Narrow Results

Modifications in the integration of the source terms in hyperbolic conservation laws such as those governing combustion, detonation and radiative transport, with a first order upwind differencing technique are proposed and analysed. The von Neuman stability and phase error analysis for a linear scalar equation, together with a few test problems is presented in order to compare the performance of the resulting variants of the donor cell scheme. It is established that when the source term is integrated using higher order formulae, the resulting scheme gives better resolution and has better stability limit and phase accuracy, compared to the standard single nodal value replacement. It is shown that integration by the trapezoidal rule gives sufficient accuracy and further improvement may not necessarily be achieved using better methods, such as the Simpson’s rule.

Uniqueness of solutions of genuinely nonlinear n × n strictly hyperbolic systems of balance laws is established moving from Oleïnik-type decay estimates. As middle step, the result relies on the fulfillment of a condition which controls the local oscillation of the solution in a forward neighborhood of each point in the t–x plane.

This paper deals with the analysis of some numerical difficulties related to the appearance of wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids. The fluid is supposed to be governed by the Shallow Water equations and the discretization of the equations is performed, when wet/dry fronts do not appear, by means of the Q-scheme of Roe upwinding the source terms introduced in Ref. 40. This scheme is well-balanced in the sense that it solves exactly stationary solutions corresponding to water at rest. Wet/dry fronts cannot be correctly treated with this scheme: it can produce negative values of the thickness of the fluid layer and stationary solutions corresponding to water at rest including wet/dry transitions are not exactly solved. In Refs. 3–5 some variants of this numerical scheme have been proposed that partially solve these difficulties. Here we propose a new variant: at intercells where wet/dry transitions occur, a Nonlinear Riemann Problem is considered instead of a Linear one. The exact solutions of these nonlinear problems, which are easy to calculate, are used in order to define the numerical fluxes. We investigate the properties of the resulting scheme and present some comparisons with the numerical results obtained with some other modified numerical schemes proposed previously.

In this work we present some moment preserving finite volume schemes (FVS) for solving population balance equations. We are considering unified numerical methods to simultaneous aggregation, breakage, growth and source terms, e.g. for nucleation. The criteria for the preservation of different moments are given. The property of conservation is a special case of preservation. First we present a FVS which shows the preservation with respect to one-moment depending upon the processes under consideration. In case of the aggregation and breakage it satisfies first-moment preservation whereas for the growth and nucleation we observe zeroth-moment preservation. This is due to the well-known property of conservativity of FVS. However, coupling of all the processes shows no preservation for any moments. To overcome this issue, we reformulate the cell average technique into a conservative formulation which is coupled together with a modified upwind scheme to give moment preservation with respect to the first two-moments for all four processes under consideration. This allows for easy coupling of these processes. The preservation is proven mathematically and verified numerically. The numerical results for the first two-moments are verified for various coupled processes where analytical solutions are available.

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

We introduce a distinction among the classical three point finite difference schemes for nonlinear systems of conservation laws : those which are flux schemes and those which are not. We discuss at the continuous level why we think that these schemes are of interest. We also study some of their properties and their extensions to the multidimensional (finite volumes) case.