Introduction to Numerical Methods in Differential Equations

by Holmes, Mark H.

• ISBN13: 9780387308913
• ISBN10: 0387308911
• eBook ISBN(s): 9780387681214
• Format: Hardcover
• Copyright: 2006-10-30
• Publisher: Springer Nature
• More Book Details
• Purchase Benefits

Summary

This is a textbook for upper division undergraduates and beginning graduate students. Its objective is that students learn to derive, test and analyze numerical methods for solving differential equations, and this includes both ordinary and partial differential equations. In this sense the book is constructive rather than theoretical, with the intention that the students learn to solve differential equations numerically and understand the mathematical and computational issues that arise when this is done. An essential component of this is the exercises, which develop both the analytical and computational aspects of the material. The importance of the subject of the book is that most laws of physics involve differential equations, as do the modern theories on financial assets. Moreover many computer animation methods are now based on physics based rules and are heavily invested in differential equations. Consequently numerical methods for differential equations are important for multiple areas. The author currently teaches at Rensselaer Polytechnic Institute and is an expert in his field. He has previously published a book with Springer, Introduction to Perturbation Methods.

Author Biography

Mark H. Holmes is Professor of Mathematical Sciences at Renssaelaer Polytechnic Institute, New York.

Table of Contents

Preface	vii
1 Initial Value Problems	1	(44)
1.1 Introduction	1	(4)
1.1.1 Examples of IVPs	2	(3)
1.2 Methods Obtained from Numerical Differentiation	5	(13)
1.2.1 The Five Steps	5	(10)
1.2.2 Additional Difference Methods	15	(3)
1.3 Methods Obtained from Numerical Quadrature	18	(4)
1.4 Runge–Kutta Methods	22	(2)
1.5 Extensions and Ghost Points	24	(2)
1.6 Conservative Methods	26	(5)
1.6.1 Velocity Verlet	27	(2)
1.6.2 Symplectic Methods	29	(2)
1.7 Next Steps	31	(2)
Exercises	33	(12)
2 Two-Point Boundary Value Problems	45	(38)
2.1 Introduction	45	(1)
2.1.1 Birds on a Wire	45	(1)
2.1.2 Chemical Kinetics	45	(1)
2.2 Derivative Approximation Methods	46	(16)
2.2.1 Matrix Problem	49	(1)
2.2.2 Tridiagonal Matrices	50	(2)
2.2.3 Matrix Problem Revisited	52	(3)
2.2.4 Error Analysis	55	(3)
2.2.5 Extensions	58	(4)
2.3 Residual Methods	62	(7)
2.3.1 Basis Functions	63	(3)
2.3.2 Residual	66	(3)
2.4 Shooting Methods	69	(3)
2.5 Next Steps	72	(2)
Exercises	74	(9)
3 Diffusion Problems	83	(44)
3.1 Introduction	83	(5)
3.1.1 Heat Equation	83	(5)
3.2 Derivative Approximation Methods	88	(17)
3.2.1 Implicit Method	100	(2)
3.2.2 Theta Method	102	(3)
3.3 Methods Obtained from Numerical Quadrature	105	(7)
3.3.1 Crank–Nicolson Method	106	(3)
3.3.2 L-Stability	109	(3)
3.4 Methods of Lines	112	(1)
3.5 Collocation	113	(5)
3.6 Next Steps	118	(1)
Exercises	119	(8)
4 Advection Equation	127	(28)
4.1 Introduction	127	(5)
4.1.1 Method of Characteristics	127	(3)
4.1.2 Solution Properties	130	(1)
4.1.3 Boundary Conditions	131	(1)
4.2 First-Order Methods	132	(7)
4.2.1 Upwind Scheme	132	(1)
4.2.2 Downwind Scheme	132	(2)
4.2.3 Numerical Domain of Dependence	134	(4)
4.2.4 Stability	138	(1)
4.3 Improvements	139	(7)
4.3.1 Lax–Wendroff Method	140	(4)
4.3.2 Monotone Methods	144	(1)
4.3.3 Upwind Revisited	145	(1)
4.4 Implicit Methods	146	(2)
Exercises	148	(7)
5 Numerical Wave Propagation	155	(26)
5.1 Introduction	155	(9)
5.1.1 Solution Methods	155	(5)
5.1.2 Plane Wave Solutions	160	(4)
5.2 Explicit Method	164	(7)
5.2.1 Diagnostics	167	(2)
5.2.2 Numerical Experiments	169	(2)
5.3 Numerical Plane Waves	171	(5)
5.3.1 Numerical Group Velocity	174	(2)
5.4 Next Steps	176	(1)
Exercises	176	(5)
6 Elliptic Problems	181	(42)
6.1 Introduction	181	(6)
6.1.1 Solutions	183	(3)
6.1.2 Properties of the Solution	186	(1)
6.2 Finite Difference Approximation	187	(9)
6.2.1 Building the Matrix	190	(2)
6.2.2 Positive Definite Matrices	192	(4)
6.3 Descent Methods	196	(8)
6.3.1 Steepest Descent Method	198	(1)
6.3.2 Conjugate Gradient Method	199	(5)
6.4 Numerical Solution of Laplace's Equation	204	(3)
6.5 Preconditioned Conjugate Gradient Method	207	(5)
6.6 Next Steps	212	(2)
Exercises	214	(9)
A Appendix	223	(8)
A.1 Order Symbols	223	(1)
A.2 Taylor's Theorem	224	(1)
A.3 Round-Off Error	225	(2)
A.3.1 Function Evaluation	225	(1)
A.3.2 Numerical Differentiation	226	(1)
A.4 Floating-Point Numbers	227	(4)
References	231	(4)
Index	235