P-stable hybrid linear multistep methods (HLMMs) have been an interesting focus for the numerical solution of second order initial value problems (IVPs) in ordinary di_erential equations (ODEs), because of their high order of accuracy. In this thesis, we present a new class of P-stable HLMMs with order p = 2 and p = 4 respectively for the numerical solution of second order systems. The hybrid schemes which are obtained via Pade 0 approximation approach have minimum Phase-lag error. Numerical experiments are carried out to show the accuracy of the proposed schemes. Nevertheless, the desire in this work is on high order P-stable schemes (p > 4). We give a proposition with proof, stating the limitation of the approach in search for higher order P-stable formulas. Key words: P-stability, Phase-lag error (PLE) constant, Hybrids, order, Interval of periodicity, Pade 0 approximation, Principal local truncation error (PLTE).
Table of Contents
1 INTRODUCTION
1.1 Overview
1.2 Statement of the Problem
1.3 Justification of the Study
1.4 Motivation of the Study
1.5 Aim and Objectives of the Mathematical Research
1.6 Methodology
1.7 Significance of the Study
1.8 Scope of the Thesis
2 PRELIMINARIES
2.1 Introduction
2.2 Linear Multi Step Methods (LMMs)
2.3 Hybrid Linear Multistep Method
2.3.1 One-Leg Linear Multistep Method
2.3.2 One-Leg Hybrid Linear Multistep Method
2.4 Super-Implicit Hybrid Linear Multistep Method
2.5 Existence and Uniqueness Theorem
2.5.1 Theorem
2.6 Basic Definitions
2.6.1 Order, Consistency and Zero-Stability
2.6.2 Definition
2.6.3 Definition
2.6.4 Definition
2.6.5 Definition
2.7 Stability Analysis
2.7.1 Definition
2.7.2 Definition
2.7.3 Definition
2.7.4 Definition
2.7.5 Theorem
2.7.6 Theorem
2.7.7 Stability Polynomial
2.8 Pade Approximation
2.9 Phase-Lag Analysis
2.9.1 Definition
2.9.2 Definition
3 LITERATURE REVIEW
3.1 Introduction
3.2 The Fourth Order P-stable Method of Hairer (1979)
3.3 The Hybrid Schemes of Fatunla (1983-1995)
3.4 The Hybrid Linear Multistep Formulas of Fatunla et al (2007)
4 PROPOSED P-STABLE HYBRID LINEAR MULTISTEP METHODS AND THEIR NUMERICAL IMPLEMENTATION
4.1 Development of the Proposed P-stable Formulas
4.1.1 Derivation Of 2nd Order P-stable Hybrid Scheme
4.1.2 Construction of Fourth Order P-stable Formulas
4.1.3 Construction of Symmetric P-stable Hybrid Linear Multistep Methods
4.1.4 Proposition
4.1.5 Corollary
4.1.6 Corollary
4.2 Phase-lag and Stability Analysis of the new Schemes
4.2.1 Consistency
4.2.2 Zero-Stability
4.2.3 Convergence
4.2.4 Interval of Periodicity
4.2.5 Phase-Lag Properties
4.3 Numerical Implementation and Experiments
4.4 Implementation Issues
4.4.1 Functional Iteration Method
4.4.2 Newton-Raphson Iteration Method
4.4.3 Example
4.4.4 Example
4.4.5 Example
4.5 Discussion of Numerical Results
5 SUMMARY
5.1 Conclusion
5.2 Findings
5.3 Contribution to Knowledge
5.4 Future Work
Research Objectives and Core Topics
The primary objective of this thesis is to construct a new family of P-stable Hybrid Linear Multistep Methods (HLMMs) that exhibit minimal phase-lag error for the numerical solution of periodic second-order initial value problems, while addressing the limitations of existing higher-order P-stable formulas.
- Development of new P-stable HLMMs with orders p=2 and p=4.
- Implementation of Pade approximation for the derivation of numerical schemes.
- Comprehensive stability and phase-lag analysis to ensure numerical accuracy.
- Numerical experimentation on oscillatory and orbital problems to validate the schemes.
Excerpt from the Book
3.2 The Fourth Order P-stable Method of Hairer (1979)
In the first instance, we consider the P-stable two step HLMMs describe by
k Σ j=0 αjyn+j = h^2 k Σ j=0 βjf (tn+j, yn+j) + h^2βf (t~n, y~n) (3.2.1)
y~n = k Σ j=0 γjyn+j + h^2 k Σ j=0 δjf (tn+j, yn+j) (3.2.2)
where h is the step size and
tj = t0 + jh, t~n = tn + ( k Σ j=0 γj j ) h. (3.2.3)
when applied to the scalar test (2.7.1) yields a difference equation with the following characteristics polynomial
k Σ j=0 (αj - z^2 (βj + βγj) - z^4βδj) R^j = 0 (3.2.4)
Summary of Chapters
1 INTRODUCTION: Covers the background, problem statement, research objectives, and the scope of the thesis regarding numerical schemes for second-order IVPs.
2 PRELIMINARIES: Introduces foundational mathematical concepts, including linear multistep methods, hybrid methods, stability theory, and Pade approximation.
3 LITERATURE REVIEW: Reviews existing P-stable methods, with a focus on work by Hairer (1979) and Fatunla (1983-1995), evaluating their stability and phase-lag properties.
4 PROPOSED P-STABLE HYBRID LINEAR MULTISTEP METHODS AND THEIR NUMERICAL IMPLEMENTATION: Details the derivation of new P-stable formulas, their stability analysis, and implementation strategies using functional and Newton-Raphson iterations.
5 SUMMARY: Provides concluding remarks on the research, highlights significant findings, discusses contributions to the field, and suggests future research directions.
Keywords
P-stability, Phase-lag error, Hybrid Linear Multistep Methods, Order of accuracy, Interval of periodicity, Pade approximation, Principal local truncation error, Second order IVPs, Numerical implementation, Oscillatory problems, Stability analysis, Convergence, Functional iteration, Newton-Raphson method.
Frequently Asked Questions
What is the core focus of this research?
The research focuses on the development of a new class of P-stable Hybrid Linear Multistep Methods (HLMMs) to provide efficient and accurate numerical solutions for second-order initial value problems, specifically those characterized by oscillatory behavior.
What are the primary thematic areas of this work?
The study spans numerical analysis, specifically the derivation of P-stable formulas, stability and phase-lag analysis, and the practical implementation of these methods on complex physical problems such as orbital and undamped Duffing systems.
What is the main research objective?
The objective is to derive a family of HLMMs with maximum phase-lag order or minimum phase-lag error using the Pade approximation approach, while also investigating the limitations of searching for higher-order P-stable formulas.
Which scientific methodology is utilized?
The research relies on the Pade approximation approach as a powerful tool to guarantee a high degree of accuracy and unconditional P-stability. The methods are applied to scalar test problems and tested against existing literature on nearly periodic IVPs.
What is addressed in the main body of the work?
The main body involves the construction of 2nd and 4th order P-stable hybrid schemes, rigorous stability and phase-lag analysis, and numerical experimentation using Matlab to compare the performance of proposed methods against existing schemes.
Which keywords define this thesis?
Key terms include P-stability, Phase-lag error, Hybrid Linear Multistep Methods (HLMMs), Pade approximation, and interval of periodicity.
How is the P-stability of the proposed methods ensured?
P-stability is ensured by utilizing free parameters in the derivation process and rigorously analyzing the stability polynomials and periodicity intervals, which are validated through graphical plots demonstrating indefinite progression along the positive real axis.
What is the significance of the "Phase-lag" in this study?
Phase-lag is a critical metric for evaluating the accuracy of numerical schemes when solving oscillatory problems. This study seeks to develop methods that possess minimal phase-lag error to prevent accumulation of errors in periodic solutions.
- Arbeit zitieren
- Isaac Felix (Autor:in), 2018, A Class of P-Stable Hybrid Linear Multistep Methods with Minimal Phase-Lag Error for Second Order Initial Value Problems, München, GRIN Verlag, https://www.grin.com/document/442178
