A Limiting Strategy for the BFECC Method for Solving Advection Equations

1. A Limiting Strategy for the BFECC Method for Solving Advection Equations • Yingjie Liu, Georgia Institute of Technology • Dupont and Liu, ’03, ’07 • Kim, Liu, Llamas, Jiao and Rossignac, ’05,’07 • Selle,Fedkiw,Kim,Liu and Rossignac,’07 • Hu, Li and Liu, in preparation for submission.

2. Some classical methods for solving the hyperbolic equation • Upwind scheme (for a>0) • (2) Lax-Friedrichs scheme • Both are 1st order, CFL number < 1

3. Some classical methods for solving the hyperbolic equation • (3) Lax-Wendroff scheme • (4) MacCormack scheme (a>0) • Upwind: • Downwind: • Averaging: • Both are 2nd order, CFL number < 1. MacCormack scheme coincides with Lax-Wendroff scheme for constant a.

4. Some classical methods for solving the hyperbolic equation • (5) “Classical” unstable center difference • (6) CIR (Courant-Isaacson-Ree) or semi Lagrangian scheme or (backward) characteristic method • where the RHS is approximated by a local linear interpolation. Therefore it’s unconditionally stable (without CFL restriction). It’s 1st order accurate.

5. Convenience of the CIR Scheme on Multi Dimensions, on Unstructured Meshes, or on Locally Refined Meshes without Worrying about CFL Restriction. characteristic line t y The local linear interpolation can be along a line. x

6. Level Set Method (Osher and Sethian ’88)

9. Implementation of BFECC • Suppose we have a 1st order CIR subroutine solving the equation • CIR: • BFECC: • Forward: • Backward: • Modify n-level values: • Forward again

12. Error Flow • BFECC: • Modified MacCormack: • (Selle,Fedkiw,Kim,Liu,Rossignac,’08)

13. BFECC for the level set computation of rotating Zalesak’s slotted disk, CFL=3. Top: 100X100. Bottom: 200X200

14. Zalesak’s Slotted Disk Moving on Triangulated Sphere (ByungMoon Kim et al, IEEE trans Visual Comput Graphics, ’07)

15. Top: CIR; Bottom: BFECCLeft: One revolution; Right: Two Revolutions

16. BFECC for Non Smooth Velocity Field (Dupont and Liu, Math. Comp. ’07) • For the first two back-and forth advection steps, wherever the non smoothness of the velocity field is detected, say at grid point i, locally recompute the back-and forth error at i with a fictitious constant velocity field • Apply the final advection step with the original velocity field.

17. (3)locally compute the back-and-forth error, incorporate it into the solution. (4) Final advection with the CIR scheme Local Constant Velocity

18. Remarks on Re-distancing • Solve the redistancing equation (Sussman, Smereka, Osher’94) with a 1st order upwind scheme • only at grid points where : (1)they are away from the interface; or (2) they are adjacent to the interface but is not close to 1. • This method though 1st order, essentially maintains the 2nd order interface accuracy computed from previous BFECC. • Russo and Smereka’2000 proposed a redistancing method which does not modify the values at nodes adjacent to the interface.

19. Shrinking Disk: BFECC for Level Set Advection. Left: Turn off BFECC near corners; Right: Local constant velocity fix of BFECC

20. The predicted vanishing time is at 31<T<32. The last figure on the right is computed at T=31.

21. Expanding Circles. • Left: locally turn off • BFECC at corners. • Right: local const vel fix • for BFECC

22. Fluid Simulations • Navier-Stokes equations: • Projection method (Chorin ’68, Temam ’68). • Use operator splitting to solve the following sequentially within a time step.

23. (Stam, SIGGRAPH ’99) The first advection equation • can be solved with a semi Lagrangian scheme (or CIR scheme) to remove the CFL restriction, thus speed up the fluid simulations for graphics. • Use BFECC to further improve the accuracy of the semi Lagrangian scheme with very little extra effort. (Kim et al, ’05, ’07).

24. Movie

25. The Level Set Method (Osher-Sethian ‘88) Interfaces (points in 1D)

26. The Regional Level Set Method(Zheng,Yong and Paul, SIGGRAPH 2006.) interfaces (points in 1D) Regions: I II III If regions I & II are merging, the local Φ is I II If regions II & III stay separated by membrane II III

27. Regional Level Set Equation Solved by the CIR Method and Then Improved by BFECC characteristic line CIR Method: t y The local linear interpolation is along a line, thus The regional local modification of level set function Works. x BFECC: simply call the above procedure 3 times as described before.

28. Simulation of Bubbles in Foam With The Volume Control Method (Kim, Liu, Llamas, Jiao and Rossignac, SIGGRAPH 2008) More than 400 bubbles rising. Coarse grid 64X64X64, locally half the mesh size. Simulated by a single 4ghz Pentium 4 processor.

31. What if there are discontinuities in the solution ?

32. What happen if BFECC is applied recursively ? • Let L be the linear scheme that updates forward in time while L* is obtained by applying L to the time reversed equation. Assume • Definition L is time-reversible if • Lemma L is time-reversible iff

33. Applying BFECC recursively approaches a time-reversible scheme • Theorem. If L is time-reversible and at least first order, then it is at least second order accurate. • Let be the scheme by applying BFECC recursively k times to L. • Theorem. If or , then

34. Forward and Then Backward Error • The forward-and-then-backward error for L is • The forward-and-then-backward error for is • similarly defined. Note that is zero if L is time-reversible, and the forward-and-then-backward error for tends to zero in 2-norm as k increases. This is implied from the previous theorem.

35. Simplified Forward-and-Then-Backward Error for the Second Iteration • is obtained by applying BFECC to L, then what is ? • It’s unnecessary to compute it, we only need an approximation • Define

36. Relation Between the 1st and the 2nd (Simplified) Forward-and-Then-Backward Errors • Theorem. • Corollary. If then • This means on average, the 2nd forward-and-then-backward error is smaller. This property is used to determine where to apply a limiting procedure.

37. Limiting Strategy • If at a grid point i • a careful study suggests that the forward-and-then backward errors adjacent to grid point i are large and could possibly produce overshoots. We can limit them to no larger than the one at grid i. • This modification to the error is not going to change the order of accuracy if it’s applied to the smooth area.

38. BFECC with Limiting • Forward advection • Backward advection • Forward advection with • Backward advection to compute • Limit locally and do final advection

44. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for HamiltonJacobi equations, SIAM J. Numer. Anal., 28 (1991. • R. Abgrall, Numerical discretization of the rst-order Hamilton-Jacobi equation on triangular meshes, Comm. Pure Appl. Math.,49(1996). • A. Kurganov & Eitan Tadmor, New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, J. Conput. Phys, 160(2000). • D. Levy, G. Puppo and G. Russo, A fourth order central WENO scheme for multi-dimensional hyperbolic systems of conservation laws, SIAM J. Sci Comput., 24(2002). • Y.-T. Zhang and C.-W. Shu, High order WENO schemes for Hamilton-Jacobi equations on triangular meshes,SIAM J. Sci. Comput., 24 (2003).

47. Thank You !