D. Hollevoet

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

Fourth-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.

Exponentially-fitted methods and their stability functions

We investigate the properties of stability functions of exponentially-fitted Runge-Kutta methods, and we show that it is possible (to some extent) to determine the stability function of a method without actually constructing the method itself. To focus attention, examples are given for the case of one-stage methods. We also make the connection with so-called integrating factor methods and exponential collocation methods. Various approaches are given to construct these methods.

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 multiple 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.

Exponentially-fitted methods: searching the frequency

Exponential fitted algorithms for initial value and boundary value methods and for the calculation of quadrature rules are introduced. Special attention is paid to the form of the leading order term of the error and to the global error term in particular. It is shown in which way a “best value” for the occurring frequency can be selected. Numerical experiments illustrate the proposed strategy.

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.

Exponentially‐Fitted Methods Applied to Fourth‐Order Boundary Value Problems

We solve a linear fourth‐order boundary value problem by means of a set of five‐point finite difference equations whose coefficients are determined by a couple (K,P), where K+1 indicates the number of conditions related to polynomials and P+1 indicates the number of conditions related to exponentials (or trigonometric functions). These methods, which we call exponentially‐fitted, contain a parameter which can be tuned for the problem at hand. Starting from the expression for the error, we will discuss ways to choose this parameter. It will also be shown that the obtained accuracy can heavily depend upon the accuracy to which the parameter is computed.

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.

The Use of Exponentially Fitted Methods in a Deferred Correction Framework

The combination of exponential fitting and deferred correction based on mono‐implicit Runge‐Kutta methods is investigated. The structure of the B‐series coefficients of exponentially fitted deferred correction (EFDC) schemes is compared to that of classical counterparts. It is shown how the leading error term of an EFDC scheme can be annihilated or minimized.