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.     

Semidiscrete central-upwind scheme for conservation laws with a discontinuous flux function in space

In this paper, a modified semidiscrete central-upwind scheme is derived for the scalar conservation laws with a discontinuous flux function in space. The new scheme is based on dealing with the phase transition at the stationary discontinuity, where the unknown variable function is not continuous, but the flux function is continuous. The main advantages of the new scheme are the same as them of the original semidiscrete central-upwind scheme. Numerical results are displayed to illustrate the efficiency of the methods.     

A second order explicit finite element scheme to multidimensional conservation laws and its convergence

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L ^p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL ² is proved under a stronger condition, Δt≤Ch ^2/4     

On the convergence of a local third order shock capturing method for hyperbolic conservation laws

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892–915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior.     

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

For the scalar conservation laws with discontinuous flux, an infinite family of (A, B)‐interface entropies are introduced and each one of them is shown to form an L¹‐contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned (A, B) pairs the solution will have bounded total variation in the case of strictly convex fluxes. © 2010 Wiley Periodicals, Inc.     

On the symmetry classification and conservation laws for quasilinear differential equations of second order

We obtain a sufficient condition for the absence of tangent transformations admitted by quasilinear differential equations of second order and a sufficient condition for the linear autonomy of the operators of the Lie group of transformations admitted by weakly nonlinear differential equations of second order. We prove a theorem concerning the structure of conservation laws of first order for weakly nonlinear differential equations of second order. We carry out the classification by first-order conservation laws for linear differential equations of second order with two independent variables.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.     

Strong uniqueness and second order convergence in nonlinear discrete approximation

Summary Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.     

A relaxation scheme for conservation laws with a discontinuous coefficient

We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient . If , we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat-Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact Riemann solver.     

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016     

Conservation laws with discontinuous flux functions and boundary condition

In a bounded domain we study the existence and uniqueness of entropy solutions of , where is allowed to have some discontinuities of first type.     

Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux

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.     

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.     

On the optimization of flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the approximation of the solutions of the Riemann problem for a discontinuous conservation law

For a class of discontinuous flux functions introduced in [3] (cf. also [4]), we prove, for the Riemann problem, an extension of the existence result proved in [2] for a Lipschitz continuous flux function. In the last section, and based in the previous results, we apply the Lax-Friedrichs approximation method and the limiters technique (cf.[6]) to compute the quoted solution in a numerical example. For related results see [5].     

On convergence of solutions of a second order nonlinear difference equations

The object of the present paper is to establish some criteria of the convergence of all bounded solutions of (1), (1').