Introduction to Numerical Methods in Differential Equations

By: Mark H. Holmes

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Hardcover | 24 October 2006

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252 Pages

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The title gives a reasonable ?rst-order approximation to what this book is about. To explain why, let's start with the expression "di?erential equations." These are essential in science and engineering, because the laws of nature t- ically result in equations relating spatial and temporal changes in one or more variables.Todevelopanunderstandingofwhatisinvolvedin?ndingsolutions, the book begins with problems involving derivatives for only one independent variable, and these give rise to ordinary di?erential equations. Speci?cally, the ?rst chapter considers initial value problems (time derivatives), and the second concentrates on boundary value problems (space derivatives). In the succeeding four chapters problems involving both time and space derivatives, partial di?erential equations, are investigated. This brings us to the next expression in the title: "numerical methods." This is a book about how to transform differential equations into problems that can be solved using a computer.The fact is that computers are only able to solve discrete problems and generally do this using ?nite-precision arithmetic. What this means is that in deriving and then using a numerical algorithmthecorrectnessofthediscreteapproximationmustbeconsidered,as must the consequences of round-o? error in using ?oating-point arithmetic to calculatetheanswer.Oneoftheinterestingaspectsofthesubjectisthatwhat appears to be an obviously correct numerical method can result in complete failure. Consequently, although the book concentrates on the derivation and use of numerical methods, the theoretical underpinnings are also presented andusedinthedevelopment.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Numerical Analysis

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Published: 29th November 2010

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Preface	p. vii
Initial Value Problems	p. 1
Introduction	p. 1
Examples of IVPs	p. 2
Methods Obtained from Numerical Differentiation	p. 5
The Five Steps	p. 5
Additional Difference Methods	p. 15
Methods Obtained from Numerical Quadrature	p. 18
Runge-Kutta Methods	p. 22
Extensions and Ghost Points	p. 24
Conservative Methods	p. 26
Velocity Verlet	p. 27
Symplectic Methods	p. 29
Next Steps	p. 31
Exercises	p. 33
Two-Point Boundary Value Problems	p. 45
Introduction	p. 45
Birds on a Wire	p. 45
Chemical Kinetics	p. 45
Derivative Approximation Methods	p. 46
Matrix Problem	p. 49
Tridiagonal Matrices	p. 50
Matrix Problem Revisited	p. 52
Error Analysis	p. 55
Extensions	p. 58
Residual Methods	p. 62
Basis Functions	p. 63
Residual	p. 66
Shooting Methods	p. 69
Next Steps	p. 72
Exercises	p. 74
Diffusion Problems	p. 83
Introduction	p. 83
Heat Equation	p. 83
Derivative Approximation Methods	p. 88
Implicit Method	p. 100
Theta Method	p. 102
Methods Obtained from Numerical Quadrature	p. 105
Crank-Nicolson Method	p. 106
L-Stability	p. 109
Methods of Lines	p. 112
Collocation	p. 113
Next Steps	p. 118
Exercises	p. 119
Advection Equation	p. 127
Introduction	p. 127
Method of Characteristics	p. 127
Solution Properties	p. 130
Boundary Conditions	p. 131
First-Order Methods	p. 132
Upwind Scheme	p. 132
Downwind Scheme	p. 132
Numerical Domain of Dependence	p. 134
Stability	p. 138
Improvements	p. 139
Lax-Wendroff Method	p. 140
Monotone Methods	p. 144
Upwind Revisited	p. 145
Implicit Methods	p. 146
Exercises	p. 148
Numerical Wave Propagation	p. 155
Introduction	p. 155
Solution Methods	p. 155
Plane Wave Solutions	p. 160
Explicit Method	p. 164
Diagnostics	p. 167
Numerical Experiments	p. 169
Numerical Plane Waves	p. 171
Numerical Group Velocity	p. 174
Next Steps	p. 176
Exercises	p. 176
Elliptic Problems	p. 181
Introduction	p. 181
Solutions	p. 183
Properties of the Solution	p. 186
Finite Difference Approximation	p. 187
Building the Matrix	p. 190
Positive Definite Matrices	p. 192
Descent Methods	p. 196
Steepest Descent Method	p. 198
Conjugate Gradient Method	p. 199
Numerical Solution of Laplace's Equation	p. 204
Preconditioned Conjugate Gradient Method	p. 207
Next Steps	p. 212
Exercises	p. 214
Appendix	p. 223
Order Symbols	p. 223
Taylor's Theorem	p. 224
Round-Off Error	p. 225
Function Evaluation	p. 225
Numerical Differentiation	p. 226
Floating-Point Numbers	p. 227
References	p. 231
Index	p. 235
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Introduction to Numerical Methods in Differential Equations

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Hardcover

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