Miglena N. Koleva

Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics☆

Abstract In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H . In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h , h ≪ H . On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies h = O ( H 2 r ) , r = 1 , 2 , …  , where r is the r th Newtons iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher–Kolmogorov, Blasius and Emden–Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations.

Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance

Abstract In this paper, on the basis of Newton’s method, we propose a fast quasilinearization numerical scheme, coupled with Rothe’s method, for fully nonlinear parabolic equations. General conditions that provide quadratic, uniform and monotone convergence of the quasilinearization method (QLM) of solving fully nonlinear ordinary differential equations that arise on each time level, are formulated and elaborated. The convergence of QLM and its rate are examined numerically, on a simple test example with an exact solution. The first few iterations already provide extremely accurate and stable numerical results. The second goal is to consider three applications of the proposed schemes in financial mathematics. Namely, numerical results for three nonlinear problems of optimal investment are presented and discussed. The numerical experiments of the last problem are based on the data from statistic information of the Bulgarian National Bank and Bratislava Interbank.

Numerical solution of the heat equation in unbounded domains using quasi-uniform grids

Numerical solutions of the heat equation on the semi-infinite interval in one dimension and on a strip in two dimensions with nonlinear boundary conditions are investigated. At the space discretization with respect to the variable on the semi-infinite interval, we use quasi-uniform mesh with finite number of intervals. Convergence results are formulated. Numerical experiments demonstrate the efficiency of the approximations. The results are compared with those, obtained by the well known method of artificial boundary conditions.

On the blow-up of finite difference solutions to the heat-diffusion equation with semilinear dynamical boundary conditions

In this paper we analyse three known finite difference schemes applied to the heat-diffusion equation with semilinear dynamical boundary conditions. We prove that the numerical blow-up times converge to the continuous ones. Also, the number of peaks of the solutions is studied. Numerical experiments are discussed and at the same time, certain interesting properties of the continuous solutions are predicted.

On splitting-based numerical methods for nonlinear models of European options

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations.

Numerical solution of a nonlinear evolution equation for the risk preference

A singular nonlinear partial differential equation (PDE) for the risk preference was derived by the first author in previous publications. The PDE is related to the Arrow-Pratt coefficient of relative risk aversion. In the present paper, we develop a Rothe-Bellman & Kalaba quasilinearization method on quasi-uniform space mesh to numerically investigate such PDE. Numerical experiments are discussed.

Numerical Solution via Transformation Methods of Nonlinear Models in Option Pricing

Based on transformation techniques used in analysis of initial boundary value problems, we propose and discuss two numerical approaches for nonlinear models in option pricing. The first one exploits difference schemes for a degenerate parabolic problem on a finite interval. The second one solves the problem on infinite interval by Rothe’s method. An appropriate substitution reduces the matter to a semilinear integral equation at each time step. Numerical experiments are discussed.

On the Computation of Blow-Up Solutions of Elliptic Equations with Semilinear Dynamical Boundary Conditions

In this paper we study numerically blow-up solutions of elliptic equations with nonlinear dynamical boundary conditions. First, we formulate a result for blow-up, when dynamical boundary condition is posed on the part of the boundary. Next, by semidiscretization, we obtain a system of ordinary differential equations (ODEs), the solution of which also blows up. Under certain assumptions we prove that the numerical blow-up time converges to the corresponding real blow-up time, when the mesh size goes to zero. We investigate numerically the blow-up set (BUS) and the blow-up rate. Numerical experiments with local mesh refinement technique are also discussed.

Fourth-order compact schemes for a parabolic-ordinary system of European option pricing liquidity shocks model

This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed.

Positivity Preserving Numerical Method for Non-linear Black-Scholes Models

A motivation for studying the nonlinear Black- Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investors preferences, feedback and illiquid markets effects and risk from a volatile unprotected portfolio. In this work we develop positivity preserving algorithm for solving a large class of non-linear models in mathematical finance on the original infinite domain. Numerical examples are discussed.