ON CONSTRUCTION OF HIGH ORDER EXPONENTIALLY FITTED METHODS BASED ON PARAMETERIZED RATIONAL APPROXIMATIONS TO exp

By the discussion of the formula and properties of (4,4) parametric form rational approximation to function exp(q), the fourth order derivative one_step exponentially fitted method and the third order derivative hybrid one_step exponentially fitted method are presented, their order p satisfying 6≤p≤8. The necessary and sufficient conditions for the two methods to be A_ stable are given. Finally, for the fourth order derivative method, the error bound and the necessary and sufficient conditions for it to be median are discussed.

Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.

A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory

In this paper,a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable.Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory.Here,the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach.When the shift parameter is smaller than the perturbation parameter,the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed.The proposed finite difference scheme is unconditionally stable.When the shift parameter is larger than the perturbation parameter,a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed.This scheme is also unconditionally stable.The applicability of the proposed methods is demonstrated by means of two examples.

Improvement of scattering correction for in situ coastal and inland water absorption measurement using exponential fitting approach

The absorption coefficient of water is an important bio-optical parameter for water optics and water color remote sensing. However, scattering correction is essential to obtain accurate absorption coefficient values in situ using the nine-wavelength absorption and attenuation meter AC9. Establishing the correction always fails in Case 2 water when the correction assumes zero absorption in the near-infrared(NIR) region and underestimates the absorption coefficient in the red region, which affect processes such as semi-analytical remote sensing inversion. In this study, the scattering contribution was evaluated by an exponential fitting approach using AC9 measurements at seven wavelengths(412, 440, 488, 510, 532, 555, and 715 nm) and by applying scattering correction. The correction was applied to representative in situ data of moderately turbid coastal water, highly turbid coastal water, eutrophic inland water, and turbid inland water. The results suggest that the absorption levels in the red and NIR regions are significantly higher than those obtained using standard scattering error correction procedures. Knowledge of the deviation between this method and the commonly used scattering correction methods will facilitate the evaluation of the effect on satellite remote sensing of water constituents and general optical research using different scatteringcorrection methods.

EXPONENTIAL FITTED METHODS FOR THE NUMEMCAL SOLUTION OF THE SCHRDINGER EQUATION

A new sixth-order Runge-Kutta type method is developed for the numericalintegration of the radial Schrodinger equation and of the coupled differential equa-tions of the Schrodinger type. The formula developed contains certain free pa-rameters which allows it to be fitted automatically to exponential functions. Wegive a comparative error analysis with other sixth order exponentially fitted meth-ods. The theoretical and numerical results indicate that the new method is moreaccurate than the other exponentially fitted methods.

Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations(PDE)in the convectiondominated case,i.e.,for European options,if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as P´eclet number-is high.For Asian options,additional similar problems arise when the"spatial"variable,the stock price,is close to zero.Here we focus on three methods:the exponentially fitted scheme,a modification of Wang’s finite volume method specially designed for the Black-Scholes equation,and the Kurganov-Tadmor scheme for a general convection-diffusion equation,that is applied for the first time to option pricing problems.Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence.For the reduction technique proposed by Wilmott,a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options.Finally,we present experiments and comparisons with different(non)linear Black-Scholes PDEs.