Juri D. Kandilarov

The Immersed Interface Method for a Nonlinear Chemical Diffusion Equation with Local Sites of Reactions

A diffusion equation with nonlinear localized chemical reactions is considered in this paper. As a result of the reactions, although the equation is parabolic, the derivatives of the solution are discontinuous across the interfaces (local sites of reactions). A second-order accurate immersed interface method is constructed for the diffusion equation involving interfaces. The new method is more accurate than the standard approach and it does not require the interfaces to be grid points. Several experiments that confirm second-order accuracy are presented. The efficiency of the proposed algorithm is also demonstrated for solving blow up problems. The proposed technique could be extended for construction of efficient numerical algorithms on uniform grids for the present equations with moving interfaces [9] but more analysis is required.

Analysis of Immersed Interface Difference Schemes for Reaction-diffusion Problems with Singular Own Sources

Abstract This paper analyzes immersed interface difference schemes for onedimensional reaction-diffusion equations with singular own linear and nonlinear sources. Error bounds in the infinity norm based on the maximum principle are derived. Sharper bounds and a more detailed structure of the error are obtained using the asymptotic error expansion analysis method. Numerical examples confirm the theoretical results.

Immersed-Boundary Level Set Approach for Numerical Solution of Elliptic Interface Problems

In this paper we consider elliptic problems with variable discontinuous coefficients and interface jump conditions, in which the solution is continuous, but the jump of the flux depends on the solution. A new numerical method, based on immersed-boundary approach combined with level set method, is developed. Using regular grids it is robust and easy to implement for curvelinear interface problems. Numerical experiments are presented.

Construction and Convergence of Difference Schemes for a Model Elliptic Equation with Dirac-delta Function Coefficient

We first discuss the difficulties that arise at the construction of difference schemes on uniform meshes for a specific elliptic interface problem. Estimates for the rate of convergence in discrete energetic Sobolevs norms compatible with the smoothness of the solution are also presented.

A rothe-immersed interface method for a class of parabolic interface problems

A technique combining the Rothe method with the immersed interface method (IIM) of R. Leveque and Z. Li, [8] for numerical solution of parabolic interface problems in which the jump of the flux is proportional to a given function of the solution is developed. The equations are discretized in time by Rothes method. The space discretization on each time level is performed by the IIM. Numerical experiments are presented.

Comparison of two numerical methods for computation of american type of the floating strike asian option

We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear time-dependent term is involved in the resulting equation. Two new numerical algorithms are proposed. In the first algorithm a predictor-corrector scheme is used. The second one is based on the Newton method. Computational experiments, confirming the accuracy of the algorithms, are presented and discussed.

A Method of Lines Approach to the Numerical Solution of Singularly Perturbed Elliptic Problems

Two elliptic equations with power and interior boundary layers, respectively, in a square are considered. The elliptic problems are reduced to systems of ordinary differential equations by the method of lines. For construction of difference schemes fitted operator technique is used. Uniform convergence for the scheme of the first problem is proved.

Immersed Interface Difference Schemes for a Parabolic-Elliptic Interface Problem

Second order immersed interface difference schemes for a parabolic-elliptic interface problem arising in electromagnetism is presented. The numerical method uses uniform Cartesian meshes. The standard schemes are modified near the interface curve taking into account the specific jump conditions for the solution and the flux. Convergence of the method is discussed and numerical experiments, confirming second order of accuracy are shown.

Time Discretization/Linearization Approach Based on HOC Difference Schemes for Semilinear Parabolic Systems of Atmosphere Modelling

We implement implicit-explicit (IMEX) linear multistep time-discretization to HOC difference schemes for weakly coupled nonlinear parabolic systems with desirable time-step restrictions and positivity preservation of the numerical solution. Numerical experiments are performed with IMEX-BDF1 (backward difference method of order one), IMEX-BDF2 (backward difference method of order two) and CN-LF (Crank-Nicolson Leap Frog) to check the properties of the methods.

Numerical Method for Solving Free Boundary Problem Arising from Fixed Rate Mortgages

In this paper a mortgage contract with a given duration and a fixed mortgage interest rate is considered. The borrower is allowed to terminate the contract at any time at his choice by paying off the outstanding sum to the issuer. The mathematical model leads to a free boundary problem where the moving boundary is the optimal time of termination. A new numerical method, based on the immersed interface method (IIM) and integral representation of the solution is proposed. Using Thomas algorithm the nonlinear equation for the free boundary position is obtained and solved iteratively. Numerical analysis is presented and discussed.