Collocation Solution Research Articles

Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra–Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we discuss the relationships between necessary conditions for blow-up of collocation solutions and exact solutions.

A geoid model for Egypt based on GOCO03S and logarithmic covariance function

In this study, the logarithmic least-squares collocation technique was applied to model the geoid surface over Egypt, using the satellite only model GOSO03S as a reference field. Firstly, several combinations of the logarithmic covariance fitting parameters were tried and the best parameters were selected to solve for the geoid. Four different types of gravitational data were input into the collocation algorithm. The resulting geoid model was evaluated at GPS/Levelling control and showed a much better spatial precision than the GOCO03S geoid. After the removal of a long wavelength trend, the final geoid model showed a spatial precision of 11 cm. Such precision was found to lie within the mean geoid noise at the check points, which was obtained from the collocation solution. So, the logarithmic covariance model proved to work well with precise satellite only geopotential models, when used in a remove-restore scheme. Furthermore, such technique resulted in consistent spatial precision and estimated noise for the final geoid model.

Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation

The solution and analysis of Kuramoto–Sivashinsky equation by cubic Hermite collocation method is performed and a bound for maximum norm of the semi-discrete solution is derived by using Lyapunov functional. Error estimates are also obtained for semi-discrete solutions and verified by numerical experiments.

Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations

In this paper, Legendre Collocation method, an easy-to-use variant of the spectral methods for the numerical solution of a class of fractional integro-differential equations (FIDE’s), is researched. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Legendre Gauss quadrature formula. An efficient convergence analysis of the proposed method is given and rate of convergence is established in the L2-norm. Due to the fact that the solutions of FIDE’s usually have a weak singularity at origin, we use a variable transformation to change the original equation into a new equation with a smooth solution. We prove that after this regularization technique, numerical solution of the new equation by adopting the Legendre collocation method has exponentially rate of convergence. Numerical results are presented which clarify the high accuracy of the proposed method.

Legendre Collocation Solution to Fractional Ordinary Differential Equations

In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

Discrete Legendre spectral projection methods for Fredholm–Hammerstein integral equations

In this paper we discuss the discrete Legendre Galerkin and discrete Legendre collocation methods for Fredholm–Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both discrete Legendre Galerkin and discrete Legendre collocation solutions in both infinity and L2-norm. Numerical examples are given to illustrate the theoretical results.

A multi-level spectral deferred correction method

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.

Error estimation based on locally weighted defect for boundary value problems in second order ordinary differential equations

We investigate efficient asymptotically correct a posteriori error estimates for the numerical approximation of two-point boundary value problems for second order ordinary differential equations by piecewise polynomial collocation methods. Our error indicators are based on the defect of the collocation solution with respect to an associated exact difference scheme and backsolving using a cheap, low order finite-difference scheme. We prove high asymptotical correctness of this error indicator and illustrate the theoretical analysis by numerical examples.

Vibration analysis and robust control of highly deformable beams in a heavy pinched loop configuration

A heavy pinched loop is formed by bringing and clamping the two ends of a highly deformable slender beam, elastica. A collocation solution technique is implemented for studying the formation statically and dynamically, i.e. small vibrations around the large deformed static solutions, and the earlier work using a shooting method is validated. A new and clear Galerkin formulation capable of modelling damping is established for finding transients, and a new theoretical multi-point boundary value problem approach is used for numerically obtaining the frequency response function. Lastly, the obtained dynamic model is used for active vibration control, wherein a controller is designed using H ∞ algorithm for active damping in a heavy pinched loop for two simplified cases, and the simulated results are shown.

On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.

Three Dimensional Vibration Analysis of Rectangular Plates with Undamaged and Damaged Boundaries by the Spectral Collocation Method

This paper presents a new numerical technique for the free vibration analysis of isotropic three dimensional elastic plates with damaged boundaries. In the study, it is assumed that the plates have free lateral surfaces, and two opposite simply supported edges, while the other edges could be clamped, simply supported or free. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of three dimensional plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged three dimensional plates indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for damage detection of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.Copyright © 2011 by ASME

Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method

In this paper, we present a new idea to approximate high-dimensional nonlinear stochastic partial differential equations (SPDEs) by a kernel-based collocation method, which is a meshfree approximation method. A reproducing kernel is used to construct an approximate basis. The kernel-based collocation solution is a linear combination of the reproducing kernel with the differential and boundary operators of SPDEs at the given collocation points placed in the related high-dimensional domains. Its random expansion coefficients are computed by a random optimisation problem with constraint conditions induced by the nonlinear SPDEs. For a fixed kernel function, the convergence of kernel-based collocation solutions only depends on the fill distance of the chosen collocation points for the bounded domain of SPDEs. The numerical experiments of Sobolev-spline kernels for Klein-Gordon SPDEs show that the kernel-based collocation method produces the well-behaved approximate probability distributions of the SPDE solutions.

Design of a Simulator for Enhanced Oil Recovery Process Using a Nigerian Reservoir as a Case Study

This study involves the applications of different numerical techniques in a more general way to the design of a simulator for an enhanced oil recovery process with surfactant assisted water flooding. The data from a hypothetical oil well and a Nigerian oil well were used for the validation of the simulator. The process is represented by a system of nonlinear partial differential equations: the continuity equation for the transport of the components and Darcy’s equation for the phase flow. The orthogonal collocation, finite difference and coherence theory techniques were used in solving the equations that characterized the multidimensional, multiphase and multicomponent flow problem. Matlab computer programs were used for the numerical solution of the model equ- ations. The predicted simulator, obtained from the resulting numerical exercise confers uncondi- tional stability and more insight into the physical reservoir description. The results of the ortho- gonal collocation solution were compared with those of finite difference and coherence solutions. The results indicate that the concentration of surfactants for orthogonal collocation show more features than the predictions of the coherence solution and the finite difference, offering more opportunities for further understanding of the physical nature of the complex problem. We have found out that the partition of the three components between the two-phases determines other physical property data and hence the oil recovery. The oil recovery for the Nigerian oil reservoir is higher than the recovery predicted for the hypothetical crude. The displacement mechanism for the multicomponent and multiphase system is stable in the Nigerian oil reservoir due to the mod- erate value of the oil/water viscosity instead of the hypothetical reservoir with high value of oil/water ratio.

Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method

The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.

A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix

In this paper, a well-conditioned collocation method is constructed for solving general $p$th order linear differential equations with various types of boundary conditions. Based on a suitable Birkhoff interpolation, we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features: the condition number of the linear system is independent of the number of collocation points, and the underlying boundary conditions are imposed exactly. Moreover, the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative (at interior collocation points), which is therefore called the pseudospectral integration matrix (PSIM). We show that PSIM produces the optimal integration preconditioner and stable collocation solutions with even thousands of points.

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.

Collocation methods for impulsive differential equations

By choosing a suitable piecewise continuous collocation space, the convergence, global superconvergence and local superconvergence of the collocation solution for linear impulsive differential equations are derived. The conditions that the collocation solution is asymptotical stable are obtained and some numerical experiments are given.

Legendre spectral projection methods for Urysohn integral equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n12−r) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n−2r) in both L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n−r) in both L2-norm and infinity norm, n being the highest degree of the Legendre polynomial employed in the approximation and r being the smoothness of the kernel. We are able to obtain similar superconvergence rates for the iterated Galerkin solution for Urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions.

Tropospheric profiles of wet refractivity and humidity from the combination of remote sensing data sets and measurements on the ground

Abstract. We reconstruct atmospheric wet refractivity profiles for the western part of Switzerland with a least-squares collocation approach from data sets of (a) zenith path delays that are a byproduct of the GPS (global positioning system) processing, (b) ground meteorological measurements, (c) wet refractivity profiles from radio occultations whose tangent points lie within the study area, and (d) radiosonde measurements. Wet refractivity is a parameter partly describing the propagation of electromagnetic waves and depends on the atmospheric parameters temperature and water vapour pressure. In addition, we have measurements of a lower V-band microwave radiometer at Payerne. It delivers temperature profiles at high temporal resolution, especially in the range from ground to 3000 m a.g.l., though vertical information content decreases with height. The temperature profiles together with the collocated wet refractivity profiles provide near-continuous dew point temperature or relative humidity profiles at Payerne for the study period from 2009 to 2011. In the validation of the humidity profiles, we adopt a two-step procedure. We first investigate the reconstruction quality of the wet refractivity profiles at the location of Payerne by comparing them to wet refractivity profiles computed from radiosonde profiles available for that location. We also assess the individual contributions of the data sets to the reconstruction quality and demonstrate a clear benefit from the data combination. Secondly, the accuracy of the conversion from wet refractivity to dew point temperature and relative humidity profiles with the radiometer temperature profiles is examined, comparing them also to radiosonde profiles. For the least-squares collocation solution combining GPS and ground meteorological measurements, we achieve the following error figures with respect to the radiosonde reference: maximum median offset of relative refractivity error is −16% and quartiles are 5% to 40% for the lower troposphere. We further added 189 radio occultations that met our requirements. They mostly improved the accuracy in the upper troposphere. Maximum median offsets have decreased from 120% relative error to 44% at 8 km height. Dew point temperature profiles after the conversion with radiometer temperatures compare to radiosonde profiles as to: absolute dew point temperature errors in the lower troposphere have a maximum median offset of −2 K and maximum quartiles of 4.5 K. For relative humidity, we get a maximum mean offset of 7.3%, with standard deviations of 12–20%. The methodology presented allows us to reconstruct humidity profiles at any location where temperature profiles, but no atmospheric humidity measurements other than from GPS are available. Additional data sets of wet refractivity are shown to be easily integrated into the framework and strongly aid the reconstruction. Since the used data sets are all operational and available in near-realtime, we envisage the methodology of this paper to be a tool for nowcasting of clouds and rain and to understand processes in the boundary layer and at its top.

Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium

A completely spectral collocation method (CSCM) is developed to solve radiative transfer equation in anisotropic scattering medium with graded index. Different from the Chebyshev collocation spectral method based on the discrete ordinates method (SP-DOM), the CSCM is used to discretize both the angular domain and the spatial domain of radiative transfer equation. In this approach, the angular derivative term and the integral term are approximated by the high order spectral collocation scheme instead of the low order finite difference approximations. Compared with those available data in literature, the CSCM has a good accuracy for a wide range of the extinction coefficient, the scattering albedo, the scattering phase function, the gradient of refractive index and the boundary emissivity. The CSCM can provide exponential convergence for the present problem. Meanwhile, the CSCM is much more economical than the SP-DOM. Moreover, for nonlinear anisotropic scattering and graded index medium with space-dependent albedo, the CSCM can provide smoother results and mitigate the ray effect.