scholarly journals Scalar conservation laws with general boundary condition and continuous flux function

2006 ◽  
Vol 228 (1) ◽  
pp. 111-139 ◽  
Author(s):  
Kaouther Ammar ◽  
Petra Wittbold ◽  
Jose Carrillo
Keyword(s):  
Boundary Condition ◽  
Conservation Laws ◽  
General Boundary ◽  
General Boundary Condition ◽  
Scalar Conservation Laws ◽  
Flux Function ◽  
Scalar Conservation

scholarly journals EXISTENCE RESULT FOR THE COUPLING PROBLEM OF TWO SCALAR CONSERVATION LAWS WITH RIEMANN INITIAL DATA

2010 ◽  
Vol 20 (10) ◽  
pp. 1859-1898 ◽  
Author(s):  
BENJAMIN BOUTIN ◽  
CHRISTOPHE CHALONS ◽  
PIERRE-ARNAUD RAVIART
Keyword(s):  
Conservation Laws ◽  
Weak Sense ◽  
Scalar Conservation Laws ◽  
Existence Of A Solution ◽  
Coupling Condition ◽  
Coupling Problem ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Fixed Interface ◽  
Scalar Conservation

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x = 0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove the existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non-uniqueness of both continuous and discontinuous (at interface x = 0) solutions. The behavior of some numerical scheme is also investigated.

scholarly journals Regularity estimates for scalar conservation laws in one space dimension

2018 ◽  
Vol 15 (04) ◽  
pp. 623-691 ◽  
Author(s):  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Space Dimension ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Flux Function ◽  
Regularity Estimates ◽  
Degeneracy Condition ◽  
Inflection Points ◽  
Scalar Conservation ◽  
One Space Dimension

We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function [Formula: see text] has on the entropy solution. More precisely, if the set [Formula: see text] is dense, the regularity of the solution can be expressed in terms of [Formula: see text] spaces, where [Formula: see text] depends on the nonlinearity of [Formula: see text]. If moreover the set [Formula: see text] is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that [Formula: see text] for every [Formula: see text] and that this can be improved to [Formula: see text] regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

HIGH-FIELD LIMIT FROM A KINETIC EQUATION TO MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

2007 ◽  
Vol 04 (01) ◽  
pp. 123-145 ◽  
Author(s):  
F. BERTHELIN ◽  
N. J. MAUSER ◽  
F. POUPAUD
Keyword(s):  
Kinetic Equation ◽  
Conservation Laws ◽  
High Field ◽  
Distribution Functions ◽  
Scalar Conservation Laws ◽  
Field Limit ◽  
Flux Function ◽  
Transport Operator ◽  
Scalar Conservation

We study the relaxation of kinetic BGK models involving a high-field term within the transport operator. They lead us to multidimensional scalar conservation laws with a flux-function which is perturbed with respect to classical relaxation. The proof of the relaxation limit makes modified Maxwellians to appear. We consider "pseudo distribution functions" which can take negative values and we introduce appropriate admissible states. This is a first step towards adapting this analysis to quantum kinetic BGK models.

scholarly journals On the spreading of characteristics for non-convex conservation laws

2001 ◽  
Vol 131 (4) ◽  
pp. 909-925 ◽  
Author(s):  
Helge Kristian Jenssen ◽  
Carlo Sinestrari
Keyword(s):  
Conservation Laws ◽  
Error Term ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Flux Function ◽  
Characteristic Speed ◽  
Convex Case ◽  
Lipschitz Estimate ◽  
Scalar Conservation ◽  
Hölder Estimate

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

scholarly journals High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1079
Author(s):  
Tingting Xiang ◽  
Guodong Wang ◽  
Suping Zhang
Keyword(s):  
Conservation Laws ◽  
High Order ◽  
Splitting Scheme ◽  
Scalar Conservation Laws ◽  
Flux Function ◽  
Flux Splitting ◽  
Discontinuous Flux Function ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Accurate Scheme

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

Sign in / Sign up

Export Citation Format

Share Document