Cambridge University Press
Computational Methods for Geodynamics describes all the numerical methods typically used to solve problems related to the dynamics of the Earth and other terrestrial planets – including lithospheric deformation, mantle convection and the geodynamo. It starts with a discussion of the fundamental principles of mathematical and numerical modeling, which is then followed by chapters on finite difference, finite volume, finite element, and spectral methods; methods for solving large systems of linear algebraic equations and ordinary differential equations; data assimilation methods in geodynamics; and the basic concepts of parallel computing. The final chapter presents a detailed discussion of specific geodynamic applications in order to highlight key differences between methods and demonstrate their respective limitations. Readers learn when and how to use a particular method in order to produce the most accurate results. This combination of textbook and reference handbook brings together material previously only available in specialist journals and mathematical reference volumes, and presents it in an accessible manner assuming only a basic familiarity with geodynamic theory and calculus. It is an essential text for advanced courses on numerical and computational modeling in geodynamics and geophysics, and an invaluable resource for researchers looking to master cutting-edge techniques.
“An outstanding synthesis of contemporary
issues in geodynamics with a rigorous but highly accessible treatment of modern
methods in numerical modeling. I have no doubt that this book will be an
invaluable resource for students and researchers entering the field of
computational geophysics for years to come.”- Professor David Bercovici, Yale University
“This is the most current and complete book on computational geodynamics. I would recommend this book to every aspiring student or researcher interested in computations.” - Professor David Yuen, University of Minnesota
CONTENT
Foreword by Professor
Gerald Schubert, University of California
Preface
Acknowledgements
1.
Basic concepts of computational geodynamics
1.1. Introduction to scientific computing and computational
geodynamics
1.2. Mathematical models of geodynamic problems. Direct and
inverse problems
1.3. Governing equations
1.4. Boundary and initial conditions
1.5. Analytical and numerical solutions
1.6. Rationale of numerical modelling
1.7. Numerical methods: Possibilities and limitations
1.8. Components of numerical modelling
1.9. Properties of numerical methods
1.10. Concluding remarks
2.
Finite difference methods
2.1. Introduction.
Basic concepts
2.2. Convergence,
accuracy and stability
2.3. Finite
difference sweep method
2.4. Principle
of the maximum
2.5.
Application
of a finite difference method to two-dimensional heat equation
2.5.1
Statement
of the problem
2.5.2
Finite
difference discretization
2.5.3
Monotonic
finite difference scheme
2.5.4
Solution
method
2.5.5
Verification
of the finite difference scheme
3. Finite volume method
3.1. Introduction
3.2 Grids and control volumes: structured and unstructured grids
3.3 Comparison
to finite difference and finite element methods
3.4 Treatment
of advection-diffusion problems
3.4.1 Diffusion
3.4.2 Advection
3.5
Treatment
of momentum-continuity equations
3.5.1 Discretisation
3.5.2 Solution methods
3.5.3 Multigrid
3.6 Modelling
convection and extensions
3.6.1 Overall solution strategy
3.6.2 Extension to compressible equations
3.6.3 Extension to spherical geometry
4.
Finite element methods
4.1. Introduction
4.2. Lagrangian
versus Eulerian description of motion
4.3. Mathematical
preliminaries
4.4. Weighted
residual methods. Variational problem
4.5. Simple
finite element problem
4.6. The
Petrov-Galerkin method for convection-dominated problems
4.7. Penalty-function
formulation of Stokes flow
4.8. FE
discretization
4.9. High
order interpolation functions. Cubic splines
4.10. Two- and three-dimensional
finite element problems
4.10.1. Two-dimensional problem of gravitational
advection
4.10.2. Three-dimensional problem of gravitational
advection
4.11. FE solution refinements
4.12. Concluding remarks.
5. Spectral methods
5.1. Introduction
5.2.
Basic
functions and transforms
5.2.1. Overview
5.2.2.
Trigonometric
5.2.3.
Chebyshev
polynomials
5.2.4.
Spherical
harmonics
5.3. Solution
method
5.3.1. Poisson's equation
5.3.2. Galerkin, Tau and pseudo-spectral methods
5.4. Mantle
convection
5.4.1.
Constant
viscosity, three-dimensional Cartesian geometry
5.4.2.
Constant viscosity, spherical
geometry
5.4.3.
Compressibility
5.4.4.
Self-gravitation and
geoid
5.4.5.
Tectonic plates and
laterally-varying viscosity
6. Numerical methods for solving linear algebraic equations
6.1. Introduction
6.2. Direct
methods
6.2.1. Gauss elimination
6.2.2. LU-factorization
6.2.3. Cholesky method
6.3. Iterative methods
6.3.1. Jacobi
method
6.3.2. Gauss-Seidel
method
6.3.3. Conjugate
gradient method
6.3.4. Method
of distributive iterations
6.4. Multigrid
methods
6.4.1 Two grid cycle
6.4.2 Restriction, prolongation and coarse grid
operators
6.4.3 Multigrid cycle
6.4.4 Full approximation storage (for nonlinear
problems or grid refinement)
6.4.5 Full multigrid
6.4.6 Algebraic multigrid
6.5. Iterative
methods for the Stokes equations
6.5.1. Distributive iterations
6.5.2. SIMPLE method
6.5.3. Uzawa-type methods
6.6 Alternating
direction implicit method
6.7 Coupled
equations solving
6.8 Non-linear
equations solving
6.9 Convergence
and iteration errors
7. Numerical methods for solving ordinary differential equations
7.1. Introduction
7.2 Euler
method
7.3 Runge-Kutta
methods
7.4. Multi-step
methods
7.4.1. The midpoint rule
7.4.2. The trapezoidal rule
7.5. Crank-Nicolson
method
7.6. Predictor-corrector
methods
7.6.1. The Adams-Bashforth-Moulton method
7.7. Method
of characteristics
7.8. Semi-Lagrangian
method
7.9. Total
variation diminishing method
7.10. Lagrangian methods
7.10.1 Lagrangian meshes
7.10.2 Particle-in-cell method
8. Data assimilation methods
8.1. Introduction
8.2. Data
assimilation
8.3. Backward
advection (BAD) method
8.4. Application
of the BAD method: restoration of the evolution of salt diapirs
8.5. Variational
(VAR) method
8.6.
Application
of the VAR method: restoration of mantle plume evolution
8.6.1 Forward modelling
8.6.2 Backward modelling
8.6.3 Performance of the numerical algorithm
8.7. Challenges in VAR data assimilation
8.7.1 Smoothness of observational data
8.7.2 Smoothness of the target temperature
8.7.3 Numerical noise
8.8. Quasi-reversibility
(QRV) method
8.8.1 The QRV method for reconstruction of
thermo-convective flow
8.8.2 Optimisation problem
8.8.3 Numerical algorithm for QRV data
assimilation
8.9. Application
of the QRV method: restoration of mantle plume evolution
8.10. Application of the QRV method: restoration of descending lithosphere evolution
8.10.1 The Vrancea seismicity and the relic
descending slab
8.10.2 Temperature model
8.10.3 QRV data assimilation
8.10.4 What the past tells us
8.10.5 Limitations and uncertainties
8.11. Comparison of data assimilation methods
8.12. Errors in forward and backward modelling
9. Parallel computing
9.1. Introduction
9.2. Parallel
versus sequential processing
9.3. Terminology
of parallel processing
9.4. Shared and distributed memory
9.4.1 Shared
memory
9.4.2 Distributed
memory
9.5. Domain decomposition
9.6. Message Passing
9.7. Basics of Message Passing Interface
9.8. Cost of parallel processing
9.9. Concluding remarks
10.
Modelling of geodynamic problems
10.1. Introduction and overview
10.2. Review
of numerical methods used
10.2.1.
Mantle
convection
10.2.2.
Crust
and lithosphere dynamics
10.3. Compressible
flow
10.3.1. Importance, extended Boussinesq and anelastic
approximations
10.3.2. Continuity equation
10.3.3. Momentum equation
10.3.4. Energy equation
10.3.5. Nondimensionalisation and dissipation number
10.4. Phase
transitions
10.4.1. Background
10.4.2. Effective expansivity and heat capacity
10.4.3. Fixed depth approximations
10.4.4. Phase equilibria approach
10.5. Compositional
variations
10.5.1. Field-based approaches
10.5.2. Tracer (marker) based approaches
10.6. Complex
rheologies
10.6.1. Non-Newtonian creep and plasticity
10.6.2. Visco-elasticity
10.7. Continents
and lithospheric plates in mantle convection models
10.7.1. Overview
10.7.2. Rigid block approach
10.7.3. Rheological approach
10.8. Treatment
of a free surface and surface processes
10.9. Porous
flow in a deformable matrix
10.10. Geodynamo modelling
Appendix I. Definitions and relations from vector and matrix algebra
I1.
Introduction
I2.
Matrix
multiplication
I3.
Transposed,
square, and symmetric matrices. Determinants
I4.
Upper
and lower triangular matrices
I5.
Identity
matrix. Inverse matrix
I6.
Matrix
addition and scalar multiplication
I7.
Linear
combination of vectors
I8.
Vector
and matrix norm
I9.
Divergence,
gradient, curl, and Laplasian operators
Appendix II. Spherical coordinates
II1. Spherical grids
II2. Vector operators, strain rates and stress equations in spherical polar coordinates
II3. Two-dimensional
approximations of spherical geometry
Appendix III. List of computer codes and how to access them
References
Author Index
Subject Index