Conservative Compact Schemes

This repository exemplifies the most clean and minimalistic way to program conservative compact schemes for solving systems of equations such as Navier-Stokes or Euler-Equations.

This short and extremely versatile implementation of conservative compact schemes is possible thanks to three ingredients:

• A Taylor Table algorithm recently presented in here.
• An ingenious spare technique to produce highly performing discrete finite-difference operations in higher-dimensions. Such example is available here.
• The conservative boundary schemes by Brady and Livescu in [1]

Here we reproduce some of the invicid benchmark tests proposed in [1]:

Test 1.1: Euler equations 1d, Gaussian wave with slip-walls boundaries

N	Lele643	E8	T8
101
q_int

Test 2.1: Euler equations 2d, Gaussian wave with slip-walls boundaries

NxM	Lele643	E8	T8
61x61
q_int

Test 3.1: Euler equations 3d, Gaussian wave with slip-walls boundaries

NxMxR	Lele643	E8	T8
41x41x41
q_int

Conclusion:

• Because the compact schemes presented in [1] have been taylored to work well with Dirichlet (homogeneous) boundary conditions. There is no need for using a skew-symmetric formulation of the convective operators. (The skew-symmetric from of the convective terms in reality adds small amounts of numerical diffusion in the scheme).
• Classical schemes such as Lele643 [2] or Pade43 [3], fail in the short term due spurios oscilations that emanate from the boundaries. This is the reason why compact schemes normaly rely on filters to keep the solution stable.
• By integrating the conserved quantities over time, is easy to see that the schemes proposed in [1] are truly conservative and stable for very long time simulations (Or at least until the formation of a shockwave) and do not need any filtering or the addition of artificial viscosity. Therefore, they are high-order conservatives compact schemes in the pure finite-difference sense.

Happy coding ;D

-- Manuel A. Diaz, 2021.

Future work:

• In a similar fassion, perhaps, one could taylor a compact-scheme for the Navier-Stokes equations. But this time using some basic characteristic boundary-conditions: inflow, outflow, isothermal-walls.
• This boundaries are not suitalbe for studying cases where shock from. How thery compare to classical BCs for WENO methods?

References:

[1] Brady, P. T., & Livescu, D. (2019). High-order, stable, and conservative boundary schemes for central and compact finite differences. Computers & Fluids, 183, 84-101.

[2] Laizet, S., & Lamballais, E. (2009). High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy. Journal of Computational Physics, 228(16), 5989-6015.

[3] Moin, P. (2010). Fundamentals of engineering numerical analysis. Cambridge University Press.