Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Samuel N Jator,Kindyl L King

doi:10.1155/2018/3750274

Abstract

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

Highlights

General second-order differential equations frequently arise in several areas of science and engineering, such as celestial mechanics, quantum mechanics, control theory, circuit theory, astrophysics, and biological sciences
In this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering
The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and implemented in a block-by-block manner

Summary

Introduction

General second-order differential equations frequently arise in several areas of science and engineering, such as celestial mechanics, quantum mechanics, control theory, circuit theory, astrophysics, and biological sciences. Equation (2) is conventionally solved by converting it into an equivalent first-order system of double dimension and solved using standard methods that are available in the literature for solving systems of firstorder IVPs (see Lambert [16], Hairer et al [17], and Brugnano et al [18]) These methods are implemented in a step-by-step fashion in which, on the partition SN, an approximation yn is obtained at tn only after an approximation at tn−1 has been computed, where for some constant step-size h = (tN − t0 )/N and integer N > 0, SN fl {t0 < t1 < .

Derivation

Properties of the Method

Numerical Examples

Application to a Stiff Problem

Conclusion