共查询到20条相似文献,搜索用时 0 毫秒
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On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law
A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero. 相似文献
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Matthias Kunik 《Mathematical Methods in the Applied Sciences》1992,15(7):495-509
We examine the existence and regularity results for a scalar conservation law with a convexity condition and solve its weak solution with shocks by using a special method of characterization combined with a representation formula for the weak solution. 相似文献
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In this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics. The generalization has two aspects: The new entropy functional is defined not for one but for two solutions. It is defined in terms of the L1 distance between the two solutions as well as the variations of each separate solution. In addition, it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero. © 1999 John Wiley & Sons, Inc. 相似文献
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Małgorzata Peszyńska 《Numerical Methods for Partial Differential Equations》2014,30(1):239-264
We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014 相似文献
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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
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Graziano Guerra 《Journal of Differential Equations》2004,206(2):438-469
We consider the Cauchy problem for the 2×2 strictly hyperbolic system
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Travelling waves of a viscous conservation law (so-called viscous profiles) are shown to be stable in polynomialy weighted L∞ spaces. There is no assumption of convexity on the nonlinear term and thus earlier results of Il'in and Oleǐnik, Sattinger, and Kawashima and Matsumura are generalized. The method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula. Estimates are derived for the semigroup through a new technique for estimating the resolvent. © 1993 John Wiley & Sons, Inc. 相似文献
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Philippe Le Floch 《Mathematical Methods in the Applied Sciences》1988,10(3):265-287
We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ? ? . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz. 相似文献
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In this paper, we solve the Riemann problem with the initial data containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. Under suitably generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves are proved. Further, four kinds of different structure for solutions are established uniquely. 相似文献
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In this paper we consider a boundary-value problem for one-dimensional hyperbolic equation with nonlocal initial data in integral form. We prove the existence and uniqueness of the generalized solution. 相似文献
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Daniel N. Ostrov 《Journal of Mathematical Analysis and Applications》2007,335(2):996-1012
We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur. 相似文献
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We derive an algorithm for solving the initial value problem for ut = ½σ2uxx + f(u)ux. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for ut = f(u)ux. A comparison with the random choice method is illustrated by the numerical example. 相似文献
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In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions \(\rho \) to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe \(\rho (x,t)\) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of \(\rho (x,t)\) over \(0 \le x \le L\) for \(L > 0\), with a suitable random boundary condition at \(x = L\). 相似文献
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V. M. Goloviznin A. A. Kanaev 《Computational Mathematics and Mathematical Physics》2012,52(3):400-410
The CABARET computational algorithm is generalized to one-dimensional scalar quasilinear hyperbolic partial differential equations
with allowance for inequality constraints on the solution. This generalization can be used to analyze seepage of liquid radioactive
wastes through the unsaturated zone. 相似文献
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Julien Jimenez 《Journal of Evolution Equations》2011,11(3):553-576
We deal in this paper with a scalar conservation law, set in a bounded multidimensional domain, and such that the convective
term is discontinuous with respect to the space variable. First, we introduce a weak entropy formulation for the homogeneous
Dirichlet problem associated with the first-order reaction-convection equation that we consider. Then, we establish an existence
and uniqueness property for the weak entropy solution. The method of doubling variables and a pointwise reasoning along the
curve of discontinuity are used to state uniqueness. Finally, the vanishing viscosity method allows us to prove the existence
result. Another method to obtain the existence of a solution, which relies on the regularization of the flux, is also detailled,
at least for a particular case. 相似文献