Front Tracking for Scalar Balance Equations

Abstract

Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL-condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady-state solutions (or achieving them in the long time limit) with good accuracy.

Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

“…To conclude from this that (2.39) holds for all k ∈ R, i.e., that u is an entropy solution to (3.1) in the sense of Definition 1.1, we repeat the argument in [11]. Again, convergence of the whole sequence {u ∆t } ∆t>0 follows from the uniqueness of entropy solutions.…”

Kinetic description of bottleneck effects in traffic flow

“…Otherwise, if N ≤ a 0 Lρ * , the evolution will lead to the trivial solution ρ(x, t) = N/(a 0 L) ≤ ρ * . This conclusion can also be verified through the front tracking method [17][18][19] . However, it is indicated here through a numerical example only in Section 3.…”

“…(7)- (12). The conclusion could be mathematically proven through the front tracking method (see [17][18][19] and the references therein for a discussion of this method). However, it is indicated here by a numerical example only in Section 3.…”

Well-balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation

“…Comparing this with the last term in the double integral in (4.12) and repeating the argument from [15] to extend the set of permissible c's to R, we conclude that the limit u is an entropy solution in the sense of Definition 2.3…”

“…A variety of well-balanced schemes can be found in literature; see [13,11,5,6] and the references cited therein. For a partial overview, see also the introductory part of [15].…”