A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.     

A multidomain spectral method for solving elliptic equations

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.     

A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations

In this article, we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Péclet number. From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement. Two main features about the results obtained in this work are worthy of special attention: First, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.     

A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers

A numerical scheme for a class of two-point singularly perturbed boundary value problems with an interior turning point having an interior layer or twin boundary layers is proposed. The solution of this type of problem exhibits a transition region between rapid oscillations and the exponential behaviour. The problem with interior turning point represents a one-dimensional version of stationary convection–diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. To solve these problems numerically, we consider a scheme which comprises quintic B-spline collocation method on an appropriate piecewise-uniform mesh, which is dense in the neighbourhood of the interior/boundary layer(s). The method is shown to be parameter-uniform with respect to the singular perturbation parameter ?. Some relevant numerical examples are illustrated to verify the theoretical aspects computationally. The results compared with other existing methods show that the proposed method provides more accurate solutions.     

Spline methods for solving system of second-order boundary-value problems

In this paper, we use cubic polynomial splines to derive some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

Shape optimization of continua using NURBS as basis functions

The present paper introduces a numerical solution to shape optimization problems of domains in which boundary value problems of partial differential equations are defined. In the present paper, the finite element method using NURBS as basis functions in the Galerkin method is applied to solve the boundary value problems and to solve a reshaping problem generated by the H1 gradient method for shape optimization, which has been developed as a general solution to shape optimization problems. Numerical examples of linear elastic continua illustrate that this solution works as well as using the conventional finite element method.     

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

ATOMFT: solving ODEs and DAEs using Taylor series

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODEs into a FORTRAN 77 object program that is compiled, linked with the ATOMFT runtime library, and run to solve the problem. We review the use of the ATOMFT system for nonstiff and stiff ODEs, the propagation of global errors, and applications to differential algebraic equations arising from certain control problems, to boundary value problems, to numerical quadrature, and to delay problems.     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

Efficient classes of Runge-Kutta methods for two-point boundary value problems

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

High-order scheme for determination of a control parameter in an inverse problem from the over-specified data

The problem of finding the solution of partial differential equations with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, thermo-elasticity, chemical diffusion and control theory. In this paper we present a high order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximate the spatial derivative with a fourth order compact scheme and reduce the problem to a system of ordinary differential equations (ODEs). Then we apply a fourth order boundary value method for the solution of resulting system of ODEs. So the proposed method has fourth order accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. Several numerical examples and also some comparisons with other methods in the literature will be investigated to confirm the efficiency of the new procedure.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises a B-spline collocation method on a uniform mesh, which leads to a tridiagonal linear system. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The analysis is done on a uniform mesh, which permits its extension to the case of adaptive meshes which may be used to improve the solution. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

Development,evaluation and selection of methods for elliptic partial differential equations

We present a framework within which to evaluate and compare computational methods to solve elliptic partial differential equations. We then report on the results of comparisons of some classical methods as well as a new one presented here. Our main motivation is the belief that the standard finite difference methods are almost always inferior for solving elliptic problems and our results are strong evidence that this is true. The superior methods are higher order (fourth or more instead of second) and we describe a new collocation finite element method which we believe is more efficient and flexible than the other well known methods, e.g., fourth order finite differences, fourth order finite element methods of Galerkin, Rayleigh-Ritz or least squares type.Our comparisons are in the context of the relatively complicated problems that arise in realistic applications. Our conclusion does not hold for simple model problems (e.g., Laplaces equation on a rectangle) where very specialized methods are superior to the generally applicable methods that we consider. The accurate and relatively simple treatment of boundary conditions involving curves and derivations is a feature of our collocation method.     

The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.