This work departs from the classical van der Waals type phase change mod- elling which usually requires additional terms related to very small scale effects (see ). These are intended to correct the core system intrinsic lack of hyperbolicity. A drawback of this approach is that it requires numerical strategies to use very fine discretizing grids. On the contrary, the present system is fully compliant with standard numerical relaxation techniques for hyperbolic systems although the model equilibrium states are compatible with the equilibium Maxwell points for a van der Waals law.
Abstract. This work is concerned by the numerical approximation of the weak solutions of a system of partial differential equations arising when modeling the movements of cells according to a chemoattractant concentration. The adopted PDE system turns out to couple a hyperbolicsystem with a diffusive equation. The solutions of such a model satisfy several properties to be preserved at the numerical level. Indeed, the solutions may contain vacuum, satisfy steady regimes and asymptotic regimes. By deriving a judicious approximate Riemann solver, a finitevolumemethod is designed in order to exactly preserve the steady regimes of particular physical interest. Moreover, the scheme is able to deal with vacuum regions and it preserves the asymptotic regimes. Numerous numerical experiments illustrate the relevance of the proposed numerical procedure.
A broad literature is available on the design of robust and accurate, shock–capturing schemes for general hyperbolic systems posed on a flat geometry like the Minkowski spacetime. In the present work, we intend to also take a curved background geometry into account, by following recent work by the first author and his collaborators; cf. [2–5,15]. To this end, we introduce a finitevolume scheme which is based on the geometric formulation (1.1), rather than on the corresponding partial differential equations in a specific local coordinate chart. In order to achieve the well–balanced property, we extend the approach in Russo et al. [23, 25, 26] and LeFloch et al. , and we introduce a discretization which accurately takes into account the family of steady solutions to the balance laws and, therefore, the geometric e ffects induced by the Lorentzian geometry ( M,g ) . To implement this strategy, it is necessary to first investigate the class of steady solutions to the Euler system on the curved background under consideration.
Compressible fluid dynamics provide a large variety of problems which involve geomet- rical features. The prototype example is the system of shallow water on the sphere with topography, which describe fluid flows on the surface of the Earth for instance, in connec- tions with weather predictions . Motivated by numerous applications in fluid dynamics, the study of hyperbolic conservation laws posed on curved manifolds were recently initiated in the mathematical and numerical literature. We build here on the work by Ben-Artzi and LeFloch  who proposed to rely on an analogue of the inviscid Burgers equation for curved geometry and, more generally, various classes of hyperbolic conservation laws on manifolds. Since Burgers equations has played such an important role in the development of shock-capturing schemes for compressible fluid problems, it is also expected that the class of “geometric Burgers models” should provide an ideal simplified setup in order to design and test geometric-preserving shock-capturing scheme. The mathematical properties of en- tropy solutions to conservation laws on manifolds (including on spacetimes, that is, with time-dependent (Lorentzian) metrics) were then extensively investigated by LeFloch and co-authors [1, 2, 7, 6] and –. Subsequently, hyperbolic conservation laws on evolving
Figure 5. Bump on tail test: time evolution of the total energy for ”Banks” methods (CD4 and up5), for the unsplit Vfsl methods (Vfsl3-ns and Vfsl5-ns) and for semi-Lagrangian method (Lag3 and Lag5). N x = N v = 128, ∆t = 0.01.
see that the details have been eliminated by the scheme. When a higher order is used (as for up5 or Vfsl5- ns), additional small structures are described. On Figures 10 , 11 , we again see the link between LAG3/up3, LAG5/up5 and also CD4/PPM1 for small ∆t as depicted in Remark 3.3 . In particular, the bad oscillations of the centered reconstruction PPM1 are emphasized, when (very) small time steps are used, whereas the uncentered reconstructions LAG3 and LAG5 are insensitive to the decrease of time step. Note also that the PPM1 reconstruction behaves well when the time step is not too small, which is possible for a semi-Lagrangian scheme.
rise to the most efficient way to describe M. More precisely, we want to compare the complexity of 3-manifolds and their fundamental groups.
The study of the complexity of 3-manifolds goes back to the classical work of H. Kneser [K]. Recall that the Kneser complexity invariant k(M) is defined to be the minimal number of simplices of a triangulation of the manifold M. The main result of Kneser is that this complexity serves as a bound of the number of embedded incompressible 2-spheres in M, and bounds the numbers of factors in a decomposition of M as a connected sum. A version of this complexity was used by W. Haken to prove the existence of hierarchies for a large class of compact 3-manifolds (called since then Haken manifolds). Another measure of the complexity c(M) for the 3-manifold M is due to S. Matveev. It is the minimal number of vertices of a special spine of M [Ma]. It is shown that in many important cases (e.g. if M is a non-compact hyperbolic 3-manifold of finitevolume) one has k(M) = c(M) [Ma].
The study of a complexity of 3-manifolds goes back to the classical work of H. Kneser [K]. Recall that the Kneser complexity invariant k(M) is defined to be the minimal number of sim- plices of a triangulation of the manifold M. The main result of Kneser is that this complexity serves as a bound of the number of embedded incompressible 2-spheres in M, and bounds the numbers of factors in a decomposition of M as a connected sum. A version of this complexity was used by W. Haken to prove the existence of hierarchies for a large class of compact 3-manifolds (called since then Haken manifolds). Another complexity c(M) of the 3-manifold M was defined by S. Matveev as the minimal number of vertices of a special spine of M. It is shown in [Ma] that k(M) = c(M) if M is a non-compact hyperbolic 3-manifold of finitevolume.
1.2.2. Consideration on the discontinuities of the viscosity. Even for scalar diffusion problems, it is known that such discontinuities in the coefﬁcients imply a consistency defect in the numerical ﬂuxes of usual ﬁnite volume schemes. It is needed to modify the scheme in order to take into account the jumps of the coefﬁcients of the problem and then to recover the optimal ﬁrst order convergence rate. As in the scalar case , we need to introduce a modiﬁed gradient operator (see Deﬁnition 2.5) and ﬁnally deﬁne a modiﬁed approximate viscous stress tensor D η, D N u T (see Deﬁnition 2.7) on each diamond cell. We derive a modiﬁed DDFV
Brezzi-Pitkaranta  in the mass conservation equation. Convergence analysis and a priori estimates for the two-dimensional method are available in .
The new DDFV method that we study in this work extends the discrete 2-D setting proposed in  for the steady Stokes equation with variable viscosity to the 3-D framework for polyhedral meshes of . More precisely, the degrees of freedom of the components of the velocity are defined for the control volumes of the primal mesh, the dual mesh of the vertices, and the dual mesh of faces and edges. On its turn, the pressure variable is approximated by a piecewise constant function defined on the mesh of the diamond cells. We emphasize the fact that the present DDFV scheme is not a simple extension to three spatial dimensions of the 2-D scheme originally developed in , because it is based on a construction for the dual meshes and the diamond mesh that is very specific to the 3-D case. For the present scheme, we prove a discrete analog of the Korn inequality, the uniform stability, the well-posedness and the convergence. Moreover, we derive a priori estimates for the degrees of freedom of velocity and pressure using suitably defined mesh dependent norms, and a priori estimates for the approximation errors in the continuous setting using standard Sobolev norms.
The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), FiniteVolume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a ”black-box” solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finitevolume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meanings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to FiniteVolume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid- Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).
In this paper we require the two following hypotheses, that are always satis- fied in the case of a knot complement.
Assumption 1.2. For each peripheral torus T i 2 , α(π 1 (T i 2 )) ∼ = Z.
This assumption is a necessary condition for the acyclicity (over the field of rational fractions) of the Z r -cover associated with α, so that the twisted Alexan- der polynomials do not vanish. Assumption 1.2 holds true for the abelianization map of a knot in a homology sphere, or more generally for the abelianization map of a link in a homology sphere having the property that the linking number of pairwise different components vanish. Furthermore, for any cusped, oriented, hyperbolic 3-manifold M , there exists an epimorphism α : π 1 (M ) Z satisfy-
estimates for finitevolume approximation of diffusion equations on unstructured grids has only been achieved recently [ 29 ]. The extension of this optimal result to our much more complex cross-diffusion system appears to be an interesting and challenging issue.
3. Numerical analysis on a fixed mesh. This section is devoted to the proof of Theorem 2.2 . In Section 3.1 , we establish a priori estimates on a slightly modified scheme that will be shown to reduce to the original scheme ( 2.4 ). Then in Section 3.2 , we apply a topological degree argument to prove the existence of solutions to our scheme. Section 3.3 is devoted to the proof of the entropy dissipation property.
these oscillations are visible. 5 CONCLUSIONS
It has been presented in this work the use and extension of a finitevolumemethod for elastic- viscoplastic solids with curvilinear structured meshes of quadrangles. The formulation is based on the flux-difference splitting method, and uses the second order accurate Strang splitting approach for the computation of the right hand side of the differential system of equations. Comparison is performed with a finite element solution obtained with the finite element code Cast3M  on a suddenly loaded then unloaded heterogeneous volume. The comparison shows identical cumulated viscoplastic strains computed. However, the finitevolume solution enables to overcome spurious numerical oscillations on the stress and velocity fields thanks to flux limiters, applied on the homogeneous part of (7), satisfying a non-increasing total variation.
To test the method, we have considered the Riemann problem and we present an new approach based on the configuration identification where we propose a classification of the Riemann problem solutions. Indeed, the main difficulty with the nonconservative problems is the nonstrictly hyperbolicity character of the system and eigenvalues can cross or merge. We develop a technique where we describe all the admissible configurations which can appear and we use the classification to solve the inverse Riemann problem introduced by Andrianov & Warnecke (2004). In particularly we are able to compute complex situations including the resonant cases and prove that rarefaction can only reach the sonic point from the lower porosity side. New Sod tube tests corresponding to particular difficult situations are proposed to check the solvers. Numerical tests have been performed to compare the approximated solution with the exact solution computed with the inverse Riemann problem algorithm. Other simulations of a nontrivial steady-state solution are also proposed to measured the method accuracy.
We give some details of the Vofire method, which is a multidimensional non- linear FV scheme. The geometrical idea relies on the following observation: in dimension greater than 2, the numerical diffusion can be decomposed into two different diffusions: the longitudinal diffusion, along the velocity field, which is typically one-dimensional, and the transverse diffusion, which is really due to the fact that the mesh is multi-dimensional. This distinction between the two phenomena could appear arbitrary, but is in accordance with basic numer- ical tests. Consider for example an initial condition which is the characteristic function of the square ]0.25, 0.75[×]0.25, 0.75[. This profile is advected with the upwind scheme. The velocity direction u has a great influence on the result. It is illustrated on figure 1.
French Atomic Energy Commission (CEA), Centre de Cadarache , France
In this paper we present the analysis of a full three dimensional transient two-phase flow with strong desequilibria. This analysis uses a finitevolume scheme with a density pertur- bation method recently developed. The application test case is a three dimensional exten- sion of a one dimensional experiment on blowdown pipe. The results are stable with mesh refinements and show realistic three dimensional effects.
As far as discontinuous Galerkin methods are concerned (see  for a more de- tailed review of this class of methods), we can cite for instance  where the LDG method in variables velocity-velocity gradient-pressure is analysed in details, in particular its Inf-Sup stability properties. Since this method is locally conservative, it can be understood in some sense as a higher order finitevolume approximation. However, this method requires a pressure stabilization term in the mass conserva- tion equation. Another DG method in the velocity/pressure formulation without pressure stabilization (at least on matching simplicial grids) is analyzed in . None of these methods is able to cope with general grids without pressure stabiliza- tion contrary to the DDFV method presented here. This is an important feature of staggered methods.