Computational Methods for Geodynamics

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 8
Foreword......Page 13
Preface......Page 15
Acknowledgements......Page 19
1.1 Introduction to scientific computing and computational geodynamics......Page 21
1.2 Mathematical models of geodynamic problems......Page 22
1.3.1 The equation of continuity......Page 23
1.3.2 The equation of motion......Page 24
1.3.3 The heat equation......Page 26
1.3.4 The rheological law......Page 27
1.3.5 Other equations......Page 30
1.3.7 Stream function formulation......Page 31
1.3.8 Poloidal and toroidal decomposition......Page 32
1.4 Boundary and initial conditions......Page 33
1.5 Analytical and numerical solutions......Page 34
1.6 Rationale of numerical modelling......Page 35
1.7 Numerical methods: possibilities and limitations......Page 36
1.8 Components of numerical modelling......Page 37
1.9 Properties of numerical methods......Page 40
1.10 Concluding remarks......Page 42
2.1 Introduction: basic concepts......Page 44
2.2 Convergence, accuracy and stability......Page 49
2.3 Finite difference sweep method......Page 50
2.4 Principle of the maximum......Page 51
2.5.1 Statement of the problem......Page 52
2.5.2 Finite difference discretisation......Page 53
2.5.3 Monotonic finite difference scheme......Page 55
2.5.4 Solution method......Page 56
2.5.5 Verification of the finite difference scheme......Page 60
3.2 Grids and control volumes: structured and unstructured grids......Page 63
3.3 Comparison to finite difference and finite element methods......Page 64
3.4.1 Diffusion......Page 65
3.4.2 Advection......Page 67
3.5.1 Discretisation......Page 69
3.5.2 Solution methods......Page 75
3.5.3 Multigrid......Page 77
3.6.1 Overall solution strategy......Page 80
3.6.3 Extension to spherical geometry......Page 81
4.1 Introduction......Page 83
4.2 Lagrangian versus Eulerian description of motion......Page 84
4.3 Mathematical preliminaries......Page 85
4.4 Weighted residual methods: variational problem......Page 86
4.5 Simple FE problem......Page 89
4.6 The Petrov–Galerkin method for advection-dominated problems......Page 91
4.8 FE discretisation......Page 95
4.9 High-order interpolation functions: cubic splines......Page 96
4.10.1 Two-dimensional problem of gravitational advection......Page 99
4.10.2 Three-dimensional problem of gravitational advection......Page 107
4.11 FE solution refinements......Page 111
Step 3. Post-processing phase......Page 112
5.2.1 Overview......Page 113
5.2.2 Trigonometric......Page 114
5.2.3 Chebyshev polynomials......Page 115
5.2.4 Spherical harmonics......Page 116
5.3.1 Poisson\'s equation......Page 118
5.3.2 Galerkin, Tau and pseudo-spectral methods......Page 119
5.4.1 Constant viscosity, three-dimensional Cartesian geometry......Page 120
5.4.3 Compressibility......Page 125
5.4.4 Self-gravitation and geoid......Page 126
5.4.5 Tectonic plates and laterally varying viscosity......Page 127
6.2 Direct methods......Page 129
6.2.1 Gauss elimination......Page 130
6.2.2 LU-factorisation......Page 131
6.2.3 Cholesky method......Page 133
6.3.1 Jacobi method......Page 134
6.3.2 Gauss–Seidel method......Page 135
6.3.3 Conjugate gradient method......Page 137
6.3.4 Method of distributive iterations......Page 138
6.4 Multigrid methods......Page 139
6.4.1 Two-grid cycle......Page 140
6.4.3 Multigrid cycle......Page 142
6.4.4 Full approximation storage (for non-linear problems or grid refinement)......Page 143
6.4.6 Algebraic multigrid......Page 145
6.5.1 Distributive iterations......Page 146
6.5.2 SIMPLE method......Page 147
6.6 Alternating direction implicit method......Page 148
6.7 Coupled equations solving......Page 150
6.8 Non-linear equation solving......Page 151
6.9 Convergence and iteration errors......Page 152
7.2 Euler method......Page 154
7.3 Runge–Kutta methods......Page 155
7.4.1 The midpoint rule (leap-frog method)......Page 157
7.4.2 The trapezoidal rule......Page 158
7.5 Crank–Nicolson method......Page 159
7.6 Predictor–corrector methods......Page 160
7.7 Method of characteristics......Page 161
7.8 Semi-Lagrangian method......Page 162
7.9 Total variation diminishing methods......Page 164
7.10.2 Particle-in-cell method......Page 166
8.1 Introduction......Page 168
8.2 Data assimilation......Page 171
8.3 Backward advection (BAD) method......Page 172
8.4 Application of the BAD method: restoration of the evolution of salt diapirs......Page 173
8.5 Variational (VAR) method......Page 176
8.6 Application of the VAR method: restoration of mantle plume evolution......Page 182
8.6.1 Forward modelling......Page 183
8.6.3 Performance of the numerical algorithm......Page 185
8.7 Challenges in VAR data assimilation......Page 188
8.7.2 Smoothness of the target temperature......Page 189
8.7.3 Numerical noise......Page 190
8.8 Quasi-reversibility (QRV) method......Page 191
8.8.1 The QRV method for restoration of thermo-convective flow......Page 194
8.8.2 Optimisation problem......Page 195
8.8.3 Numerical algorithm for QRV data assimilation......Page 196
8.9 Application of the QRV method: restoration of mantle plume evolution......Page 197
8.10.1 The Vrancea seismicity and the relic descending slab......Page 200
8.10.2 Temperature model......Page 202
8.10.3 QRV data assimilation......Page 204
8.10.4 What the past tells us......Page 205
8.10.5 Limitations and uncertainties......Page 210
8.11 Comparison of data assimilation methods......Page 212
8.12 Errors in forward and backward modelling......Page 215
9.2 Parallel versus sequential processing......Page 217
9.3 Terminology of parallel processing......Page 219
9.4.1 Shared memory......Page 221
9.5 Domain decomposition......Page 223
9.6 Message passing......Page 227
9.7 Basics of the Message Passing Interface......Page 229
9.8 Cost of parallel processing......Page 233
9.9 Concluding remarks......Page 235
10.1 Introduction and overview......Page 236
10.2.1 Mantle convection......Page 237
10.2.2 Crust and lithosphere dynamics......Page 241
10.3.1 Importance, extended Bounssinesq and anelastic approximations......Page 243
10.3.2 Continuity equation......Page 244
10.3.4 Energy equation......Page 245
10.3.5 Non-dimensionalisation and dissipation number......Page 246
10.4.1 Background......Page 247
10.4.2 Effective expansivity and heat capacity......Page 248
10.4.4 Phase equilibria approach......Page 249
10.5.2 Tracer (marker)-based approaches......Page 251
10.6.1 Non-Newtonian creep and plasticity......Page 255
10.6.2 Viscoelasticity......Page 256
10.7.1 Overview......Page 258
10.7.2 Rigid block approach......Page 276
10.7.3 Rheological approach......Page 277
10.8 Treatment of a free surface and surface processes......Page 280
10.9 Porous flow through a deformable matrix......Page 281
10.10 Geodynamo modelling......Page 283
A1 Introduction......Page 286
A3 Transposed, square, and symmetric matrices. Determinants......Page 287
A4 Upper and lower triangular matrices......Page 288
A6 Matrix addition and scalar multiplication......Page 289
A8 Vector and matrix norm......Page 290
A9 Divergence, gradient, curl, and Laplacian operators......Page 291
B1 Spherical grids......Page 294
B2 Vector operators, strain rates and stress equations in spherical polar coordinates......Page 296
B3 Two-dimensional approximations of spherical geometry......Page 298
Appendix C: Freely available geodynamic modelling codes......Page 300
References......Page 303
Author index......Page 337
Subject index......Page 343
Colour plates\n......Page 259