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Numerical methods
Revised March 2001
By R. W. Riddaway and revised by
M. Hortal
| NUMERICAL METHODS | |||||||
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Table of contents
1 . Some introductory ideas
| 1.1 Introduction |
| 1.2 Classification of PDE's |
| 1.3 Existence and uniqueness |
| 1.4 Discretization |
| 1.5 Convergence, consistency and stability |
2 . Finite differences
| 2.1 Introduction |
| 2.2 The linear advection equation: Analytical solution |
| 2.3 Space discretization: Dispersion and round-off error |
| 2.4 Time discretization: Stability and computational mode |
| 2.5 Stability analysis of various schemes |
| 2.6 Group velocity |
| 2.7 Choosing a scheme |
| 2.8 The two-dimensional advection equation . |
3 . The non-linear advection equation
| 3.1 Introduction |
| 3.2 Preservation of conservation properties |
| 3.3 Aliasing |
| 3.4 Non-linear instability |
| 3.5 A necessary condition for instability |
| 3.6 Control of non-linear instability |
4 . Towards the primitive equations
| 4.1 Introduction |
| 4.2 The one-dimensional gravity-wave equations |
| 4.3 Staggered grids |
| 4.4 The shallow-water equations. |
| 4.5 Increasing the size of the time step |
| 4.6 Diffusion |
5 . The semi-Lagrangian technique
| 5.1 Introduction |
| 5.2 Stability in one-dimension |
| 5.3 Cubic spline interpolation |
| 5.4 Cubic Lagrang interpolation and shape preservation |
| 5.5 Various quasi-Lagrangian schemes in 2D |
| 5.6 Stability on the shallow water equations |
| 5.7 Computation of the trajectory |
| 5.8 Two-time-level schemes |
6 . The spectral method
| 6.1 Introduction |
| 6.2 The one-dimensional linear advection equation |
| 6.3 The non-linear advection equation |
| 6.4 The one-dimensional gravity wave equations |
| 6.5 Stability of various time stepping schemes |
| 6.6 The spherical harmonics |
| 6.7 The reduced Gaussian grid |
| 6.8 Diffusion in spectral space |
| 6.9 Advantages and disadvantages |
| 6.10 Further reading |
7 . The finite-element technique
| 7.1 Introduction |
| 7.2 Linear advection equation |
| 7.3 Second-order derivatives |
| 7.4 Boundaries, irregular grids and asymmetric algorithms |
| 7.5 Treatment of non-linear terms |
| 7.6 Staggered grids and two-dimensional elements |
| 7.7 Two dimensional elements |
| 7.8 The local spectral technique |
| 7.9 Application for the computation of vertical integrals in the ECMWF model |
8 . Solving the algebraic equations
| 8.1 Introduction |
| 8.2 Gauss elimination |
| 8.3 Iterative methods |
| 8.4 Decoupling of the equations |
| 8.5 The Helmholtz equation |
References
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13.06.2002 |
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