 | Ismail-Zadeh A. Computational methods for geodynamics / A.Ismail-Zadeh, P.J.Tackley. - Cambridge: Cambridge University Press, 2010. - xv, 313 p., [16] p. of col. plates: ill. - Ref.: p.267-300. - Auth. ind.: p.301-306. - Sub. ind.: p.307-313. - ISBN 978-0-521-86767-2
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Foreword by Gerald Schubert .................................... xi
Preface ...................................................... xiii
Acknowledgements ............................................. xvii
1 Basic concepts of computational geodynamics .................. 1
1.1 Introduction to scientific computing and computational
geodynamics ............................................. 1
1.2 Mathematical models of geodynamic problems .............. 2
1.3 Governing equations ..................................... 3
1.4 Boundary and initial conditions ........................ 13
1.5 Analytical and numerical solutions ..................... 14
1.6 Rationale of numerical modelling ....................... 15
1.7 Numerical methods: possibilities and limitations ....... 16
1.8 Components of numerical modelling ...................... 17
1.9 Properties of numerical methods ........................ 20
1.10 Concluding remarks ..................................... 22
2 Finite difference method .................................... 24
2.1 Introduction: basic concepts ........................... 24
2.2 Convergence, accuracy and stability .................... 29
2.3 Finite difference sweep method ......................... 30
2.4 Principle of the maximum ............................... 31
2.5 Application of a finite difference method to a two-
dimensional heat equation .............................. 32
3 Finite volume method ........................................ 43
3.1 Introduction ........................................... 43
3.2 Grids and control volumes: structured and
unstructured grids ..................................... 43
3.3 Comparison to finite difference and finite element
methods ................................................ 44
3.4 Treatment of advection-diffusion problems .............. 45
3.5 Treatment of momentum-continuity equations ............. 49
3.6 Modelling convection and model extensions .............. 60
4 Finite element method ....................................... 63
4.1 Introduction ........................................... 63
4.2 Lagrangian versus Eulerian description of motion ....... 64
4.3 Mathematical preliminaries ............................. 65
4.4 Weighted residual methods: variational problem ......... 66
4.5 Simple FE problem ...................................... 69
4.6 The Petrov-Galerkin method for advection-dominated
problems ............................................... 71
4.7 Penalty-function formulation of Stokes flow ............ 75
4.8 FE discretisation ...................................... 75
4.9 High-order interpolation functions: cubic splines ...... 76
4.10 Two-and three-dimensional FE problems .................. 79
4.11 FE solution refinements ................................ 91
4.12 Concluding remarks ..................................... 92
5 Spectral methods ............................................ 93
5.1 Introduction ........................................... 93
5.2 Basis functions and transforms ......................... 93
5.3 Solution methods ....................................... 98
5.4 Modelling mantle convection ........................... 100
6 Numerical methods for solving linear algebraic equations ... 109
6.1 Introduction .......................................... 109
6.2 Direct methods ........................................ 109
6.3 Iterative methods ..................................... 114
6.4 Multigrid methods ..................................... 119
6.5 Iterative methods for the Stokes equations ............ 126
6.6 Alternating direction implicit method ................. 128
6.7 Coupled equations solving ............................. 130
6.8 Non-linear equation solving ........................... 131
6.9 Convergence and iteration errors ...................... 132
7 Numerical methods for solving ordinary and partial
differential equations ..................................... 134
7.1 Introduction .......................................... 134
7.2 Euler method .......................................... 134
7.3 Runge-Kutta methods ................................... 135
7.4 Multi-step methods .................................... 137
7.5 Crank-Nicolson method ................................. 139
7.6 Predictor-corrector methods ........................... 140
7.7 Method of characteristics ............................. 141
7.8 Semi-Lagrangian method ................................ 142
7.9 Total variation diminishing methods ................... 144
7.10 Lagrangian methods .................................... 146
8 Data assimilation methods ................................. 148
8.1 Introduction .......................................... 148
8.2 Data assimilation ..................................... 151
8.3 Backward advection (BAD) method ....................... 152
8.4 Application of the BAD method: restoration of the
evolution of salt diapirs ............................. 153
8.5 Variational (VAR) method .............................. 156
8.6 Application of the VAR method: restoration of mantle
plume evolution ....................................... 162
8.7 Challenges in VAR data assimilation ................... 168
8.8 Quasi-reversibility (QRV) method ...................... 171
8.9 Application of the QRV method: restoration of mantle
plume evolution ....................................... 177
8.10 Application of the QRV method: restoration of
descending lithosphere evolution ...................... 180
8.11 Comparison of data assimilation methods ............... 192
8.12 Errors in forward and backward modelling .............. 195
9 Parallel computing ......................................... 197
9.1 Introduction .......................................... 197
9.2 Parallel versus sequential processing ................. 197
9.3 Terminology of parallel processing .................... 199
9.4 Shared and distributed memory ......................... 201
9.5 Domain decomposition .................................. 203
9.6 Message passing ....................................... 207
9.7 Basics of the Message Passing Interface ............... 209
9.8 Cost of parallel processing ........................... 213
9.9 Concluding remarks .................................... 215
10 Modelling of geodynamic problems ........................... 216
10.1 Introduction and overview ............................. 216
10.2 Numerical methods used ................................ 217
10.3 Compressible flow ..................................... 223
10.4 Phase transitions ..................................... 227
10.5 Compositional variations .............................. 231
10.6 Complex rheologies .................................... 235
10.7 Continents and lithospheric plates in mantle
convection models ..................................... 238
10.8 Treatment of a free surface and surface processes ..... 244
10.9 Porous flow through a deformable matrix ............... 245
10.10Geodynamo modelling ................................... 247
Appendix A Definitions and relations from vector and matrix
algebra ............................................ 250
Appendix В Spherical coordinates .............................. 258
Appendix С Freely available geodynamic modelling codes ........ 264
References .................................................... 267
Author index .................................................. 301
Subject index ................................................. 307
Colour plate section between pages 238 and 239.
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