Narrow Results

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.

A hierarchy of centered (non-upwind) difference schemes is identified for solving hyperbolic equations. The bottom of the hierarchy is the classical Lax–Friedrichs scheme, which is the least accurate in computation, and the top of the hierarchy is the FORCE scheme, which is the optimal scheme in the family. The FORCE scheme is optimal in the sense that it is monotone, has the optimal stability condition for explicit methods, and has the smallest numerical viscosity. It is shown that the FORCE scheme is consistent with the Lax entropy inequality, that is, the limit functions of the FORCE approximate solutions are entropy solutions. The convergence of the FORCE scheme is also established for the isentropic Euler equations and the shallow water equations. Some related centered difference schemes are also surveyed and discussed.

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and "hyperbolic" L^∞ weak-⋆ compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, …). Our results cover the case of non-Lipschitz nonlinearities.

We study a scalar conservation law whose flux has a single spatial discontinuity. There are many notions of (entropy) solution, the relevant concept being determined by the application. We focus on the so-called vanishing viscosity solution. We utilize a Kružkov-type entropy inequality which generalizes the one in [K. H. Karlsen, N. H. Risebro and J. D. Towers, $L^1$-stability for entropy solutions of nonlinear degenerate parabolic convection–diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk.3 (2003) 1–49], singles out the vanishing viscosity solution whether or not the crossing condition is satisfied, and has a discrete version satisfied by the Godunov variant of the finite difference scheme of [S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal.26(6) (1995) 1425–1451]. We show that the solutions produced by that scheme converge to the unique vanishing viscosity solution. The scheme does not require a Riemann solver for the discontinuous flux problem. This makes its implementation simple even when the flux is multimodal, and there are multiple flux crossings.

We consider the Godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time. The time-and space-dependence of the flux occurs through a positive multiplicative coefficient. That coefficient has a spatial discontinuity along a fixed interface at $x=0$. Time discontinuities occur in the coefficient independently on either side of the interface. This setup applies to the Lighthill–Witham–Richards (LWR) traffic model in the case where different time-varying speed limits are imposed on different segments of a road. We prove that the approximate solutions produced by the Godunov scheme converge to the unique entropy solution, as defined by Coclite and Risebro in 2005. Convergence of the Godunov scheme in the presence of spatial flux discontinuities alone is a well-established fact. The novel aspect of this paper is convergence in the presence of additional temporal flux discontinuities.

This paper deals with the convergence of finite volume scheme (FVS) for solving coagulation and multiple fragmentation equations having locally bounded coagulation kernel but singularity near the origin due to fragmentation rates. Thanks to the Dunford–Pettis and De La Vall$é$e-Poussin theorems, we establish that numerical solution is converging to the weak solution of the continuous model using a weak $L^1$ compactness argument. A suitable stable condition on time step is taken to achieve the result. Furthermore, when kernels are in $W_{\mathrm{\text{loc}}}^{1,\infty }$ space, first-order error approximation is demonstrated for a uniform mesh. It is numerically validated by taking four test problems of coupled coagulation–fragmentation models.