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1.
A new class of Godunov-type numerical methods (called here weakly nonoscillatory or WNO) for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class generalizes the classical nonoscillatory schemes. In particular, it contains modified versions of Min-Mod and UNO. Under certain conditions, convergence and error estimates for WNO methods are proved.
2.
1.IntroductionIn[1],JinandXinconstructedaclassofthelocalrelaxationapproximation--TheRelaxingSchemesforsystemsofnonlinearcollservationlawsorsinglenonlinearconservationlawson,dofi(u)~ ;ac~0,(1l)otMoxil=1zwithinitialdatau(0,x)~\"o(x),x~(xl,...,-cd),byusingtheideaofthelocalrelaxationapproximation[1--4].Theschemeisobtainedinthefollowingway:firstalinearhyperbolicsystemwithastiffsourcetermisconstructedtoapproximatetheoriginalsystem(1.1)withasmalldissipativecorrection;Then,thenewlinearhyperbolicsyste… 相似文献
3.
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. 相似文献
4.
Benjamin Gess Xavier Lamy 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(2):505-521
We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained. 相似文献
5.
Hua-zhong Tang 《计算数学(英文版)》1999,(4)
1.IntroductionLetusconsidertheCauchyproblemsfornonlinearhyperbolicscalarconservationlaws:wheref:W~WisLipschitzcontinuousfunctions,andtheinitialdata"o(x)isagivenfunctioninLI(R)nLoo(R).Asitiswell--known,thisproblemingeneraldoesnotadmitsmoothsolution,so... 相似文献
6.
Hua-Zhong Tang & Ning Zhao 《计算数学(英文版)》1999,17(4):369-378
In this paper, following the paper [7], we analyze the "sharp" estimate of the rate of entropy dissipation of the fully discrete MUSCL type Godunov schemes by the general compact theory introduced by Coquel-LeFloch [1, 2], and find: because of small viscosity of the above schemes, in the vicinity of shock wave, the estimate of the above schemes is more easily obtained, but for rarefaction wave, we must impose a "sharp" condition on limiter function in order to keep its entropy dissipation and its convergence. 相似文献
7.
G. Bretti R. Natalini B. Piccoli 《Journal of Computational and Applied Mathematics》2007,210(1-2):71-77
We introduce a simulation algorithm based on a fluid-dynamic model for traffic flows on road networks, which are considered as graphs composed by arcs that meet at some junctions. The approximation of scalar conservation laws along arcs is made by three velocities Kinetic schemes with suitable boundary conditions at junctions. Here we describe the algorithm and we give an example. 相似文献
8.
１．引言本文研究如下非线性刚性守恒律方程组的全隐式差分逼近． 方程（1．１）中的源项ｇ（ｕ,ｖ）定义为 ｇ（ｕ,ｖ）＝ｖ－（１－μ）ｆ（ｕ）,（１．２）其中ｆ是ｕ的一个给定函数,δ是一个小正参数,称为松弛时间,μ是参数．方程组（１．１）频繁出现于粘弹性力学中． 在零松弛时间限（δ→０）下,从（１．１）可得到如下方程组该方程组通常称为“平衡”模型,而方程组（１．１）称为“非平衡”模型． 文中将假设μ满足 ０＜ μ＜ １,（１．４）以便保证拟稳定性条件［１９,２０］和次特征条件［１１,２,３］： λ１≤λ＊… 相似文献
9.
Haitao Fan. 《Mathematics of Computation》2001,70(235):1043-1069
When shock speed times is rational, the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved. A sufficient condition under which the above solutions can be extended to , implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with Lax-Friedrichs' splitting, and the Godunov scheme for scalar conservation laws are found to satisfy this condition. Thus, the existence of discrete shock profiles for these schemes is established when is rational.
10.
In this paper we introduce a simulation algorithm based on fluid dynamic models to reproduce the behavior of traffic in a portion of the urban network in Rome. Numerical results, obtained comparing experimental data with numerical solutions, show the effectiveness of our approximation. 相似文献
11.
Silvia Jerez 《Numerical Methods for Partial Differential Equations》2013,29(6):2133-2145
In this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L1 and L∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
12.
Christophe Chalons 《Numerical Methods for Partial Differential Equations》2008,24(4):1127-1147
This paper presents a new numerical strategy for computing the nonclassical weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of concave–convex and convex–concave flux functions. In such situations the so‐called nonclassical shocks, violating the classical Oleinik entropy criterion and selected by a prescribed kinetic relation, naturally arise in the resolution of the Riemann problem. Enforcing the kinetic relation from a numerical point of view is known to be a crucial but challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic relation at the discrete level. The proposed strategy is based on the use of a numerical flux function and random numbers. We prove that the resulting scheme enjoys important consistency properties. Numerous numerical evidences illustrate the validity of our approach. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
13.
Huanan Yang. 《Mathematics of Computation》1996,65(213):45-67
We develop a new approach, the method of wavewise entropy inequalities for the numerical analysis of hyperbolic conservation laws. The method is based on a new extremum tracking theory and Volpert's theory of BV solutions. The method yields a sharp convergence criterion which is used to prove the convergence of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.
14.
A criterion for algebraic convergence of the entropy is presented and an algebraic convergence result for the entropy of an exclusion process is improved. A weak entropy inequality is considered and its relationship to entropic convergence is discussed. 相似文献
15.
Vuk Milisic 《Proceedings of the American Mathematical Society》2003,131(6):1727-1737
We present some new convergence results for a discrete velocities BGK approximation to an initial boundary value problem for a single hyperbolic conservation law. In this paper we show stability and convergence toward a unique entropy solution in the general framework without any restriction either on the data of the limit problem or on the set of velocity of the BGK model.
16.
本文用WENO算法解决双曲型守恒律方程组初（边值）问题．给出一种满足熵条件、Sδ熵条件和边界熵条件的WENO算法．通过这个算法就能得到守恒律方程组的数值解,数值解和理论解是非常吻合的． 相似文献
17.
Based on kinetic formulation for scalar conservation laws, we present implicit kinetic schemes. For time stepping these schemes require resolution of linear systems of algebraic equations. The scheme is conservative at steady states. We prove that if time marching procedure converges to some steady state solution, then the implicit kinetic scheme converges to some entropy steady state solution. We give sufficient condition of the convergence of time marching procedure. For scalar conservation laws with a stiff source term we construct a stiff numerical scheme with discontinuous artificial viscosity coefficients that ensure the scheme to be equilibrium conserving. We couple the developed implicit approach with the stiff space discretization, thus providing improved stability and equilibrium conservation property in the resulting scheme. Numerical results demonstrate high computational capabilities (stability for large CFL numbers, fast convergence, accuracy) of the developed implicit approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 26–43, 2002 相似文献
18.
Ramaz Botchorishvili Benoit Perthame Alexis Vasseur. 《Mathematics of Computation》2003,72(241):131-157
We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.
19.
Christophe Berthon;Yves Coudière;Vivien Desveaux 《Mathematical Modelling and Numerical Analysis》2014,48(2):583-602
We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.https://doi.org/10.1051/m2an/2013105 相似文献
20.
Hua-zhong Tang 《计算数学(英文版)》2000,18(3):313-324
1. IntroductionIn [6], Jin and adn constructed a class of uPWind relaxing schemes for nonlinearconservation lawswith initial data u(0, x) ~ \"o(x), x ~ (xl, ...t -cd), by using the idea of the local relaxation approximation [2,3,6,10].The relaxing scheme is obtained in the following way: A linear hyperbolic systemwith a stiff source term is first constructed to approximate the original equation (1.1)with a small dissipative correction. Then this linear hyperbolic system is solved easilyby und… 相似文献