Françoise Foucher

Efficient well-balanced hydrostatic upwind schemes for shallow-water equations

The proposed work concerns the numerical approximations of the shallow-water equations with varying topography. The main objective is to introduce an easy and systematic technique to enforce the well-balance property and to make the scheme able to deal with dry areas. To access such an issue, the derived numerical method is obtained by involving the free surface instead of the water height and this produces the scheme well-balanced independently from the numerical flux function associated with the homogeneous problem. As a consequence, we obtain an easy well-balanced scheme which preserves non-negative water height. When compared with the well-known hydrostatic reconstruction, the presented topography discretization does not involve any max function known to introduce some numerical errors as soon as the topography admits very strong variations or discontinuities. A second-order MUSCL accurate reconstruction is adopted. The proposed hydrostatic upwind scheme is next extended for considering 2D simulations performed over unstructured meshes. Several 1D and 2D numerical experiments are performed to exhibit the relevance of the scheme.

Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes

Given a bivariate function f defined in a rectangular domain @W, we approximate it by a C^1 quadratic spline quasi-interpolant (QI) and we take partial derivatives of this QI as approximations to those of f. We give error estimates and asymptotic expansions for these approximations. We also propose a simple algorithm for the determination of stationary points, illustrated by a numerical example.

Quadratic spline quasi-interpolants and collocation methods

Univariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated and compared with finite difference methods on some numerical examples of elliptic boundary value problems.

A well-balanced scheme for the shallow-water equations with topography

A non-negativity preserving and well-balanced scheme that exactly preserves all the smooth steady states of the shallow water system, including the moving ones, is proposed. In addition, the scheme must deal with vanishing water heights and transitions between wet and dry areas. A Godunov-type method is derived by using a relevant average of the source terms within the scheme, in order to enforce the required well-balance property. A second-order well-balanced MUSCL extension is also designed. Numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states.

Superheating field for the Ginzburg–Landau equations in the case of a large bounded interval

We study the asymptotic behavior of the local superheating field for a film of width 2d in the regime κ small, κd large, where κ is the Ginzburg–Landau parameter. This gives a mathematical justification for the introduction of the semi-infinite model as a good approximation for this regime.

Hydrostatic Upwind Schemes for Shallow–Water Equations

We consider the numerical approximation of the shallow–water equations with non–flat topography. We introduce a new topography discretization that makes all schemes to be well–balanced and robust. At the discrepancy with the well–known hydrostatic reconstruction, the proposed numerical procedure does not involve any cut–off. Moreover, the obtained scheme is able to deal with dry areas. Several numerical benchmarks are performed to assert the interest of the method.

A Well-Balanced Finite Volume Scheme for a Mixed Hyperbolic/Parabolic System to Model Chemotaxis

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 hyperbolic system 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 finite volume method 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 well-balanced scheme for the shallow-water equations with topography or Manning friction

We consider the shallow-water equations with Manning friction or topography, as well as a combination of both these source terms. The main purpose of this work concerns the derivation of a non-negativity preserving and well-balanced scheme that approximates solutions of the system and preserves the associated steady states, including the moving ones. In addition, the scheme has to deal with vanishing water heights and transitions between wet and dry areas. To address such issues, a particular attention is paid to the study of the steady states related to the friction source term. Then, a Godunov-type scheme is obtained by using a relevant average of the source terms in order to enforce the required well-balance property. An implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights. A second-order well-balanced MUSCL extension is designed, as well as an extension for the two-dimensional case. Numerical experiments are performed in order to highlight the properties of the scheme.

Analysis of a positive CVFE scheme for simulating breast cancer development, local treatment and recurrence

In this paper, a positive CVFE scheme for simulating an anisotropic breast cancer development is analyzed. The mathematical model includes reaction–diffusion-convection terms with an anisotropic heterogeneous diffusion tensor. The diffusion term is discretized using a finite element method combined with the use of Godunov scheme over a primal triangular mesh. The convective term is discretized using a nonclassical upwind finite volume scheme over a barycentric dual mesh. The scheme ensures the validity of the discrete positivity preserving and other discrete properties without any restriction on the transmissibility coefficients. Finally, a numerical simulation is provided to simulate the spread of tumor cells before and after applying a local treatment using the surgery.

Convergence of a positive nonlinear Control Volume Finite Element scheme for solving an anisotropic degenerate breast cancer development model

Abstract In this paper, a nonlinear control volume finite element (CVFE) scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed. This model includes both ordinary differential equations and convection–diffusion–reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the Godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh, where the dual volumes are constructed around the vertices of the original mesh. This technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients. The convergence of the scheme is proved, only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the Kolmogorov relative compactness theorem. The proposed scheme is robust, locally conservative, efficient, and stable, which is confirmed by numerical experiments over a general mesh.