HIGH ORDER ONE-STEP A-STABLE EXPONENTIALLY FITTED METHODS

In this paper,the necessary and sufficient conditions for generalone-step m ethods to be exponentially fitted atq0∈C aregiven.A classofm ultiderivative hybrid one-step m ethods of order at leasts+ 1 is constructed w ith s+ 1 param eters,w here sis the order of derivative.The necessary and sufficient conditions for these m ethods to be A-stable and exponentially fitted is proved.Furtherm ore,a class ofA-stable 2 param eters hybrid one-step m ethods oforderatleast 8 are constructed,w hich use 4th order derivative.These m ethods are exponentially fitted atq0 if and only if its fitted function f(q) satisfies f(q0)= 0.Finally,an A-stable exponentially fitted m ethod oforder 8 is obtained.

AN EXPONENTIALLY FITTED DIFFERENCE SCHEME FOR THE HYPERBOLIC-HYPERBOLIC SINGULARLY PERTURBED INITIAL-BOUNDARY VALUE PROBLEM

In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.

EXPONENTIALLY FITTED LOCAL DISCONTINUOUS GALERKIN METHOD FOR CONVECTION-DIFFUSION PROBLEMS

In this paper, we study the local discontinuous Galerkin （LDG） method for one- dimensional singularly perturbed convection-diffusion problems by an exponentially fitted technique. We prove that the method is uniformly first-order convergent in the energy norm with respect to the small diffusion parameter.

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.

EXPONENTIALLY FITTED TRAPEZOIDAL SCHEME FOR A STOCHASTIC OSCILLATOR

This paper applies exponentially fitted trapezoidal scheme to a stochastic oscillator. The scheme is convergent with mean-square order 1 and symplectic. Its numerical solution oscillates and the second moment increases linearly with time. The numerical example verifies the analysis of the scheme.

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.

DIFFERENCE SCHEME FOR AN INITIAL-BOUNDARY VALUE PROBLEM FOR LINEAR COEFFICIENT-VARIED PARABOLIC DIFFERENTIAL EQUATION WITH A NONSMOOTH BOUNDARY LAYER FUNCTION

In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.

FINITE ELEMENT ANALYSIS OF A LOCAL EXPONENTIALLYFITTED SCHEME FOR TIME-DEPENDENTCONVECTION-DIFFUSION PROBLEMS

In [16], Stynes and O' Riordan(91) introduced a local exponentially fitted finite element (FE) scheme for a singularly perturbed two-point boundary value problem without turning-point. An E-uniform h(1/2)-order accuracy was obtain for the epsilon-weighted energy norm. And this uniform order is known as an optimal one for global exponentially fitted FE schemes (see [6, 7, 12]). In present paper, this scheme is used to a parabolic singularly perturbed problem. After some subtle analysis, a uniformly in epsilon convergent order h\ln h\(1/2) + tau is achieved (h is the space step and tau is the time step), which sharpens the results in present literature. Furthermore, it implies that the accuracy order in [16] is actuallay h\ln h\(1/2) rather than h(1/2).

Modeling height–diameter relationship for Populus euphratica in the Tarim riparian forest ecosystem, Northwest China

Modeling height–diameter relationships is an important component in estimating and predicting forest development under different forest management scenarios. In this paper, ten widely used candidate height–diameter models were fitted to tree height and diameter at breast height（DBH）data for Populus euphratica Oliv. within a 100 ha permanent plots at Arghan Village in the lower reaches of the Tarim River, Xinjiang Uyghur Autonomous Region of China. Data from 4781 trees were used and split randomly into two sets：75 % of the data were used to estimate model parameters（model calibration）, and the remaining data（25 %） were reserved for model validation. All model performances were evaluated and compared by means of multiple model performance criteria such as asymptotic t-statistics of model parameters, standardized residuals against predicted height,root mean square error（RMSE）, Akaike’s informationcriterion（AIC）, mean prediction error（ME） and mean absolute error（MAE）. The estimated parameter a for model（6） was not statistically significant at a level of a = 0.05. RMSE and AIC test result for all models showed that exponential models（1）,（2）,（3） and（4） performed significantly better than others. All ten models had very small MEs and MAEs. Nearly all models underestimated tree heights except for model（6）. Comparing the MEs and MAEs of models, model（1） produced smaller MEs（0.0059） and MAEs（1.3754） than other models. To assess the predictive performance of models, we also calculated MEs by dividing the model validation data set into 10-cm DBH classes. This suggested that all models were likely to create higher mean prediction errors for tree DBH classes[20 cm. However, no clear trend was found among models.Model（6） generated significantly smaller mean prediction errors across all tree DBH classes. Considering all the aforementioned criteria, model（1）： TH ？ 1：3 t a= e1 t b？eàc？DBHT and model（6）： TH ？ 1：3 t DBH2= ea t b？DBH t c ？ DBH2T are recommended as suitable models for describing the height–diameter relationship of P. euphratica. The limitations of other models showing poor performance in predicting tree height are discussed. We provide explanations for these shortcomings.

Spatial distribution of soil hydraulic parameters estimated by pedotransfer functions for the Jialing River Catchment,Southwestern China

Soil hydraulic parameters θs,α,n,Ks,L and θr of the van Genuchten-Mualem model were estimated using three pedotransfer functions(PTFs) based on soil properties for surface soils of the largest main tributary catchment(the Jialing River) of the upper Yangtze River in China.The soil database was from the second national soil survey of China with a spatial 30 × 30 arc-second resolution.According to the statistical analysis of the differences between the continuous-PTFs-estimated values of soil hydraulic parameters for the study catchment and the reference values for a specific texture class provided in the development database of a specific PTF,best estimations were obtained using the W?sten PTF.The Rawls & Brakenssiek PTF was good estimation for parameter θr that was assumed as zero by W?sten PTF.The established higher θr(0.08%) and lower Ks(20 cm/d) and θs(0.43%) in the mid-downstream area relative to the other areas of the catchment could lead to larger amounts of surface runoff andconsequently provide higher energy to erode soil.Thus,these factors provide a supporting explanation for previously reported severe soil erosion occurring in this area.Spatial heterogeneity analysis for estimated hydraulic parameters in terms of semivariogram showed that the spatial correlation distance was in the range of 50-80 km and that the spatial variability(sill) was not large except for parameters Ks and L.The semi-variance with the exponential model at the zero distance(nugget) was 30%-50% of the sill.This study provided a practical PTF approach for estimating soil hydraulic properties from soil survey data at a large watershed scale.The estimation results could provide better insight into the mechanism of surface runoff and soil erosion,which is important to better understand and manage erosion in the catchment.

Modeling frequency distributions of tree height, diameter and crown area by six probability functions for open forests of Quercus persica in Iran

The prediction of the distribution of quantitative variables in a forest stand is of great interest to forest managers, for the evaluation of forest resources and scheduling of future silvicultural treatments. The aim of this research was to model the distribution of quantitative variables for Quercus persica in open forests in Iran. To investigate the probability distribution of trees in natural stands, 642 trees were selected for measurement using a systematic random sampling method. Selected trees were measured and data were analyzed. Gamma, beta, normal,lognormal, exponential and Weibull probability distributions were fitted to the height distribution of trees. Variables of distribution functions were estimated using the maximum likelihood estimation method. Actual probability and probability which derived from functions was compared using Kolmogorov–Smirnov and Anderson–Darling tests. Beta, Weibull and Weibull probability distributions explained the distributions of tree height, DBH and crown area.

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.

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.

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.

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.

Synthesis and thermometric properties of Yb^3＋-Er^3＋ co-doped K2GdF5 up-conversion phosphors

Yb^3＋-Er^3＋ co-doped K2GdF5 up-conversion phosphor was successfully synthesized by a solid-state reaction method. The phase purity and structure of the sample were characterized by powder X-ray diffraction. The sample emitted orange light at room temperature and its up-conversion spectra at different temperatures were recorded under the excitation of a 980 nm diode laser. The energy transfer from Yb^3＋ to Er^3＋ notably enhanced the up-conversion luminescence intensity. The possible up-conversion mechanisms and processes were proposed based on the power dependence of the luminescence intensities. The temperature-dependent up-conversion luminescence and temperature sensing performances of the sample were discussed according to the fluorescence intensity ratio of green emissions originating from ~2H（11/2）/~4S（3/2）→~4I（15/2） transitions of Er^3＋ in the range from 307 K to 570 K under the excitation of 980 nm laser with power of 260 mW. The dependence of the fluorescence intensity ratio on temperature was fitted with an exponential function and the effective energy difference obtained was 690 cm^（–1）, which further gave a relative temperature sensitivity of 1.1%/K at 307 K. The results suggested that the Yb^3＋-Er^3＋ co-doped K2GdF5 sample is a promising candidate for optical temperature sensor.

Thermal time-constant spectrum extraction method in AlGaN/GaN HEMTs

The transient temperature rise in the active region in AlGaN/GaN high electron mobility transistors （HEMTs） is measured using an electrical method. The original data are smoothed and denoised by a nonparametric fitting algorithm, called locally weighted scatterplot smoothing （LOWESS）. The thermal time-constant spectrum is extracted to analyze the physical structure of the heat-conduction path in A1GaN/GaN HEMTs. The thermal time- constant spectra extracted using the LOWESS algorithm are richer and the RC network obtained is greater compared with those with the traditional denoising method （multi-exponential fitting）. Thus, the analysis of the heat-flow path is more precise. The results show that the LOWESS nonparametric fitting algorithm can remove noise from measured data better than other methods and can retain the subtle variation tendency of the original discrete data. The thermal time-constant spectra extracted using this method can describe the subtle temperature variations in the A1GaN/GaN HEMT active region. This will help researchers to precisely analyze the layer composition of the heat-flow path.

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.