Michel Fournié

High Order Compact Finite Difference Schemes for A Nonlinear Black-Scholes Equation

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

High-order compact finite difference scheme for option pricing in stochastic volatility models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.

A fictitious domain approach for the Stokes problem based on the extended finite element method

SUMMARY In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright

High order conservative difference methods for 2D drift-diffusion model on non-uniform grid

Abstract A new accurate compact finite difference scheme for solving the 2D drift-diffusion system is introduced. The scheme is based on the computation on staggered grids of the current densities given by advection-dominated equations. Conservativity is preserved and the compactness of the scheme leads to a good treatment of boundary conditions. The discretization is realized on uniform and non-uniform grids. This last grid is analytically defined using mapping techniques (spline functions). A new accurate compact finite difference scheme for solving the 2D drift-diffusion system is introduced. The scheme is based on the computation on staggered grids of the current densities given by advection-dominated equations. Conservativity is preserved and the compactness of the scheme leads to a good treatment of boundary conditions. The discretization is realized on uniform and non-uniform grids. This last grid is analytically defined using mapping techniques (spline functions). Several numerical tests show the robustness of the method.

Numerical discretization of energy-transport model for semiconductors using high-order compact schemes

Abstract This paper deals with numerical discretization of energy-transport model for nondegenerate semiconductors With a parabolic structure. The scheme is based on high-order computations using compact stencil. Numerical simulations of a ballistic diode in 1D are performed for different energy relaxation time and are compared with the results obtained by a drift-diffusion model.

High-Order ADI Schemes for Convection-Diffusion Equations with Mixed Derivative Terms

We consider new high-order Alternating Direction Implicit (ADI) schemes for the numerical solution of initial-boundary value problems for convection-diffusion equations with cross derivative terms. Our approach is based on the unconditionally stable ADI scheme proposed by Hundsdorfer. Different numerical discretizations which lead to schemes which are fourth-order accurate in space and second-order accurate in time are discussed.

High-Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-Uniform Grids

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme.

Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

We derive a fourth-order compact finite difference scheme for a two-dimensional elliptic problem with a mixed derivative and constant coefficients. We conduct experimental study on numerical solution of the problem discretized by the present compact scheme and the traditional second-order central difference scheme. We study the computed accuracy achieved by each scheme and the performance of the Gauss-Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from the difference schemes.

A fictitious domain finite element method for simulations of fluid–structure interactions: The Navier–Stokes equations coupled with a moving solid

Abstract The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier–Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by Newton׳s laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.

CFD parallel simulation using Getfem++ and mumps

We consider the finite element environment Getfem++, which is a C++ library of generic finite element functionalities and allows for parallel distributed data manipulation and assembly. For the solution of the large sparse linear systems arising from the finite element assembly, we consider the multifrontal massively parallel solver package Mumps2, which implements a parallel distributed LU factorization of large sparse matrices. In this work, we present the integration of the Mumps package into Getfem++ that provides a complete and generic parallel distributed chain from the finite element discretization to the solution of the PDE problems. We consider the parallel simulation of the transition to turbulence of a flow around a circular cylinder using Navier Stokes equations, where the nonlinear term is semi-implicit and requires that some of the discretized differential operators be updated and with an assembly process at each time step. The preliminary parallel experiments using this new combination of Getfem++ and Mumps are presented.