Incorporating compact finite difference

[sijieh23]

High-fidelity simulation of PDEs.
Solving PDEs that exhibit multi-scale behaviors (e.g., Burgers equation) requires high-fidelity numerics, preferrably with spectral-like resolution. Compact finite difference (see e.g., Lele, 1992) is one of those schemes designed for this purpose.

Incorporating compact finite difference into the findiff package
I'm wondering if it is possible, or if your team is interested in, implementing and incorporating such a scheme into the current findiff package.

The boundary points might be a little bit tricky, but still solvable. For periodic boundary condition, the scheme can easily adapt. For non-periodic condition, maybe switch to regular one-sided finite difference.

[maroba]

Sounds like an interesting new feature. At first glance, it should not be too much work. I will have a closer look and see if I have time to implement it in January.

[maroba]

Interim report: I implemented compact differences on branch "compact-differences" for periodic boundary conditions with equidistant grids according to Lele 1992.

To do's for myself:

• Adapt accuracy calculation in case of compact schemes
• Implement non-periodic case
• Implement non-uniform cases
• documentation

[sijieh23]

@maroba FYI, one can also do one-sided compact finite difference at the boundaries (see e.g., Eq.14 in Visbal & Gaitonde, 2002)