Accurate numerical discretizations of non-conservative hyperbolic systems

Abstract

Abstract. We present an alternative framework for designing efficient numerical schemes for nonconservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating the robustness of this approach are presented.Mathematics Subject Classification. 65M06, 35L65.

Numerical Simulations for Non Conservative Hyperbolic System. Application to Transient Two-Phase Flow With Cavitation Phenomenon

“…This type of equations has been investigated by many authors particularly in the domain of multiphase flow where the equations are described by a non conservative system of equations. Following the approach used in [1,4,5,7,11,12,13], the system (3.4) needs to be written in the general form:…”

Melt-blowing of viscoelastic jets in turbulent airflows: Stochastic modeling and simulation

“…integrate the quasilinear system (3.1) over the control cells [ζ i−1/2 , ζ i+1/2 ], i = 1, ..., N (t), in which we assume X(y)| [ζ i−1/2 ,ζ i+1/2 ] = X(y i ) for X = K, L, l and adopt the idea of the Lax-Friedrichs scheme for the approximation of the numerical fluxes as done in [12]. The resulting system of ordinary differential equations for the cell averages y i with respect to time has the form…”

Path-conservative in-cell discontinuous reconstruction schemes for non conservative hyperbolic systems

“…The literature is large on the topic but the proposed schemes are often not satisfying in the sense that either they work only for some very particular systems or small amplitude shocks, or they involve some random sampling techniques which are difficult to extend in several space dimensions. Without any attempt to be exhaustive, we refer for instance the reader to [7], [5], [13], [4], [11], [21], [35], [15] and the references therein where different models and numerical approaches have been considered. Among these methods, the most recent and complete theory is probably the so-called path-conservative schemes theory, developed by C. Pares [35] and collaborators.…”