Book contents
- Frontmatter
- Contents
- Foreword by Gerald Schubert
- Preface
- Acknowledgements
- 1 Basic concepts of computational geodynamics
- 2 Finite difference method
- 3 Finite volume method
- 4 Finite element method
- 5 Spectral methods
- 6 Numerical methods for solving linear algebraic equations
- 7 Numerical methods for solving ordinary and partial differential equations
- 8 Data assimilation methods
- 9 Parallel computing
- 10 Modelling of geodynamic problems
- Appendix A Definitions and relations from vector and matrix algebra
- Appendix B Spherical coordinates
- Appendix C Freely available geodynamic modelling codes
- References
- Author index
- Subject index
- Plates section
5 - Spectral methods
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword by Gerald Schubert
- Preface
- Acknowledgements
- 1 Basic concepts of computational geodynamics
- 2 Finite difference method
- 3 Finite volume method
- 4 Finite element method
- 5 Spectral methods
- 6 Numerical methods for solving linear algebraic equations
- 7 Numerical methods for solving ordinary and partial differential equations
- 8 Data assimilation methods
- 9 Parallel computing
- 10 Modelling of geodynamic problems
- Appendix A Definitions and relations from vector and matrix algebra
- Appendix B Spherical coordinates
- Appendix C Freely available geodynamic modelling codes
- References
- Author index
- Subject index
- Plates section
Summary
Introduction
Spectral methods have been widely used in geophysical modelling of different fluids including the atmosphere, ocean, outer core and mantle. Variables are expanded as a sum of orthogonal global basis functions, typically trigonometric or polynomial, in contrast to finite difference, finite element and finite volume methods in which the basis functions are local. The convergence of the method is faster than spatial methods, meaning that high mathematical accuracy can be obtained with relatively few basis functions when representing smoothly varying fields, although sharp gradients or discontinuities can cause problems. Spherical geometry is easily treated by using spherical harmonics as basis functions, giving approximately uniform resolution over the sphere.
Basis functions are typically different in the horizontal (azimuthal) and vertical (radial) directions because of the differing boundary conditions (side boundaries are often periodic). In the vertical (radial) direction, Chebyshev polynomials or finite differences are typically used. Once expanded in harmonics, spatial derivatives are given by exact analytic expressions. For linear equations, the equations for different harmonics decouple in spectral space, and each mode can be solved independently. The method is thus ideally suited for equations in which the coefficients in front of dependent variables (e.g. viscosity, thermal diffusivity, wave velocity) are spatially constant.
- Type
- Chapter
- Information
- Computational Methods for Geodynamics , pp. 93 - 108Publisher: Cambridge University PressPrint publication year: 2010