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The marker-in-cell method

The marker-in-cell method. Core and Mantle Dynamics Gregor J. Golabek. What is the marker-in-cell method?. 2. Fixed ( Eulerian ) grid points. Mobile ( Lagrangian ) markers. Cell. [modified from Gerya, 2010]. What are grid points?. 3. Temperature T. Distance x.

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The marker-in-cell method

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  1. The marker-in-cellmethod Core and Mantle Dynamics Gregor J. Golabek

  2. What is the marker-in-cell method? 2 Fixed (Eulerian) grid points Mobile (Lagrangian) markers Cell [modified from Gerya, 2010]

  3. What are grid points? 3 Temperature T Distance x

  4. Discretization of continous 1D function 4 Fixed grid points

  5. Increase number of grid points 5

  6. Higher resolution = Higher accuracy 6 [Gerya, 2010]

  7. More complex 2D problem 7 [Gerya, 2010]

  8. Discontinous problems on a grid 8 numerical diffusion! [Gerya, 2010] • Solution: • Mobile markers transport physical properties • (e.g. composition, density, temperature, ...) • Interpolation of marker properties on immobile grid points • Solution of the constitutive equations (e.g. Stokes equation => velocities) • Velocity field is used to advect the markers

  9. With marker-in-cell method 9 [Gerya, 2010]

  10. Interpolation from markers to nodes 10 N markers e.g. density r‘‘ r‘ [modified from Gerya, 2010]

  11. Averaging - Methods 11 Harmonic mean Arithmetic mean Geometric mean

  12. Averaging - Results 12 Velocity v Higher grid resolution [modified from Schmeling et al., 2008]

  13. Re-interpolation from nodes to markers 13 N markers density r‘‘ r‘ [modified from Gerya, 2010]

  14. Marker advection – Euler scheme 14 Application: Corner flow problem y error y(2) y(1) trajectory x(1) x(2) x(3) [Press et al., 1997]

  15. Marker advection – Runge-Kutta 2nd order 15 Application: Corner flow problem y error trajectory y(2) y(1) trajectory smallererror x(1) x(2) x(3) [Press et al., 1997]

  16. Marker advection – Runge-Kutta 4th order 16 A x(t) C x(t+1) B [Press et al., 1997] D

  17. Marker advection – Runge-Kutta 4th order 17 A x(t) C x(t+1) B [Press et al., 1997] D

  18. Marker advection – Runge-Kutta 4th order 18 A x(t) C x(t+1) B [Press et al., 1997] D

  19. Marker advection – Runge-Kutta 4th order 19 A x(t) C x(t+1) B [Press et al., 1997] D

  20. Geodynamical application 20 • Geodynamics: More precise Runge-Kutta 4th order scheme used • STILL: • Accumulation of advection • errors after several overturns • Formation of holes in • marker field

  21. Always check your results! 21 Entrainment Holes [Schmeling et al., 2008]

  22. Summary 22 • The marker-in-cell method is a powerful tool to advect strongly • discontinous fields in numerical models • Mobile markers are advected through an immobile grid • High order Runge-Kutta advection schemes preferred • Non-diffusive markers store physical properties • BUT: • Grid resolution has still to be sufficiently high for meaningful solution • Sufficient number of markers in each cell for averaging to mini- • mize interpolation errors • Holes in the marker field can open after several overturns • Marker refilling needed when cells are empty

  23. The end Lecture download: http://perso.ens-lyon.fr/gregor.golabek/teaching.html

  24. Numerical exercise: Nu-Ra relation 24 [Christensen,1984]

  25. Reminder: Nu and Ra number 25 Nusselt number Nu: Rayleigh number: [Turcotte and Schubert, 2002]

  26. Numerical exercise: Nu-Ra relation 26 [Christensen,1984] How to do that? Vary the Ra number in your input file Wait until steady-state is reached in the simulation Read out the heat flux qsurf and compute corresponding Nu Plot results and estimate parameters b and g

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