Table of contents for Introduction to numerical methods in differential equations / Mark H. Holmes.


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1   Initial Value Problems .............                             .
1.1  Inttoduction  .    ..    .  . .                     .   ..      1
11     Examples of VP     ...... .......                        2
1.2    Methods Obtiaind from Numerical Differentiation ...     ..
1.2.2  Additional Difference Methods ..... ....... . ..    .   15
1.3   I Methods Obtained from Numerical Quaditurem ............ 18
1.4  Runge- Kutta Methods ..             ..........             .   I22
1.5  Extensions and Ghost Poin lts .   .     .... ..... 24
1.   Conservative Methods . ........     .........                  26
1.6  1  Velocity  Veret  . ......... .                . ..    27
1.62   Symplect  Methods ........ 2
1.7  Next Steps .   ..                                           . 3
Exercises  .....  . . .... .... .....  . .  . . . . ..  ... .  .  ............... .  3
2   Two-Point Boundary Value Problems ...............                 . 4
2.1 1 Introduction ............. . . .. .5
2.1.1  Birds on  a  Wire  .. . .... .... .   ....  ... . ... . . .  . 5
2. 1 2  Chemicial Kinetics . .....            ..     '.. 45
2.2   Derivative Approxmnation Methods .. .  ....   .    . . . ..  . .  416
2 2.1  Matrix  Problem  . . ........................   .  ......  49
2.2.2  Tridihagonai Matrices . .  .  .  . . .................... .  t50
22.23  Mai Problem Revisited      ....     ..                  2
2.2.4  Error  Analysis  ....................................  55
2 .  5  Extension   .  ...  .......              ...           58
2.3  X-Residual Methods  . . . . . ... ... . .   ......  .  ..  . .  62
2.3.1  Basis  Functions  ......................  ............  63
2 .32  Residual ..   . .      . .. . ..... . .      . ........... 66
2.4  Shooting  Methods ........................          . . .  .   69
3   Diffusion Problems ....      .    ........                      8.
1. Introduction ..    ....           . .     ...             3. 3
3.1.1  Heat Equation ....   ............... 83
3.2  Derivative Approximation Methods .. . ..                  88
3.2.1  Implicit, Method .................. . . . .......... . 100
3.2.2  Theta Method ..  ......     ...     ... .......... 102
3.3  Methods Obtained from Numerical Quadrature ............ 105
3.3.1  Crank- Nicolson  Method. .. .....  .......  . ......... 106
3.3.2  L-Stabiity . . .    .....  . ....     ........ 109
3.14  Methods of Lines .. ....................                1:12
3    Collocation  .............................................  11
3.6  Next  Steps  .................. . .............. . .     118
Exercises  . . ... ... .... .... .. ... ... ... ....... .. .. ... .  ....  119
4  Advection Equation . ....................      ...........    127
4.1   troduction  .. ......... .  .............................  127
.111  Method  of Characteristics  T....... .........   ..  27
I1.2  Sohution Properties          ........    ...    .    30
t .1.3 Bomdarly Conditions ..         . . .   .          131l
12   First-Order Methods ........ ..                            132
4.2. 1  Upwind  Scheme  .......  .    .  .   ............... 132
1 2.2  Downwind  Scheme  . ............  .. .... .... . .  2
4.2.3  Numerical Domain of Depenl ete . . ....... ...... 134
1.3  Improvements .......    .   ................ ......... . . 1 39
4.3.1   ax-Wendroff Method ....        .,......          . 140
1 3.2  Monotone  Methods  .... ............ .  .......... 144
.3    Upwindp Revisited .... .......... .............     145
44   Implicit  Methods  . ....  ....................................  1416
SNumerical Wave Propagation ........                    .... .    .155
5  1.1  Solution  Methods  .. .  ..  . . .. ..  .........  .  55
5.1.2  Plane  Wav-e  Solutions  . . . ....................... . 160
5    2  Explicit Method  ...........                        . 164
5.2.1  Diagnostics  .......................             .  67
5. 2.2  Numerical  Experiments  .......................... .169
.3     Numerical Pla 1 ne  avs  ..........................1...  71
5.3  I  Numerical  Group  Velocity  .........................   171
S    Next Steps  . . .  . .          .  ............ ..... .....  176
Exercises  .........  ......  .  .               .   ...... .  . 176
6   Elliptic Problems  .. ..     ... ..      . ....   ....      18
6.1 Introduction . ........................          .......  81
6 1   Solutions  .   .  .    .     .  .  .  ..   ..    .  83
6.1.2  Properties of the Solution  ... ......  ....     . 186
6.2 Finite Difference Approximation ..    . ..... ...   .    187
6.2.1 Building the Matrix .......     ........ .....    190
6.2.2  Positive Definite Matrices ......  . ..... ..    192
6.3  Descent Methods .......    .............                196
6.3.1  Steepest Descent Method  .. 198
6.3.2  Conjugate Gradient Method  . .........       .   199
6,4  Numerical Solution of Laplace's Equation ......... ...  204
6.5  P econditioned Conjugate Gradient Method  ........   .. 207
6 6  Next  Steps  .... . .. ... . ........  .. . .... . ..........  . 214
Exercises  ................................................. . 214



Library of Congress subject headings for this publication: Differential equations Numerical solutions Textbooks