Book contents
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
14 - The multigrid method
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
Summary
Theory: Principles of multigrid method. Multigrid method for solving the Poisson equation in 2D. Coupled solving of momentum and continuity equations in 2D with multigrid for the cases with constant and variable viscosity.
Exercises: Programming of multigrid methods for solving the Poisson equation and coupled solving of the momentum and continuity equations in 2D.
Multigrid – what is it?
The use of direct methods to solve the system of equations places a strong limitation on the maximum possible number of nodal points within a numerical grid due to limitations in computer memory and computational speed. This limitation is particularly critical in 3D where, to reach the same resolution as in 2D (tens and hundreds of grid points in each direction), the amount of linear equations to be solved increases by at least two orders of magnitude and the number of computational operations required to solve the same equations grows by at least three orders of magnitude. Try to increase resolution in your visco-elasto-plastic code (Exercise 13.2) by two to three times in each direction and you will see how much slower it will compute …
Therefore, for very high resolution in 2D and for a moderate resolution in 3D, we are forced to use iterative methods which do not have such strong memory limitations. However, many iterative methods also have the problem that they require an increasing number of iterations with an increasing number of grid points (Fig. 14.1).
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- Information
- Introduction to Numerical Geodynamic Modelling , pp. 193 - 220Publisher: Cambridge University PressPrint publication year: 2009