A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests

Abstract

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a new method called δ-ziti's method for solving the governing equations. This method gives a new class of semi discrete, high-order scheme which are entropy conservative if the viscosity term is neglected. We implement a high resolution scheme for our mixed type problems that select the same viscosity solution as the Lax Friederich scheme with higher resolution. Several tests have been carried out to compare our results with those of [6] [9] [16], in the same situations, we obtained the same results but faster thanks to the CFL condition which reaches 0.8 and the simplicity of the method. We consider three types of pressure in these tests: Cubic, Van der Waals and linear in pieces. The comparison proved that the δ-ziti's method respects the generalized Liu entropy conditions, e.g. the existence of a viscous profile.

Extension of $δ_{-}ziti$ method in the unit ball: Numerical integration, resolution of Poisson's problem and Heat transfer

“…Since this last goes to Dirac function ( in the distribution sense), we can say that the δ − ziti is a mix between Galerkin and the particular method. As a conclusion, δ − ziti permits us to use two strategies in the case Ω = B(0, 1), injecting the work already done in the mono-dimensional case (see [1] and [2]). The result is impressive, in fact:…”

“…. , N, which will be reduced in the following theorem, already proved in the mono-dimensional case, (see [1] and [2]).…”

“…All the results presents in this section, are proved in [1] and [2]. The fundamental results of construction are given as follow:…”

“…Let Ψ i = Ψi || Ψi || the normalization of Ψi ( for more details see [1] and [2]). The method permits to approach a given function f and an integral, using the following relations:…”

“…The δ − ziti method is on the challenge. It was tested on several type of problems (see [1] and [2]), including the Keller-Segel model, but only in the Cartesian case ( segments, rectangle, cube • • • ).…”