Tatiana P. Chernogorova

A finite volume difference scheme for a model of settling particle dispersion from an elevated source in an open-channel flow

Abstract A finite volume fitted difference scheme is constructed to solve the unsteady convective-diffusion equation transformed on a finite domain, modeling longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough-bed surface. First we discuss the well-posedness of the differential problem and the non-negativity of its solution. Then, to overcome the degeneracy at the part of the boundary, starting from the divergent form of the equation we perform a local fitted space discretization. This approximation is determined by a set of two-point boundary value problems. Non-negativity of the numerical concentration of suspended fine particles is proved. Some results from computational experiments are presented to illustrate the properties of the constructed scheme.

Numerical investigation of solidification processes of cylindrical ingots in a metal mould at variable technological circumstances

Abstract The process of the solidification of a binary alloy in a cylindrical metal mould is investigated by the method of computational experiment. An equilibrium model is used for the mushy region. Variable technological peculiarities are taken into account. Some results of the numerical experiments are presented and discussed.

A Computational Scheme for a Problem in the Zero‐coupon Bond Pricing

In this paper we derive a finite volume difference scheme for a degenerate parabolic equation with dynamical boundary conditions of zero‐coupon bond pricing. We show that the system matrix of the discretization scheme is an M‐matrix, so that the discretization is monotone. This provides the non‐negativity of the price whit respect to time if the initial distribution is nonnegative. Then one can prove convergence of the numerical solution with rate of convergence O(h), where h denotes the mesh parameter [2]. Several numerical experiments show higher accuracy with comparison of known difference schemes near the boundary (degeneration).

Finite-volume difference scheme for the Black-Scholes equation in stochastic volatility models

We study numerically the two-dimensional Black-Scholes equation in stochastic volatility models [3]. For these models, starting from the conservative form of the equation, we construct a finite-volume difference scheme using the appropriate boundary conditions. The scheme is first order accurate in the space grid size. We also present some results from numerical experiments that confirm this.

Fitted finite volume positive difference scheme for a stationary model of air pollution

A new approach is proposed for the numerical solution of boundary value one-dimensional problem of advection-diffusion equation which arises, among others, in air pollution modeling. Since the problem is posed in an unbounded interval we use a log-transformation to confine the computational region. We discuss the well-posedness of the new problem and the properties of its solution. We derive a positive finite volume difference scheme. Some results from computational experiments are presented.

A comparison of asymptotic analytical formulae with finite-difference approximations for pricing zero coupon bond

In this paper we solve numerically a degenerate parabolic equation with dynamical boundary condition for pricing zero coupon bond and compare numerical solution with asymptotic analytical solution. First, we discuss an approximate analytical solution of the model and its order of accuracy. Then, starting from the divergent form of the equation we implement the finite-volume method of Song Wang (IMA J Numer Anal 24:699–720, 2004) to discretize the differential problem. We show that the system matrix of the discretization scheme is a M-matrix, so that the discretization is monotone. This provides the non-negativity of the price with respect to time if the initial distribution is nonnegative. Numerical experiments demonstrate second order of convergence for difference scheme when the node is not too close to the point of degeneration.

Two Splitting Methods for a Fixed Strike Asian Option

The valuation of Asian Options can often be reduced to the study of initial boundary problems for ultra-parabolic equations. Two splitting methods are used to transform the whole time-dependent problem of a fixed strike Asian option into two unsteady subproblems of a smaller complexity. The first subproblem is a time-dependent convection-diffusion and the finite volume difference method of S. Wang [6] is applied for its discretization. The second one is a transport problem and is approximated by monotone weighted difference schemes. The positivity property of the numerical methods is established. Numerical experiments are discussed.

Finite volume difference scheme for a transformed stationary air pollution problem

A new approach is proposed for the numerical solution of boundary value onedimensional problem of advection-diffusion equation, which arise, among others, in air pollution modeling. Since the problem is posed in unbounded interval we use a log-transformation to confine the computational region. We derive a finite volume scheme and show that it is monotone, i.e it preserves the nonnegativity property. Numerical experiments show higher accuracy of our scheme near the degenerate boundary and the Dirac-delta source term.

A TWO‐LEVEL SECOND‐ORDER FINITE DIFFERENCE SCHEME FOR THE SINGLE TERM STRUCTURE EQUATION

In the paper [6] the classical single factor term structure equation for models that predict non‐negative interest rates is numerically studied. For these models the authors proposed a second order accurate three‐level finite difference scheme (FDs) using the appropriate boundary conditions at zero. For the same problem we propose a two‐level second‐order accurate FDs. We also propose an effective algorithm for solving the difference schemes, for which also follows the positivity of the numerical solution. The flexibility of our FDs makes it easy to change the drift and diffusion terms in the model. The numerical experiments confirm the second‐order of accuracy of the scheme and the positivity‐convexity property.

On the Discretization of Interface Problems with Perfect and Imperfect Contact

A second-order difference scheme for a first-order elliptic system with discontinuous coefficients is derived and studied. This approximation can be viewed as an improvement of the well-known scheme with harmonic averaging of the coefficients for a second order elliptic equation, which is first-order accurate for the gradient of the solution. The numerical experiments confirm the second order convergence for the scaled gradient, and demonstrate the advantages of the new discretization, compared with the older ones.