Narrow Results

We prove the convergence of a finite volume scheme for the Richards equation β(p)_t-div(Λ(β(p))(∇p-ρg))=0 together with a Dirichlet boundary condition and an initial condition in a bounded domain Ω ×(0, T). We consider the hydraulic charge as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L² of the water saturation β(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov's theorem. This implies the convergence in L² of the approximate water mobility towards Λ(β(p)) as the time and mesh steps tend to 0, which in turn implies the convergence of the approximate pressure to a weak solution p of the continuous problem.

This paper is devoted to a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such system arises in semiconductors modelling and is composed of two degenerate parabolic equations and an elliptic one. We prove the convergence of the finite volume scheme and then the existence of solutions to the problem. Several numerical tests show the efficiency of the method.

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

We propose a relaxation framework for general fluid models which can be understood as a natural extension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibrium model. Discrete entropy inequalities are established under a natural Gibbs principle.

We propose a toy model for self-organized road traffic taking into account the state of orderliness in drivers’ behavior. The model is reminiscent of the wide family of generalized second-order models (GSOM) of road traffic. It can also be seen as a phase-transition model. The orderliness marker is evolved along vehicles’ trajectories and it influences the fundamental diagram of the traffic flow. The coupling we have in mind is non-local, leading to a kind of “weak decoupling” of the resulting $2\times 2$ system; this makes the mathematical analysis similar to the analysis of the classical Keyfitz–Kranzer system. Taking advantage of the theory of weak and renormalized solutions of one-dimensional transport equations [Panov, 2008], which we further develop on this occasion in the Appendix, we prove the existence of admissible solutions defined via a mixture of the Kruzhkov and the Panov approaches; note that this approach to admissibility does not rely upon the classical hyperbolic structure for $2\times 2$ systems. First, approximate solutions are obtained via a splitting strategy; compactification effects proper to the notion of solution we rely upon are carefully exploited, under general assumptions on the data. Second, we also address fully discrete approximation of the system, constructing a BV-stable Finite Volume numerical scheme and proving its convergence under the no-vacuum assumption and for data of bounded variation. As a byproduct of our approach, an original treatment of local GSOM-like models in the BV setting is briefly discussed, in relation to discontinuous-flux LWR models.

We prove the convergence of numerical approximations of compressive solutions for scalar conservation laws with convex flux. This new proof of convergence is fully discrete and does not use Kuznetsov's approach. We recover the well-known rate of convergence in O(Δx^½). With the same fully discrete approach, we also prove a rate of convergence in O(Δx) uniformly in time, if the initial data is a shock, or asymptotically after the compression of the initial profile. Numerical experiments confirm the theoretical analysis.

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.

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.

The two most popular equity and FX derivatives pricing models in banking practice are the local volatility model and the Heston model. While the former has the appealing property that it can be calibrated exactly to any given set of arbitrage free European vanilla option prices, the latter delivers more realistic smile dynamics. In this paper, we combine both modeling approaches to the Heston stochastic local volatility model. We build upon a theoretical framework that has been already developed and focus on the numerical model calibration which requires special care in the treatment of mixed derivatives and in cases where the Feller condition is not met in the Heston model leading to a singular transition density at zero volatility. We propose a finite volume scheme to calibrate the model after a suitable transformation of the model equation and demonstrate its accuracy in numerical test cases using real market data.