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1.
2.
We study the large-time behavior of solutions of viscous conservation laws. It is shown that solutions tend to diffusion waves, which are constructed based on the heat equation and Burgers equation. The convergence is in theL p , 1p sense and is obtained as a consequence of theL 2 decay of the difference between the solution and its asymptotic state of diffusion waves.Supported by the National Sciences Council of the Republic of China under the contract NSC-76-0208-M-001-09Supported by NSF under Grant No. DMS-84-01355  相似文献   

3.
We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.Research supported in part by Energy Dept. grant DEFG 02-88-ER25053Research supported in part by NSF grant DMS 90-0226 and Army grant DAAL 03-91-G0017  相似文献   

4.
A convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic systemv t+(vu)x=0, (>0,r>3) is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.  相似文献   

5.
The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t +f(u) x =u xx with the initial datau 0 which tend to the constant statesu ± asx±. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f(u ±). Moreover, the rate of asymptotics in time is investigated. For the casef(u+)(u), if the integral of the initial disturbance over (–,x) is small and decays at the algebraic rate as |x|, then the solution approaches the traveling wave at the corresponding rate ast. This rate seems to be almost optimal compared with the rate in the casef=u 2/2 for which an explicit form of the solution exists. The rate is also obtained in the casef(u ± =s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.  相似文献   

6.
Long-time effect of relaxation for hyperbolic conservation laws   总被引:2,自引:0,他引:2  
In processes such as invasion percolation and certain models of continuum percolation, in which a possibly random labelf(b) is attached to each bondb of a possibly random graph, percolation models for various values of a parameterr are naturally coupled: one can define a bondb to be occupied at levelr iff(b)r. If the labeled graph is stationary, then under the mild additional assumption of positive finite energy, a result of Gandolfi, Keane, and Newman ensures that, in lattice models, for each fixedr at which percolation occurs, the infinite cluster is unique a.s. Analogous results exist for certain continuum models. A unifying framework is given for such fixed-r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, uniqueness holds simultaneously for all values ofr. An example is given to show that when the site density is infinite, positive finite energy does not ensure uniqueness, even for fixedr. In addition, with finite site density but without positive finite energy, one can have fixed-r uniqueness a.s. for eachr, yet not have simultaneous uniqueness.Research supported by NSF grant DMS-9206139  相似文献   

7.
In this article we develop an improved version of the classical fifth-order weighted essentially non-oscillatory finite difference scheme of [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] (WENO-JS) for hyperbolic conservation laws. Through the novel use of a linear combination of the low order smoothness indicators already present in the framework of WENO-JS, a new smoothness indicator of higher order is devised and new non-oscillatory weights are built, providing a new WENO scheme (WENO-Z) with less dissipation and higher resolution than the classical WENO. This new scheme generates solutions that are sharp as the ones of the mapped WENO scheme (WENO-M) of Henrick et al. [A.K. Henrick, T.D. Aslam, J.M. Powers, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys. 207 (2005) 542–567], however with a 25% reduction in CPU costs, since no mapping is necessary. We also provide a detailed analysis of the convergence of the WENO-Z scheme at critical points of smooth solutions and show that the solution enhancements of WENO-Z and WENO-M at problems with shocks comes from their ability to assign substantially larger weights to discontinuous stencils than the WENO-JS scheme, not from their superior order of convergence at critical points. Numerical solutions of the linear advection of discontinuous functions and nonlinear hyperbolic conservation laws as the one dimensional Euler equations with Riemann initial value problems, the Mach 3 shock–density wave interaction and the blastwave problems are compared with the ones generated by the WENO-JS and WENO-M schemes. The good performance of the WENO-Z scheme is also demonstrated in the simulation of two dimensional problems as the shock–vortex interaction and a Mach 4.46 Richtmyer–Meshkov Instability (RMI) modeled via the two dimensional Euler equations.  相似文献   

8.
We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L1 for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.  相似文献   

9.
This paper proves that certain non-classical shock waves in a rotationally invariant system of viscous conservation laws posses nonlinear large-time stability against sufficiently small perturbations. The result applies to small intermediate magnetohydrodynamic shocks in the presence of dissipation.Research supported by Deutsche ForschungsgemeinschaftResearch supported in part by NSF Grant DMS 90-0226 and Army Grant DAAL 03-91-G-0017  相似文献   

10.
This paper proposes a new WENO procedure to compute problems containing both discontinuities and a large disparity of characteristic scales.In a one-dimensional context, the WENO procedure is defined on a three-points stencil and designed to be sixth-order in regions of smoothness. We define a finite-volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns. The reconstruction of their point-values is then ensured by a unique sixth-order Hermite polynomial. This polynomial is considered as a symmetric and convex combination, by ideal weights, of three fourth-order polynomials: a central polynomial, defined on the three-points stencil, is combined with two polynomials based on the left and the right two-points stencils.The symmetric nature of such an interpolation has an important consequence: the choice of ideal weights has no influence on the properties of the discretization. This advantage enables to formulate the Hermite interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, nonlinear weights are then defined.To deal with the peculiarities of the Hermite interpolation near discontinuities, we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights, in regions of smoothness, and one-sided weights, otherwise.The resulting scheme is a sixth-order WENO method based on central Hermite interpolation and TVD Runge–Kutta time-integration. We call this scheme the HCWENO6 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In these experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems.  相似文献   

11.
The present paper deals with an efficient and accurate limiting strategy for the multi-dimensional hyperbolic conservation laws on unstructured grids. The multi-dimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The basic idea of the proposed limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured meshes are obtained.Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the newly developed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Numerous test cases are presented to validate the basic features of the proposed approach.  相似文献   

12.
In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.  相似文献   

13.
The Osher–Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag–Tang 2D vortex problem. Total-variation-bounded (TVB) behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley–Leverett test simulations.  相似文献   

14.
Novel limiters based on the weighted average procedure are developed for finite volume methods solving multi-dimensional hyperbolic conservation laws on unstructured grids. The development of these limiters is inspired by the biased averaging procedure of Choi and Liu [10]. The remarkable features of the present limiters are the new biased functions and the weighted average procedure, which enable the present limiter to capture strong shock waves and achieve excellent convergence for steady state computations. The mechanism of the developed limiters for eliminating spurious oscillations in the vicinity of discontinuities is revealed by studying the asymptotic behavior of the limiters. Numerical experiments for a variety of test cases are presented to demonstrate the superior performance of the proposed limiters.  相似文献   

15.
A global formalism for nonlinear waves in conservation laws   总被引:1,自引:0,他引:1  
We introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws. Fundamental waves are represented as pairs of states statisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an (n+1)-dimensional manifold, the fundamental wave manifold. There is a distinguishedn-dimensional submanifold of containing a single one-dimensonal foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of itself that represent shock curves. We identify othern-dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves. This geometric framework promises to simplify greatly the study of the stability and bifurcation propertiesThis work was supported in part by: the NSF/CNPq U.S.-Latin America Cooperative Science Program under Grant INT-8612605; the Institute for Mathematics and its Applications with funds provided by the National Science Foundation; the Air Force Office of Scientific Research under Grant AFOSR 90-0075; the National Science Foundation under Grant 8901884; the U.S. Department of Energy under Grant DE-FG02-90ER25084; the U.S. Army Research Office under Grant DAAL03-89-K-0017; the Financiadora de Estudos e Projetos; the Conselho Nacional de Desenvolvimento Científico e Tecnológica (CNPq); the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ); the Coordenação de Aperfeiçamento de Pessoal de Ensino Superior (CAPES); and the Sociedade Brasileira de Matemática (SBM)  相似文献   

16.
We revisit the structure of viscous systems of conservation laws that are entropy-dissipative. We prove that the dissipated quantities are spatial derivatives of certain nonlinear coordinates that are defined only in terms of the entropy and of the linear, non-dissipated, coordinates.  相似文献   

17.
This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes.In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials.The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights.We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered.The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.  相似文献   

18.
This paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems. It is shown that the Newton–Raphson (NR) method converges for any time step size, if the flux function is convex, concave, or linear, which is, in general, the case for CFD problems. In some problems, e.g., multiphase flow in porous media, the nonlinear flux function is S-shaped (not uniformly convex or concave); as a result, a standard NR iteration can diverge for large time steps, even if an implicit discretization scheme is used to solve the nonlinear system of equations. In practice, when such convergence difficulties are encountered, the current time step is cut, previous iterations are discarded, a smaller time step size is tried, and the NR process is repeated. The criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and/or NR iteration. Here, we propose a simple modification to the NR iteration scheme for conservation laws with S-shaped flux functions that converges for any time step size. The new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver. The proposed method can be implemented in an existing simulator, e.g., for CO2 sequestration or reservoir flow modeling, quite easily. The numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media. The analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems.  相似文献   

19.
This work addresses existence of traveling waves for systems of conservation laws through an Hamiltonian formalism. A related Hamilton–Jacobi equation is given and studied. Our proofs rely on the construction of a cost functional and require a smallness condition for the dispersion with respect to diffusion. Some numerical applications are also presented, concerning a model phase transition problem.  相似文献   

20.
It is important for nonlinear hyperbolic conservation laws (NHCL) to own a simulation scheme with high order accuracy, simple computation, and non-oscillatory character. In this paper, a unified and novel lattice Boltzmann model is presented for solving n-dimensional NHCL with the source term. By introducing the high order source term of explicit lattice Boltzmann method (LBM) and the optimum dimensionless relaxation time varied with the specific issues, the effects of space and time resolutions on the accuracy and stability of the model are investigated for the different problems in one to three dimensions. Both the theoretical analysis and numerical simulation validate that the results by the proposed LBM have second-order accuracy in both space and time, which agree well with the analytical solutions.  相似文献   

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