Convergence of the Lax-Friedrichs Scheme and Stability for Conservation Laws With a Discontinuous Space-Time Dependent Flux

Abstract

Abstract. We give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the formwhere the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t) plane. In contrast to most of the existing literature on problems with discontinuous coefficients, our convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of k(x, t). Following [14], we propose a definition of entropy solution that extends the classical Kružkov definition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t) plane. We prove stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x, t) is discontinuous. We show that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.

A regulator for continuous sedimentation in ideal clarifier–thickener units

“…This property, together with the steady-state relation between the mass, the SBL and the control variable in Section 3.3, yields the first part of a control strategy, see Section 3.4. This part is realized by means of the proportional regulator (9), which controls the mass in the settler. The key idea is the following.…”

“…If, in addition, the SBL is not too close to the bottom (inequality (9) in [3] holds), optimal operation can be maintained. Furthermore, if the SBL meets the bottom, it was shown that the SBL can be restored within the thickening zone again after a finite time.…”

“…Given x r sb ∈ (0, D), arbitrary initial data and a constant feed point (u f , s) for t > 0. Assume that the regulator (9), with a given constant Q r u , connected with (1) implies that the solution is in optimal operation and converges to a stationary solution as t → ∞. Then the limit SBL is given by (15),…”

Operating Charts for Continuous Sedimentation II: Step Responses

“…In [26], existence and uniqueness of solutions were established only locally in time in the class of piecewise differentiable functions and with some further assumptions on piecewise monotonicity. It is only lately that global existence and uniqueness have been established by Bürger et al [31,32] and Karlsen and Towers [33]. In [31], the front-tracking method was utilized and approximate solutions were constructed as in [26].…”

“…These results have been established for a constant source term and constant volume flows, which is the case of the step responses considered in the present paper. The general case when the source term and volume flows are allowed to vary with time, also discontinuously, is contained in [33]. They used the Lax-Friedrich finite difference scheme and a Kružkov-type notion of entropy solution to prove existence and uniqueness, respectively.…”