Well-posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition

In this paper we give a simple proof of well-posedness of multidimensional scalar conservations laws with a strong boundary condition. The proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation.     

Two new moving boundary problems for scalar conservation laws

The application of the theory of scalar conservation laws to semiconductor device fabrication is described. This application is the source of a Stefan problem and another moving boundary problem for a class of such equations. The analogue of the Riemann problem for these problems is analyzed and solved. Conditions on the boundary values that characterize physically correct solutions are derived.     

Stochastic scalar conservation laws

We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L¹ contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

Third-order scalar evolution equations with conservation laws

Associated to the family of third-order quasilinear scalar evolution equations is the geometry of point transformations. This geometry provides a framework from which to study the structure of conservation laws of the equation, and to study the special nature of the geometry of those equations which do possess conservation laws. There is an easy and obvious normal form for equations which possess at least one conservation law. The geometric structure of the equation gives rise to a simple yet much less obvious normal form for equations which possess at least two conservation laws.     

IBVPs for scalar conservation laws with time discontinuous fluxes

The initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely dependence on time. These results ensure, for instance, the well‐posedness of a class of vehicular traffic models with time‐dependent speed limits. A traffic management problem is then shown to admit an optimal solution.     

Finite-difference schemes for scalar conservation laws with source terms

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     

Implicit kinetic schemes for scalar conservation laws

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     

Equilibrium schemes for scalar conservation laws with stiff sources

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.

     

Weakly nonoscillatory schemes for scalar conservation laws

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.

     

Kruzkov's estimates for scalar conservation laws revisited

We give a synthetic statement of Kruzkov-type estimates for multi-dimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in known for finite volume methods on unstructured grids.

Les estimations de Kruzkov pour les lois de conservation scalaires revisitées

Résumé Nous donnons un énoncé synthétique des estimations de type de Kruzkov pour les lois de conservation scalaires multidimensionnelles. Nous l'appliquons pour obtenir d'estimations nombreuses pour problèmes différents d'approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en connue pour les méthodes de volumes finis sur des maillages non structurés.

     

Diffusion-dispersion limits for multidimensional scalar conservation laws with source terms

In this paper we consider conservation laws with diffusion and dispersion terms. We study the convergence for approximation applied to conservation laws with source terms. The proof is based on the Hwang and Tzavaras's new approach [Seok Hwang, Athanasios E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (5-6) (2002) 1229-1254] and the kinetic formulation developed by Lions, Perthame, and Tadmor [P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1) (1994) 169-191].     

Second order scheme for scalar conservation laws with discontinuous flux

Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich, … satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.     

Large deviations principles for stochastic scalar conservation laws

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.     

Total oscillation diminishing property for scalar conservation laws

Summary. We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone :, we have that |(u)|_TV() is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.Mathematics Subject Classification (2000): 35L65, 35K55, 65M20     

Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we show that the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation law. Our proof is based on methodology developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] which uses the averaging lemma.     

A quantitative compactness estimate for scalar conservation laws

In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math. 7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ?‐entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L¹. © 2005 Wiley Periodicals, Inc.