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

OVERVIEW Motivation TriCubic Interpolation Proposed Implementation Examples Conclusions

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

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

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

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

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

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

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

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

Examples 𝜕 2 𝑓 𝜕𝜉𝜕𝜂 comparison

Examples 𝜕 3 𝑓 𝜕𝜉𝜕𝜂𝜕𝜁 comparison

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

Examples (tricubic vs trilinear)

Examples (tricubic vs trilinear)

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

Questions? Comments?