Tension Spline Research Articles

Interpolation methods applied to the spatialisation of monthly solar irradiation in a region of complex terrain in the state of Rio de Janeiro in the southeast of Brazil

Due to the scarcity of weather or solarimetric stations, and in the absence of observed solar irradiation data (Hg), one alternative for obtaining continuous and quality data distributed in space is the use of spatial interpolation. The aim of this study was to assess different methods of spatial interpolation applied to the monthly average daily Hg in the state of Rio de Janeiro, in the southeast of Brazil. Inverse distance weighting (IDW), tension spline (SpT) and ordinary Kriging (OKr) were assessed. Hg time series (2000–2013) were used, measured at 20 automatic weather stations located in the state of Rio de Janeiro and the states of Sao Paulo, Minas Gerais and Espirito Santo. The IDW and SpT methods showed satisfactory performance, and result in precision (coefficient of determination—r2 > 0.90 for IDW and > 0.80 for SpT) and accuracy (refined Willmott index—dref > 0.55 for IDW and SpT) for most months, while OKr showed inadequate precision (r2 0. 50) similar to that of IDW and SpT. The RMSE was less than 10% (1.64 MJ m−2 day−1) for all methods. Under conditions of heterogeneous station distribution, low station density and complex terrain, the OKr method does not produce a precise spatial interpolation of monthly Hg.

The uncertainty of gis-based interpolation methods in constructing shallow groundwater distribution map: A case study at Pleiku city, Gia Lai province

Four interpolation methods are applied to interpolate shallow groundwater level in Pleiku city, including inverse distance weighted, tension spline, universal kriging, and ordinary kriging. The cross-validation results record that the ordinary kriging is the best interpolation method which is shown by the lowest RMSE value, the highest R2 value. It is selected to assess the shallow groundwater level in spatial and seasonal change. Based on the groundwater level interpolated by the ordinary kriging, the groundwater level is divided into the northern and southern parts of the study area. The distribution of groundwater level is shallower than that in the southern part wheret groundwater depth is around less than15 m while in the southern part, most of the groundwater level is higher than 18 m. The elevation of groundwater level is found in rainy season; the elevated area accounts for 72.6% of natural area. Additionally, the groundwater level also declines in the rainy season at some region, focusing on Bien Ho lake region and two regions in the southern part of the city whose area accounts for 27.4% of the natural area.

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

PurposeThis paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.Design/methodology/approachIn this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.FindingsThe paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.Research limitations/implicationsThere are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.Practical implicationsPhysical problems with singular and non-singular coefficients are directly solved by this method.Originality/valueThe paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

Application of spline to approximate the solution of singularly perturbed boundary-value problems

We develop a class of new methods based on modification of polynomial spline function for the numerical solution of singularly perturbed boundary-value problems. The modified spline contains the exponential terms and named by tension spline, which is infinity smooth. Tension spline contain parameter, by choosing arbitrary values of such parameters the various classes of spline can be obtained. The proposed methods are accurate for solution of linear and non-linear singularly perturbed boundary-value problems. Boundary formulas are developed to associate with spline methods. These methods are converging. The analysis of convergence is shown to yield up to O(h^8 ) approximation to the solution of singularly perturbed boundary-value problems. Comparison are carried out, numerical examples are given to showing the efficiency of our methods

Computational results and analysis for a class of linear and nonlinear singularly perturbed convection delay problems on Shishkin mesh

This article presents a hybrid numerical scheme for a class of linear and nonlinear singularly perturbed convection delay problems on piecewise uniform. The proposed hybrid numerical scheme comprises with the tension spline scheme in the boundary layer region and the midpoint approximation in the outer region on piecewise uniform mesh. Error analysis of the proposed scheme is discussed and is shown $\varepsilon$-uniformly convergent. Numerical experiments for linear and nonlinear are performed to confirm the theoretical analysis.

Fourth-Order Numerical Scheme Based on Half-Step Non-Polynomial Spline Approximations for 1D Quasi-Linear Parabolic Equations

In this article, we discuss a fourth-order accurate scheme based on non-polynomial splines in tension approximations for solving quasi-linear parabolic partial differential equations (PDEs). The proposed numerical method requires only two half-step points and a central point on a uniform mesh in spatial direction. This method is derived directly from the continuity condition for the first-order derivative of the non-polynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is applicable for solving singular parabolic problems in polar systems. The proposed method is tested on the generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, and Burgers’ equations in polar coordinates.

Hargreaves ET0 Equation Regression Correction and Parameter Spatialization for Mainland China

HighlightsCorrelation coefficient between ET0 calculated by the PM and the HE, = 0.9 at 95% stations.Simple spline interpolation method more applicable than kriging interpolation for ET0 in mainland China.Linear regression correction model for the HE suggested by the FAO was further validated in large scale.Abstract. Reference crop evapotranspiration (ET0) is an important parameter employed in irrigation engineering. The Hargreaves ET0 equation (HE) is a simplified calculation method recommended by the Food and Agriculture Organization (FAO) of the United Nations and is used to determine ET0. However, the HE is empirical, and thus, the FAO suggests that linear regression correction be conducted when applying the HE in new areas. To discuss the validity of applying the HE linear regression correction model combined with spatial interpolation methods to the Chinese mainland, this study used meteorological data released by the China National Meteorological Data Service Center and considered the ET0-PM calculated by the Penman-Monteith equation to be the true value. With respect to conducting a linear regression correction with the HE, a regression analysis of the multi-year average of mean monthly daily ET0-PM and ET0-HE calculated by the HE showed that the values were significantly correlated, and correlation coefficients and R2 of 95% of stations exceeded 0.9 and 0.8, respectively. This indicates that the HE linear regression correction model recommended by the FAO is applicable for use in mainland China. The accuracy of ET0-HE corrected by the linear regression correction coefficient combined with simple tension spline interpolation was the optimal and stable at validation stations, whereas the accuracy of ET0-HE corrected by the linear regression correction coefficient combined with simple ordinary kriging interpolation contrasts with the accuracy from existing studies. This study can be used as a reference in large-scale correction research using ET0-HE, and it lays the foundation for research and development relating to spatiotemporal ET0 data products. Keywords: Coefficient modification, Linear regression model, Reference crop evapotranspiration, Spatial interpolation.

Tension spline method for the solution of elliptic equations

In this paper, two classes of methods are developed for the solution of two-dimensional elliptic partial differential equations. We have used tension spline function approximation in both x and y spatial directions and a new scheme of order has been obtained. The convergence analysis of the methods has been carried out. Numerical examples are given to illustrate the applicability and accurate nature of our approach.

Numerical Solution of a Weakly Coupled System of Singularly Perturbed Delay Differential Equations Via Cubic Spline in Tension

In this paper, a fitted numerical scheme is proposed for solving coupled system of singularly perturbed convection–diffusion delay differential equations. Similar boundary value problems are associated with expected first exit times of the membrane potential in models of neurons. The difference scheme on uniform mesh is accomplished by the method which is based on cubic spline in tension. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

Correction to: Analysis of deterministic and geostatistical interpolation techniques for mapping meteorological variables at large watershed scales

The widely scattered pattern of meteorological stations in large watersheds and remote locations, along with a need to estimate meteorological data for point sites or areas where little or no data have been recorded, has encouraged the development and implementation of spatial interpolation techniques. The various interpolation techniques featured in GIS software allow for the extraction of this new information from spatially distinct point data. Since no one interpolation method can be accurate in all regions, each method must be evaluated prior to each geographically distinct application. Many methods have been used for interpolating minimum temperature ( $$T_{ \min }$$ ), maximum temperature ( $$T_{ \max }$$ ) and precipitation data; however, only a few methods have been used in the Zayandeh-Rud River basin, Iran, and no comparison of methods has ever been carried out in the area. The accuracies of six spatial interpolation methods [Inverse Distance Weighting, Natural Neighbor (NN), Regularized Spline, Tension Spline, Ordinary Kriging, Universal Kriging] were compared in this study simultaneously, and the best method for mapping monthly precipitation and temperature extremes was determined in a large semi-arid watershed with high temperature and rainfall variation. A cross-validation technique and long-term (1970–2014) average monthly $$T_{ \min }$$ , $$T_{ \max }$$ and precipitation data from meteorological stations within the basin were used to identify the best interpolation method for each variable dataset. For $$T_{ \min }$$ , Kriging (Gaussian) proved to be the most accurate interpolation method (MAE = 1.827 °C), whereas, for $$T_{ \max }$$ and precipitation the NN method performed best (MAE = 1.178 °C and 0.5241 mm, respectively). Accordingly, these variable-optimized interpolation methods were used to define spatial patterns of newly generated climatic maps.

MAPPING OF SALINITY LEVEL USING SPLINE INTERPOLATION TECHNIQUES OVER THE WATER OF SUNGAI MERBOK, KEDAH

Interpolation map of salinity is a helpful scientific instrument for environmental monitoring and for social economic development especially for the community who lived in Sungai Merbok. This research was conducted to develop a spatial model of salinity using spline interpolation technique. 20 sampling stations were randomly set up to measure the level of salinity using YSI 650 Multiparameter Display System (MDS). Quantitative analysis of standard regression and error index were used to evaluate the developed model. The research found that the tension splines type performed better than regularized splines type. The local government and community, who live in Sungai Merbok, can use the developed map of salinity for guidelines and future development of Sungai Merbok, Kedah.

Simulation of directional well trajectory using tension spline

In order to solve the problems in the directional well trajectory simulating, the tension spline function is used for calculating the hole deviation angle and azimuth angle of arbitrary point on the borehole axis of directional well and Simpson integral formula is also used for solving its spatial coordinates. By the method, the simulation software of well trajectory is written. The results of example calculated by the software show that the simulation precision of tension spline method is higher and more reliable than the cubic spline interpolation method and can meet the requirements of actual wellbore trajectory calculation. The method can solve stress fluctuation appearing in analyzing the mechanics of pipe string using the wellbore data calculated by the cubic spline interpolation. It can also reduce errors, improve the precision of pipe string mechanics analysis and meet the engineering needs.

Tension spline technique for the solution of fourth-order parabolic partial differential equation

Tension spline technique for the solution of fourth-order parabolic partial differential equation

Tension spline method for solution of Fitzhugh–Nagumo equation

One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The Fitzhugh–Nagumo equation is a simplification of the Hodgin–Huxley model (Hodgin and Huxley, 1952) [24] for the membrane potential of a nerve axon. In this paper we developed a three time-level implicit method by using tension spline function. The resulting equations are solved by a tri-diagonal solver. We described the mathematical formulation procedure in detail. The stability of the presented method is investigated. Results of numerical experiments verify the theoretical behavior of the orders of convergence.

An improved statistical downscaling scheme of Tropical Rainfall Measuring Mission precipitation in the Heihe River basin, China

Estimating an accurate spatial distribution of precipitation with high resolution is necessary for hydrological and ecological applications, especially in data‐scarce and terrain‐complicated river basins. Satellite‐based precipitation data have been widely used to measure the spatial patterns of precipitation, but an improvement in accuracy and resolution is needed. In this article, a new statistical downscaling method is proposed to generate improved monthly precipitation fields at a higher spatial resolution of 1 km in Heihe River basin (HRB), China. The presented methods employed the geographical weighted regression (GWR) method to explore the non‐stationarity between precipitation and its factors, and used the high‐accuracy surface modelling method (HASM) to compensate for the errors produced in the GWR downscaling process. The GWR model was first established under five different spatial scales, and the optimal relation between precipitation derived from the Tropical Rainfall Measuring Mission (TRMM) and its influencing factors was found for each month. The errors caused during the scale change were modified by performing HASM as a data merging framework, which considered both the local climate characteristics and meteorological observations. Results showed that the GWR downscaling method could not generate spatial patterns of precipitation similar to those of the original TRMM products. Although the performance of the GWR method after residual interpolations using Kriging, IDW, and tension Spline was improved, there existed significant variations in some regions, and the accuracy of those methods was still not satisfactory. In comparison with the other four models, GWR‐HASM showed better performance in reproducing the precipitation field at a high spatial resolution. Results indicate that the proposed downscaling method appears feasible for precipitation estimation in data‐scarce river basins.

Analysis and numerical simulation for a class of time fractional diffusion equation via tension spline

The numerical solution for a class of time fractional diffusion equation via tension spline is studied. Time fractional derivative is considered in the Caputo sense. The numerical method is constructed by means of the Crank-Nicolson method and is proven to be conditionally stable. Convergence analysis is also discussed by using the Fourier series method. Numerical evidences are given to prove the efficiency of the proposed method.

Some splines produced by smooth interpolation

The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline.

Three level implicit tension spline scheme for solution of Convection-Reaction-Diffusion equation

Abstract In this work, the numerical approximation of Convection-Reaction-Diffusion equation is investigated using the method based on tension spline function and finite difference approximation. For nonlinear term, nonstandard finite difference method by nonlocal approximation is utilized. We describe the mathematical formulation procedure in detail and also analyze the stability and convergence of the method. Numerical results are provided to justify the good performance of the proposed scheme.

Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations via Cubic Spline in Tension

This paper deals with the singularly perturbed boundary value problem for a second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A difference scheme on a uniform mesh is accomplished by the method based on cubic spline in tension. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

Geostatistical characterization of the soil of Aguascalientes, México, by using spatial estimation techniques.

Four spatial estimation techniques available in commercial computational packages are evaluated and compared, namely: regularized splines interpolation, tension splines interpolation, inverse distance weighted interpolation, and ordinary Kriging estimation, in order to establish the best representation for the shallow stratigraphic configuration in the city of Aguascalientes, in Central Mexico. Data from 478 sample points along with the software ArcGIS (Environmental Systems Research Institute, Inc. (ESRI), ArcGIS, ver. 9.3, Redlands, California 2008) to calculate the spatial estimates. Each technique was evaluated based on the root mean square error, calculated from a validation between the generated estimates and measured data from 64 sample points which were not used in the spatial estimation process. The present study shows that, for the estimation of the hard-soil layer, ordinary Kriging offered the best performance among the evaluated techniques.