Numerical Methods for Evolutionary Differential Equations Computational Science Engineering

Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applica¬tions such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from numerous application areas. Audience: This textbook is suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Gradute students at the beginning or advanced level (depending on the discipline) and researchers in a variety of fields in science and engineering will find this book useful. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method. Contents: Preface; 1 Introduction; 2 Methods and Concepts for ODEs; 3 Finite Difference and Finite Volume Methods; 4 Stability for Constant Coefficient Problems; 5 Variable Coefficient and Nonlinear Problems; 6 Hamiltonian Systems and Long Time Integration; 7 Dispersion and Dissipation; 8 More on Handling Boundary Conditions; 9 Several Space Variables and Splitting Methods; 10 Discontinuities and Almost Discontinuities; 11 Additional Topics; Bibliography; Index. Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applica¬tions such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from numerous application areas. **Audience:** This textbook is suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Gradute students at the beginning or advanced level (depending on the discipline) and researchers in a variety of fields in science and engineering will find this book useful. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method. **Contents:** Preface; 1 Introduction; 2 Methods and Concepts for ODEs; 3 Finite Difference and Finite Volume Methods; 4 Stability for Constant Coefficient Problems; 5 Variable Coefficient and Nonlinear Problems; 6 Hamiltonian Systems and Long Time Integration; 7 Dispersion and Dissipation; 8 More on Handling Boundary Conditions; 9 Several Space Variables and Splitting Methods; 10 Discontinuities and Almost Discontinuities; 11 Additional Topics; Bibliography; Index Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This text develops, analyses, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified view. The author emphasises finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and non-smooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is also included. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues, and demonstrating methods through examples and case studies from numerous application areas. This book can be used for self-study or by any class preparing for the AP Biology exam. The book is the product of a massive effort to provide six complete practice tests based on the official AP Examinations administered by the College Board. Each practice test is prepared by test experts and contains every type of question that can be expected on the actual exam.

A complete answer key follows each practice test, along with detailed explanations for every answer. Also included is a comprehensive review course covering all major topics tested on the exam. By studying the review and completing all six practice tests, students can discover their strengths and weaknesses and become well-prepared for the actual AP Examination in Biology.

Suitable for researchers and graduate students, this text develops, analyses, and applies numerical methods for evolutionary, or time-dependent, differential problems