1. Compact Higher-Order Interpolation at Overset mesh Boundaries
Bob Tramel
Chief Scientist
10/3/2018

2. OVERVIEW
Motivation
TriCubic Interpolation
Proposed Implementation
Examples
Conclusions

3. Motivation The use of higher order schemes is growing Aero-acoustics
Aero-optics
Flow control

4. Motivation
Errors introduced by the trilinear interpolation can dominate the global error of a numerical scheme
Current implementations are typically based on dimension by dimension extension of the basic Gregory-Newton interpolation methodology
This leads to large stencils
Higher order overset interpolation still not routine

5. Motivation
Tricubic interpolation is a natural candidate to extend the trilinear methods commonly in use
The question becomes how to efficiently implement the method?

6. TriCubic Interpolation
Here we follow the outline of Leiken and Marsden1
We will work in curvilinear space
Tricubic Interpolation can be expressed as:
The following choice of variables ensures global 𝐶 1 continuity
𝑓 𝜉,𝜂,𝜁 = 𝑖,𝑗,𝑘=0 3 𝑎 𝑖𝑗𝑘 𝜉 𝑖 𝜂 𝑗 𝜁 𝑘
𝑓, 𝜕𝑓 𝜕𝜉 , 𝜕𝑓 𝜕𝜂 , 𝜕𝑓 𝜕𝜁 , 𝜕 2 𝑓 𝜕𝜉𝜕𝜂 , 𝜕 2 𝑓 𝜕𝜉𝜕𝜁 , 𝜕 2 𝑓 𝜕𝜂𝜕𝜁 , 𝜕 3 𝑓 𝜕𝜉𝜕𝜂𝜕𝜁
 1Tricubic Interpolation in Three Dimensions (2005), by F. Lekien, J. Marsden, Journal of Numerical Methods and Engineering

7. Proposed Implementation
However these must be computed at each corner
How best to compute the derivatives?
Here we explore repeated use of Padé approximations

8. Examples Consider the function 𝑓 𝑥,𝑦,𝑧 =𝑥 𝑒 − 𝑥 2 + 𝑦 2 + 𝑧 2
𝜕 3 𝑓 𝜕𝑥𝜕𝑦𝜕𝑧 𝑓

9. Examples
Here we compute 𝜕𝑓 𝜕𝜉 , 𝜕𝑓 𝜕𝜂 , 𝜕𝑓 𝜕𝜁 using the 3 point, 6th order scheme of Lele2
𝜕 2 𝑓 𝜕𝜉𝜕𝜂 is computed by applying the compact derivative in 𝜂 to 𝜕𝑓 𝜕𝜉
𝜕 2 𝑓 𝜕𝜉𝜕𝜂 can be computed by applying the compact derivative in 𝜉 to 𝜕𝑓 𝜕𝜂
2S. K. Lele. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1):16–42, November 1992

10. Examples
The derivatives agree to machine zero
𝜕 2 𝑓 𝜕𝜉𝜕𝜂 − 𝜕 2 𝑓 𝜕𝜂𝜕𝜉

11. Examples
𝜕 2 𝑓 𝜕𝜉𝜕𝜂 comparison

12. Examples
𝜕 3 𝑓 𝜕𝜉𝜕𝜂𝜕𝜁 comparison

13. Examples (tricubic vs trilinear)
Interpolate values along a partial slice through 𝑓

14. Examples (tricubic vs trilinear)

15. Examples (tricubic vs trilinear)

16. Conclusions Initial TriCubic interpolation experiments performed
Implementation in terms of Pade derivatives
Mixed Pade derivatives appear to commute
Appear to be much more accurate than trilinear
caveat auditor: Much work remains!
Things to do
Formally address order of accuracy
Can the baseline method be optimized?
Leiken/Marsden choose smoothness over formal accuracy
Use compact WENO for derivatives?
What about shocks? (Toggle to trilinear using sensor?)
Do the 𝜉,𝜂,𝜁 returned by PEGASUS/DCF need to be updated?
Robust treatment for holes must be formulated

17. Questions? Comments?