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1.
Explicit and semi-implicit finite-difference schemes approximatingnon-homogeneous scalar conservation laws are analyzed. Optimalerror bounds independent of the stiffness of the underlyingequation are presented.
This author has been supported by The Norwegian Research Council(NFR), program No 100284/431. e-mail: schroll{at}igpm.rwth-aachen.de
This author has been supported by The Norwegian Research Council(NFR), program Nos 100284/431 and STP.29643. e-mail: ragnar{at}ifi.uio.no 相似文献
2.
New first- and high-order centred methods for conservation lawsare presented. Convenient TVD conditions for constructing centredTVD schemes are then formulated and some useful results areproved. Two families of centred TVD schemes are constructedand extended to nonlinear systems. Some numerical results arealso presented. 相似文献
3.
Yousef Hashem Zahran 《Journal of Mathematical Analysis and Applications》2008,346(1):120-140
In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented. 相似文献
4.
Daniel Bouche Jean-Michel Ghidaglia Frédéric P. Pascal 《Applied Numerical Mathematics》2011,61(11):1114-1131
The purpose of this paper is to show that the cell-centered upwind Finite Volume scheme applied to general hyperbolic systems of m conservation laws approximates smooth solutions to the continuous problem at order one in space and time. As it is now well understood, there is a lack of consistency for order one upwind Finite Volume schemes: the truncation error does not tend to zero as the time step and the grid size tend to zero. Here, following our previous papers on scalar equations, we construct a corrector that allows us to prove the expected error estimate for nonlinear systems of equations in one dimension. 相似文献
5.
Ritesh Kumar Dubey 《Applied mathematics and computation》2010,215(9):3335-3342
In this work a first order accurate semi-conservative composite scheme is presented for hyperbolic conservation laws. The idea is to consider the non-conservative form of conservation law and utilize the explicit wave propagation direction to construct semi-conservative upwind scheme. This method captures the shock waves exactly with less numerical dissipation but generates unphysical rarefaction shocks in case of expansion waves with sonic points. It shows less dissipative nature of constructed scheme. In order to overcome it, we use the strategy of composite schemes. A very simple criteria based on wave speed direction is given to decide the iterations. The proposed method is applied to a variety of test problems and numerical results show accurate shock capturing and higher resolution for rarefaction fan. 相似文献
6.
《Journal of Computational and Applied Mathematics》2002,138(1):93-108
A class of semi-discrete third-order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws. These schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing (TVD) in the zero relaxation limit. A third-order TVD Runge–Kutta splitting method is developed for the temporal discretization of the semi-discrete schemes. Numerical results are given illustrating these schemes on one-dimensional nonlinear problems. 相似文献
7.
Hongxia Liu 《Journal of Differential Equations》2007,235(2):658-667
A generalized entropy functional was introduced in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the scalar hyperbolic conservation laws with convex flux function. This functional was crucially used in the functional approach to the L1 stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate. However, how to construct the generalized entropy functional for scalar conservation laws with general flux, and then how to apply the functional approach to the L1 study on general systems are still open. In this paper, we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction. 相似文献
8.
Yousef Hashem Zahran 《Applied mathematics and computation》2009,212(1):37-50
In this paper we propose a new WENO scheme, in which we use a central WENO [G. Capdeville, J. Comput. Phys. 227 (2008) 2977-3014] (CWENO) reconstruction combined with the smoothness indicators introduced in [R. Borges, M. Carmona, B. Costa, W. Sun Don, J. Comput. Phys. 227 (2008) 3191-3211] (IWENO). We use the central-upwind flux [A. Kurganov, S. Noelle, G. Petrova, SIAM J. Sci. Comp. 23 (2001) 707-740] which is simple, universal and efficient. For time integration we use the third order TVD Runge-Kutta scheme. The resulting scheme improves the convergence order at critical points of smooth parts of solution as well as decrease the dissipation near discontinuities. Numerical experiments of the new scheme for one and two-dimensional problems are reported. The results demonstrates that the proposed scheme is superior to the original CWENO and IWENO schemes. 相似文献
9.
We are interested in hyperbolic systems of conservation laws with relaxation and dissipation, particularly the zero relaxation limit. Such a limit is of interest in several physical situations, including gas flow near thermo-equilibrium, kinetic theory with small mean free path, and viscoelasticity with vanishing memory. In this article we study hyperbolic systems of two conservation laws with relaxation. For the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds, we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit. For the marginally stable case where the equilibrium speed may equal one of the frozen speeds, we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation. Our analysis includes the construction of suitable entropy pairs to derive energy estimates. We need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems. The theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit. © 1993 John Wiley & Sons, Inc. 相似文献
10.
The paper presents a well‐posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimm's existence theory and discuss the principle of nonlinear through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L1(x) distance between the two solutions and is time‐decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L1(x) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional. © 1999 John Wiley & Sons, Inc. 相似文献
11.
Summary. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented.
Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given
cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines
of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent),
in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann
solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution
content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected
third-order resolution.
Received April 10, 1996 / Revised version received January 20, 1997 相似文献
12.
Daniel Castaño Díez Max Gunzburger Angela Kunoth 《Numerical Methods for Partial Differential Equations》2008,24(6):1388-1404
We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 相似文献
13.
Viscous profiles of shock waves in systems of conservation laws can be viewed as heteroclinic orbits in associated systems of ordinary differential equations (ODE). In the case of overcompressive shock waves, these orbits occur in multi-parameter families. We propose a numerical method to compute families of heteroclinic orbits in general systems of ODE. The key point is a special parameterization of the heteroclinic manifold which can be understood as a generalized phase condition; in the case of shock profiles, this phase condition has a natural interpretation regarding their stability. We prove that our method converges and present numerical results for several systems of conservation laws. These examples include traveling waves for the Navier-Stokes equations for compressible viscous, heat-conductive fluids and for the magnetohydrodynamics equations for viscous, heat-conductive, electrically resistive fluids that correspond to shock wave solutions of the associated ideal models, i.e., the Euler, resp. Lundquist, equations.
14.
Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds 总被引:1,自引:0,他引:1
Matania Ben-Artzi Philippe G. LeFloch 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2007,24(6):989-1008
Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust Lp estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L1 contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method. 相似文献
15.
16.
Xiaohan Cheng Yufeng Nie Jianhu Feng Li Cai 《Journal of Applied Analysis & Computation》2015,5(3):453-464
A high order central-upwind scheme for approximating hyperbolic conservation laws is proposed. This construction is based on the evaluation of the local propagation speeds of the discontinuities and Peer's fourth order non-oscillatory reconstruction. The presented scheme shares the simplicity of central schemes, namely no Riemann solvers are involved. Furthermore, it avoids alternating between two staggered grids, which is particularly a challenge for problems which involve complex geometries and boundary conditions. Numerical experiments demonstrate the high resolution and non-oscillatory properties of our scheme. 相似文献
17.
This article examines the utilization of a spatial averaging technique to the nonlinear terms of the partial differential equations as an inviscid shock-regularization of hyperbolic conservation laws. A central motivation is to promote the idea of applying filtering techniques such as the observable divergence method, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence in inviscid flows while, on the other hand, generalizing and unifying previous mathematical and numerical analysis of the method applied to the one-dimensional Burgers’ and Euler equations. This article primarily concerns the mathematical analysis of the technique and examines two fundamental issues. The first is on the global existence and uniqueness of classical solutions for the regularization under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines one-dimensional scalar conservation laws and shows that the inviscid regularization method captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes. 相似文献
18.
We study high order convergence of vanishing viscosity approximation to scalar hyperbolic conservation laws in one space dimension. We prove that, under suitable assumptions, in the region where the solution is smooth, the viscous solution admits an expansion in powers of the viscosity parameter ε. This allows an extrapolation procedure that yields high order approximation to the non-viscous limit as ε→0. Furthermore, an integral across a shock also admits a power expansion of ε, which allows us to construct high order approximation to the location of the shock. Numerical experiments are presented to justify our theoretical findings. 相似文献
19.
Robert Kirby. 《Mathematics of Computation》2003,72(243):1239-1250
A class of finite volume methods based on standard high resolution schemes, but which allows spatially varying time steps, is described and analyzed. A maximum principle and the TVD property are verified for general advective flux, extending the previous theoretical work on local time stepping methods. Moreover, an entropy condition is verified which, with sufficient limiting, guarantees convergence to the entropy solution for convex flux.
20.
L. V. Gargyants 《Differential Equations》2016,52(4):458-466
A one-dimensional conservation law with a power-law flux function and an exponential initial condition is considered. We construct a generalized entropy solution with countably many shock waves. This solution is sign-alternating and one-sided periodic. 相似文献