Class Of Boundary Value Methods Research Articles

A generalized midpoint-based boundary value method for unstable partial differential equations

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

Generalized Second Derivative Linear Multistep Methods Based on the Methods of Enright

The Adams-type second derivative multistep methods of W. H. Enright is generalized to a class of boundary value methods for the numerical solution of initial value problems in ordinary differential equations. The root distribution of the stability polynomials of the new class of methods are given for the purpose of their correct implementation as boundary value methods. Computational results on some stiff problems are compared with those from some existing methods, while ODE15s of MATLAB is used as a reference numerical solution.

Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.

The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions

The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.

Boundary Value Methods as an extension of Numerov's method for Sturm–Liouville eigenvalue estimates

In this paper a class of Boundary Value Methods obtained as an extension of the Numerov's method is proposed for the numerical approximation of the eigenvalues of regular Sturm–Liouville problems subject to Dirichlet boundary conditions. It is proved that the error in the so obtained estimate of the kth eigenvalue behaves as O ( k p + 1 h p − 1 2 ) + O ( k p + 2 h p ) , where p is the order of accuracy of the method and h is the discretization stepsize. Numerical results comparing the performances of the new matrix methods with that of the corrected Numerov's method are also reported.

High-order finite difference schemes for the solution of second-order BVPs

We introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh.

Numerical approximation of nonlinear BVPs by means of BVMs

Boundary Value Methods (BVMs) would seem to be suitable candidates for the solution of nonlinear Boundary Value Problems (BVPs). They have been successfully used for solving linear BVPs together with a mesh selection strategy based on the conditioning of the linear systems. Our aim is to extend this approach so as to use them for the numerical approximation of nonlinear problems. For this reason, we consider the quasi-linearization technique that is an application of the Newton method to the nonlinear differential equation. Consequently, each iteration requires the solution of a linear BVP. In order to guarantee the convergence to the solution of the continuous nonlinear problem, it is necessary to determine how accurately the linear BVPs must be solved. For this goal, suitable stopping criteria on the residual and on the error for each linear BVP are given. Numerical experiments on stiff problems give rather satisfactory results, showing that the experimental code, called TOM, that uses a class of BVMs and the quasi-linearization technique, may be competitive with well known solvers for BVPs.

Improving performance when solving high-order and mixed-order boundary value problems in ODEs

Solving high-order or mixed-order boundary value problems by general purpose software often requires the system to be first converted to a larger equivalent first-order system. The cost of solving such problems is generally O(m3), where m is the dimension of the equivalent first-order system. In this paper, we show how to reduce this cost by exploiting the special structure the “equivalent” first-order system inherits from the original associated mixed-order system. This technique applies to a broad class of boundary value methods. We illustrate the potential benefits by considering in detail a general purpose Runge–Kutta method and a multiple shooting method.