension satisfies stability, entropy, and accuracy conditions.

Composite Schemes for Conservation Laws

Richard Liska and Burton Wendroff

SIAM J. Numer. Anal. 35, pp. 2250-2271 (22 pages) | Cited 22 times

Online Publication Date: July 25, 2006

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Global composition of several time steps of the two-step Lax--Wendroff scheme followed by a Lax--Friedrichs step seems to enhance the best features of both, although it is only first order accurate. We show this by means of some examples of one-dimensional shallow water flow over an obstacle. In two dimensions we present a new version of Lax--Friedrichs and an associated second order predictor-corrector method. Composition of these schemes is shown to be effective and efficient for some two-dimensional Riemann problems and for Noh's infinite strength cylindrical shock problem. We also show comparable results for composition of the predictor-corrector scheme with a modified second order accurate weighted essentially nonoscillatory (WENO) scheme. That composition is second order but is more efficient and has better symmetry properties than WENO alone. For scalar advection in two dimensions the optimal stability of the new predictor-corrector scheme is shown using computer algebra. We also show that the generalization of this scheme to three dimensions is unstable, but by using sampling we are able to show that the composites are suboptimally stable.

Unstable Godunov Discrete Profiles for Steady Shock Waves

Matthieu Bultelle, Magali Grassin, and Denis Serre

SIAM J. Numer. Anal. 35, pp. 2272-2297 (26 pages) | Cited 4 times

Online Publication Date: July 25, 2006

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Given a system of n conservation laws ut+f(u)x=0, the steady shock waves, when processed by the Godunov scheme, admit rather simple discrete profiles. One shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an (n-1)-dimensional space. Applying our theory to the gas dynamics with the perfect gas law p=(\ga-1)\rho e$, we construct unstable profiles for values of $\ga$ between 1 and $\ga^*$ and rather strong shocks. Since $\ga^*\in]7/5,5/3[$, our result applies to the air but not to monoatomic gases.
Finally, we illustrate this analysis by some numerical experiments, both with $\ga<\ga^*$ and $\ga>\ga^*$.

Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems

Marsha J. Berger and Randall J. LeVeque

SIAM J. Numer. Anal. 35, pp. 2298-2316 (19 pages) | Cited 34 times

Online Publication Date: July 25, 2006

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An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the AMRCLAW package, which is freely available.

Sticky Particles and Scalar Conservation Laws

Yann Brenier and Emmanuel Grenier

SIAM J. Numer. Anal. 35, pp. 2317-2328 (12 pages) | Cited 39 times

Online Publication Date: July 25, 2006

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One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum.

Stability of Steady Solutions of Burgers's Equation

Gunilla Kreiss and Heinz-Otto Kreiss

SIAM J. Numer. Anal. 35, pp. 2329-2349 (21 pages)

Online Publication Date: July 25, 2006

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We develop tools to discuss the stability of viscous shock profiles for conservation laws. The technique is based on replacing the far field regions by boundary conditions that describe the effect of the outer solution on the solution in the shock layer region. We apply the technique to Burgers's equation and prove stability for perturbations that are not localized in space.

On Families of Pointwise Optimal Finite Volume ENO Approximations

Thomas Sonar

SIAM J. Numer. Anal. 35, pp. 2350-2369 (20 pages) | Cited 6 times

Online Publication Date: July 25, 2006

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The present work is devoted to the construction of radial functions which can serve as recovery functions in essentially nonoscillatory (ENO) approximations of hyperbolic conservation laws on unstructured grids. Conditionally positive $\lambda$-definite radial functions are shown to be pointwise optimal and therefore very well suited for use in finite volume schemes. In thepresent paper we develop the theory of pointwise optimality and give examples.

Spectral Simulation of Supersonic Reactive Flows

Wai Sun Don and David Gottlieb

SIAM J. Numer. Anal. 35, pp. 2370-2384 (15 pages) | Cited 12 times

Online Publication Date: July 25, 2006

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We present in this paper numerical simulations of reactive flows interacting with shock waves. We argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long-term integrations of complicated flows, even in the presence of shock waves. A spectral code is described in detail, and the theory that leads to each of its components is explained. Results of interactions of hydrogen jetswith shock waves are presented and analyzed, and comparisons with ENO finite difference schemes are carried out.

Numerical Passage from Systems of Conservation Laws to Hamilton--Jacobi Equations, and Relaxation Schemes

Shi Jin and Zhouping Xin

SIAM J. Numer. Anal. 35, pp. 2385-2404 (20 pages) | Cited 13 times

Online Publication Date: July 25, 2006

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In this paper we study the numerical transition from a Hamilton--Jacobi (H--J) equation to its associated system of conservation laws in arbitrary space dimensions. We first study how, in a very generic setting, one can recover the viscosity solutions of the H--J equation using the numerical solutions to the system of conservation laws. We then introduce a simple, second-order relaxation scheme to solve the underlying weakly hyperbolic system of conservation laws.

Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations

Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani

SIAM J. Numer. Anal. 35, pp. 2405-2439 (35 pages) | Cited 38 times

Online Publication Date: July 25, 2006

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Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes-type parabolic equations, such as the heat equation, the porous media equations, the advection-diffusion equation, and the viscous Burgers equation. In such problems the diffusive relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter. Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term. This, however, introduces new difficulties due to the dependence of the stiff source term on the gradient. We show how to overcome this new difficulty with an adequately designed, economical discretization procedure for the relaxation term. These schemes are shown to have the correct diffusion limit. Several numerical results in one and two dimensions are presented, which show the robustness, as well as the uniform accuracy, of our schemes.

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

Bernardo Cockburn and Chi-Wang Shu

SIAM J. Numer. Anal. 35, pp. 2440-2463 (24 pages) | Cited 30 times

Online Publication Date: July 25, 2006

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In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

The Convergence Rate of Finite Difference Schemes in the Presence of Shocks

Bjorn Engquist and Björn Sjögreen

SIAM J. Numer. Anal. 35, pp. 2464-2485 (22 pages) | Cited 12 times

Online Publication Date: July 25, 2006

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Finite difference approximations generically have ${\cal O}(1)$ pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.
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