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
7 - Numerical solutions of the momentum and continuity equations
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: Types of numerical grids and their applicability for different differential equations. Staggered, half-staggered and non-staggered grids in one, two and three dimensions. Discretisation of the continuity and Stokes equations on a rectangular grid. Conservative and non-conservative discretisation schemes for Stokes equations. Mechanical boundary conditions and their numerical implementation. No slip and free slip conditions.
Exercises: Programming different mechanical boundary conditions. Solving continuity and momentum equations for the case of variable viscosity.
Grids
As we already discussed, the numerical solution of partial differential equations (PDEs) requires the definition of a grid of nodal points within the numerical model. The choice of this grid depends strongly on the type of equations to be solved. Discretisation schemes for these equations will also change with the changing types of numerical grids. The following types of numerical grid exist in numerical geodynamic modelling:
Depending on the dimension of the problem, the numerical grid can be one, two and three dimensional (1D, 2D, 3D) (Fig. 7.1).
Depending on the shape of the basic elements, the grid can be rectangular and triangular (Fig. 7.2).
Depending on the distribution of nodal points, the grid can be regular and non-regular (irregular) (Fig. 7.3).
Depending on the distribution of different variables within the grid, it can be non-staggered or staggered (Fig. 7.4).
The simplest grids are non-staggered. All variables are defined at the same nodal points (Fig. 7.5). When using finite differences (FD) with such a grid, all equations are formulated at the same nodal points.
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- Information
- Introduction to Numerical Geodynamic Modelling , pp. 83 - 104Publisher: Cambridge University PressPrint publication year: 2009