Nils Henrik Risebro

Front tracking for hyperbolic conservation laws

Preface.- Introduction.- Scalar Conservation Laws.- A Short Course in Difference Methods.- Multidimensional Scalar Conservation Laws.- The Riemann Problem for Systems.- Existence of Solutions of the Cauchy Problem.- Well-Posedness of the Cauchy Problem.- Total Variations, Compactness etc..- The Method of Vanishing Viscosity.- Answers and Hints.- References.- Index.

A mathematical model of traffic flow on a network of unidirectional roads

We introduce a model that describes heavy traffic on a network of unidirectional roads. The model consists of a system of initial-boundary value problems for nonlinear conservation laws. We uniquely formulate and solve the Riemann problem for such a system and, based on this, then show existence of a solution to the Cauchy problem.

Solution of the Cauchy problem for a conservation law with a discontinuous flux function

The Cauchy problem is solved for a conservation law arising in oil reservoir simulation where the flux function may depend discontinuously on the space variable. To do this front tracking is used as a method of analysis.

Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior

Existence and uniqueness is proved, in the class of functions satisfying a wave entropy condition, of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable. The large time limit is then studied, and explicit formulas for this limit is given in the case where the initial data as well as the x dependency of the flux vary periodically. Throughout the paper, front tracking is used as a method of analysis. A numerical example which illustrates the results and method of proof is also presented.

A front-tracking alternative to the random choice method

An alternative to Glimms proof of the existence of solutions to systems of hyperbolic conservation laws is presented. The proof is based on an idea by Dafermos for the single conservation law and in some respects simplifies Glimms original argument. The proof is based on construction of approximate solutions of which a subsequence converges. It is shown that the constructed solution satisfies Laxs entropy inequalities. The construction also gives a numerical method for solving such systems

Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

Summary.We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L1 stability) of the entropy solution in the BVt class (functions W(x,t) with ∂tW being a finite measure). The existence of a BVt entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.

A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units

We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a flux function that depends discontinuously on the spatial position. We construct a weak solution to the sedimentation model by proving the convergence of a front tracking method. The basic building block in this method is the solution of the Riemann problem, which is complicated by the fact that the flux function is discontinuous. A feature of the convergence analysis is the difficulty of bounding the total variation of the conserved variable. To overcome this obstacle, we rely on a certain non-linear Temple functional under which the total variation can be bounded. The total variation bound on the transformed variable also implies that the front tracking construction is well defined. Finally, via some numerical examples, we demonstrate that the front tracking method can be used as a highly efficient and accurate simulation tool for continuous sedimentation.

Conservation laws with a random source

We study the scalar conservation law with a noisy nonlinear source, namely,ul + f(u)x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media.

Front tracking applied to a nonstrictly hyperbolic system of conservation laws

The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension. The front tracking method is based on approximate solutions of Riemann problems. The method is compared with the random choice scheme and the upwind scheme.

A Front Tracking Method for Conservation Laws in One Dimension

Abstract We present a front tracking technique for conservation laws in one dimension. The method is based on approximations to the solution of Riemann problems where the solution is represented by piecewise constant states separated by discontinuities. The discontinuities are tracked until they interact, at this point a new Riemann problem is solved and so on. No finite differences are used. This method is tested on the system of nonstationary gas dynamics defined by the Euler equations, and three test cases are presented.