Exponentially-fitted methods and their stability functions

Is it possible to determine the stability function of an exponentially-fitted Runge-Kutta method, without actually constructing the method itself? This question was answered in a recent paper and examples were given for one-stage methods. In this paper we summarize the results and we focus on two-stage methods

Application of exponential fitting techniques to numerical methods for solving differential equations

Ever since the work of Isaac Newton and Gottfried Leibniz in the late 17th century, differential equations (DEs) have been an important concept in many branches of science. Differential equations arise spontaneously in i.a. physics, engineering, chemistry, biology, economics and a lot of fields in between. From the motion of a pendulum, studied by high-school students, to the wave functions of a quantum system, studied by brave scientists: differential equations are common and unavoidable. It is therefore no surprise that a large number of mathematicians have studied, and still study these equations. The better the techniques for solving DEs, the faster the fields where they appear, can advance. Sadly, however, mathematicians have yet to find a technique (or a combination of techniques) that can solve all DEs analytically. Luckily, in the meanwhile, for a lot of applications, approximate solutions are also sufficient. The numerical methods studied in this work compute such approximations. Instead of providing the hypothetical scientist with an explicit, continuous recipe for the solution to their problem, these methods give them an approximation of the solution at a number of discrete points. Numerical methods of this type have been the topic of research since the days of Leonhard Euler, and still are. Nowadays, however, the computations are performed by digital processors, which are well-suited for these methods, even though many of the ideas predate the modern digital computer by almost a few centuries. The ever increasing power of even the smallest processor allows us to devise newer and more elaborate methods. In this work, we will look at a few well-known numerical methods for the solution of differential equations. These methods are combined with a technique called exponential fitting, which produces exponentially fitted methods: classical methods with modified coefficients. The original idea behind this technique is to improve the performance on problems with oscillatory solutions

Multiparameter exponentially-fitted methods applied to second-order boundary value problems

Second-order boundary value problems are solved by means of a new type of exponentially-fitted methods that are modifications of the Numerov method. These methods depend upon a set of parameters which can be tuned to solve the problem at hand more accurately. Their values can be fixed over the entire integration interval, but they can also be determined locally from the local truncation error. A numerical example is given to illustrate the ideas

Three-stage two-parameter symplectic, symmetric exponentially-fitted Runge-Kutta methods of Gauss type

We construct an exponentially-fitted variant of the well-known three stage Runge-Kutta method of Gauss-type. The new method is symmetric and symplectic by construction and it contains two parameters, which can be tuned to the problem at hand. Some numerical experiments are given

On the Leading Error Term of Exponentially Fitted Numerov Methods

Second-order boundary value problems are solved with exponentially-fitted Numerov methods. In order to attribute a value to the free parameter in such a method, we look at the leading term of the local truncation error. By solving the problem in two phases, a value for this parameter can be found such that the tuned method behaves like a sixth order method. Furthermore, guidelines to choose between multi le possible values for this parameter are given

Multi-parameter exponentially fitted, P-stable Obrechkoff methods

We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for second order differential equations. An earlier result for single-parameter exponential fitting is re-examined and extended to multi-parameter, multi-order exponential fitting

Deferred correction based on exponentially fitted mono-implicit Runge-Kutta methods

The combination of exponential fitting and deferred correction based on mono-implicit Runge-Kutta (MIRK) methods is discussed. Particular attention is given to the parameter selection of the exponentially fitted deferred correction schemes to annihilate or minimize the leading error term. Several algorithms are discussed and illustrated with numerical results

Algorithm 927: the MATLAB code bvptwp.m for the numerical solution of two point boundary value problems

In this article we describe the code bvptwp.m, a MATLAB code for the solution of two point boundary value problems. This code is based on the well-known Fortran codes, twpbvp.f, twpbvpl.f and acdc.f, that employ a mesh selection strategy based on the estimation of the local error, and on revisions of these codes, called twpbvpc.f, twpbvplc.f and acdcc.f, that employ a mesh selection strategy based on the estimation of the local error and the estimation of two parameters which characterize the conditioning of the problem. The codes twpbvp.f/tpbvpc.f use a deferred correction scheme based on Mono-Implicit Runge-Kutta methods (MIRK); the other codes use a deferred correction scheme based on Lobatto formulas. The acdc.f/acdcc.f codes implement an automatic continuation strategy. The performance and features of the new solver are checked by performing some numerical tests to show that the new code is robust and able to solve very difficult singularly perturbed problems. The results obtained show that bvptwp.m is often able to solve problems requiring stringent accuracies and problems with very sharp changes in the solution. This code, coupled with the existing boundary value codes such as bvp4c.m, makes the MATLAB BVP section an extremely powerful one for a very wide range of problems

Exponentially fitted methods applied to fourth-order boundary value problems

AbstractFourth-order boundary value problems are solved by means of exponentially fitted methods of different orders. These methods, which depend on a parameter, can be constructed following a six-step flow chart of Ixaru and Vanden Berghe. Special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible. Some numerical examples are given to illustrate the practical implementation issues of these methods

The optimal exponentially-fitted Numerov method for solving two-point boundary value problems

AbstractSecond order boundary value problems are solved by means of exponentially-fitted Numerov methods. These methods, which depend on a parameter, can be constructed following a six-step flow chart of Ixaru and Vanden Berghe [L.Gr. Ixaru, G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, 2004]. Special attention is paid to the expression of the error term of such methods. An algorithm concerning the choice of the best suited method and its parameter is discussed. Several numerical examples are given to sustain the theory