I Basic Concepts and Discretizations ........................... 1
1 Advection-Diffusion-Reaction Equations ....................... 1
1.1 Nonlinear Reaction Problems from Chemistry .............. 3
1.2 Model Equations for Advection-Diffusion ................. 9
1.3 Multi-dimensional Problems ............................. 14
1.4 Examples of Applications ............................... 18
2 Basic Discretizations for ODEs .............................. 23
2.1 Initial Value Problems and Euler's Method .............. 23
2.2 Norms and Matrices ..................................... 27
2.3 Perturbations on ODE Systems ........................... 30
2.4 The θ-Method and Stiff Problems ........................ 35
2.5 Stability of the θ-Method .............................. 37
2.6 Consistency and Convergence of the θ-Method ............ 42
2.7 Nonlinear Results for the θ-Method ..................... 44
2.8 Concluding Remarks ..................................... 46
3 Basic Spatial Discretizations ............................... 48
3.1 Discrete Fourier Decompositions ........................ 49
3.2 The Advection Equation ................................. 52
3.3 The Diffusion Equation ................................. 62
3.4 The Advection-Diffusion Equation ....................... 66
4 Convergence of Spatial Discretizations ...................... 71
4.1 Stability, Consistency and Convergence ................. 71
4.2 Advection-Diffusion with Constant Coefficients ......... 74
4.3 Advection with Variable Coefficients ................... 77
4.4 Diffusion with Variable Coefficients ................... 81
4.5 Variable Coefficients and Higher-Order Schemes ......... 83
5 Boundary Conditions and Spatial Accuracy .................... 84
5.1 Refined Global Error Estimates ......................... 85
5.2 Outflow with Central Advection Discretization .......... 86
5.3 Boundary Conditions with the Heat Equation ............. 88
5.4 Boundary Conditions and Higher-Order Schemes ........... 92
6 Time Stepping for PDEs ...................................... 94
6.1 The Method of Lines and Direct Discretizations ......... 94
6.2 Stability, Consistency and Convergence ................. 99
6.3 Stability for MOL - Stability Regions ................. 103
6.4 Von Neumann Stability Analysis ........................ 111
7 Monotonicity Properties .................................... 116
7.1 Positivity and Maximum Principle ...................... 116
7.2 Positive Semi-discrete Systems ........................ 118
7.3 Positive Time Stepping Methods ........................ 121
7.4 Numerical Illustrations ............................... 124
8 Numerical Test Examples .................................... 127
8.1 The Nonlinear Schrödinger Equation .................... 128
8.2 The Angiogenesis Model ................................ 134
II Time Integration Methods .................................. 139
1 Runge-Kutta Methods ........................................ 139
1.1 The Order Conditions .................................. 140
1.2 Examples .............................................. 142
1.3 The Stability Function ................................ 144
1.4 Step Size Restrictions for Advection-Diffusion ........ 149
1.5 Rosenbrock Methods .................................... 151
2 Convergence of Runge-Kutta Methods ......................... 155
2.1 Order Reduction ....................................... 155
2.2 Local Error Analysis .................................. 158
2.3 Global Error Analysis ................................. 161
2.4 Concluding Notes ...................................... 166
3 Linear Multistep Methods ................................... 170
3.1 The Order Conditions .................................. 171
3.2 Examples .............................................. 173
3.3 Stability Analysis .................................... 174
3.4 Step Size Restrictions for Advection-Diffusion ........ 181
3.5 Convergence Analysis .................................. 182
4 Monotone ODE Methods ....................................... 185
4.1 Linear Positivity for One-Step Methods ................ 185
4.2 Nonlinear Positivity for One-Step Methods ............. 189
4.3 Positivity for Multistep Methods ...................... 192
4.4 Related Monotonicity Results .......................... 196
5 Variable Step Size Control ................................. 197
5.1 Step Size Selection ................................... 197
5.2 An Explicit Runge-Kutta Example ....................... 200
5.3 An Implicit Multistep Example ......................... 203
5.4 General Purpose ODE Codes ............................. 205
6 Numerical Examples ......................................... 206
6.1 A Model for Antibodies in Tumorous Tissue ............. 206
6.2 The Nonlinear Schrödinger Equation .................... 209
III AdvectionJDiffusion Discretizations ...................... 215
1 Non-oscillatory MOL Advection Discretizations .............. 215
1.1 Spatial Discretization for Linear Advection ........... 215
1.2 Numerical Examples .................................... 222
1.3 Positivity and the TVD Property ....................... 226
1.4 Nonlinear Scalar Conservation Laws .................... 233
2 Direct Space-Time Advection Discretizations ................ 239
2.1 Optimal-Order DST Schemes ............................. 239
2.2 A Non-oscillatory Third-Order DST Scheme .............. 243
2.3 Explicit Schemes with Unconditional Stability ......... 248
3 Implicit Spatial Discretizations ........................... 250
3.1 Order Conditions ...................................... 251
3.2 Examples .............................................. 253
3.3 Stability and Convergence ............................. 258
3.4 Monotonicity .......................................... 261
3.5 Time Integration Aspects .............................. 263
4 Non-uniform Grids - Finite Volumes (ID) .................... 264
4.1 Vertex Centered Schemes ............................... 265
4.2 Cell Centered Schemes ................................. 272
4.3 Numerical Illustrations ............................... 278
4.4 Higher-Order Methods and Limiting ..................... 281
5 Non-uniform Grids - Finite Elements (ID) ................... 283
5.1 The Basic Galerkin Method ............................. 283
5.2 Standard Galerkin Error Estimates ..................... 288
5.3 Upwinding ............................................. 291
6 Multi-dimensional Aspects .................................. 292
6.1 Cartesian Grid Discretizations ........................ 293
6.2 Diffusion on Cartesian Grids .......................... 295
6.3 Advection on Cartesian Grids .......................... 303
6.4 Transformed Cartesian Grids ........................... 308
6.5 Unstructured Grids .................................... 311
7 Notes on Moving Grids and Grid Refinement .................. 316
7.1 Dynamic Regridding .................................... 316
7.2 Static Regridding ..................................... 321
IV Splitting Methods ......................................... 325
1 Operator Splitting ......................................... 325
1.1 First-Order Splitting ................................. 325
1.2 Second-Order Symmetrical Splitting .................... 329
1.3 Higher-Order Splittings ............................... 330
1.4 Abstract Initial Value Problems ....................... 331
1.5 Advection-Diffusion-Reaction Splittings ............... 335
1.6 Dimension Splitting ................................... 337
1.7 Boundary Values and Stiff Terms ....................... 344
2 LOD Methods ................................................ 348
2.1 The LOD-Backward Euler Method ......................... 348
2.2 LOD Crank-Nicolson Methods ............................ 351
2.3 The Trapezoidal Splitting Method ...................... 359
2.4 Boundary Correction Techniques ........................ 365
2.5 Numerical Comparisons ................................. 367
3 ADI Methods ................................................ 369
3.1 The Peaceman-Rachford Method .......................... 369
3.2 The Douglas Method .................................... 373
4 IMEX Methods ............................................... 383
4.1 The IMEX-θ Method ..................................... 383
4.2 IMEX Multistep Methods ................................ 386
4.3 Notes on IMEX Runge-Kutta Methods ..................... 391
4.4 Concluding Remarks and Tests .......................... 393
5 Rosenbrock AMF Methods ..................................... 398
5.1 One-Stage Methods of Order One and Two ................ 398
5.2 Two-Stage Methods of Order Two and Three .............. 400
5.3 A Three-Stage Method of Order Two ..................... 403
5.4 Concluding Remarks and Tests .......................... 405
6 Numerical Examples ......................................... 409
6.1 Two Chemo-taxis Problems from Biology ................. 409
6.2 The Numerical Methods ................................. 411
6.3 Numerical Experiments ................................. 412
V Stabilized Explicit Runge-Kutta Methods .................... 419
1 The RKC Family ............................................. 420
1.1 Stability Polynomials ................................. 420
1.2 Integration Formulas .................................. 426
1.3 Internal Stability and Full Convergence Properties .... 430
2 The ROCK Family ............................................ 433
2.1 Stability Polynomials ................................. 433
2.2 Integration Formulas .................................. 435
2.3 Internal Stability and Convergence .................... 436
3 Numerical Examples ......................................... 438
3.1 A Combustion Model .................................... 439
3.2 A Radiation-Diffusion Model ........................... 441
Bibliography ............................................... 447
Index ......................................................... 465
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