Is there a simple explanation for why the degree of the polynomial is now part of the CFL constraint?


In Section 3.1 of the 2017 ASPECT paper

Heister, T., Dannberg, J., Gassmöller, R., Bangerth, W., 2017. High accuracy mantle convection simulation through modern numerical methods. II: Realistic models and problems. Geophys. J. Int. 210 (2), 833–851.

the CFL constraint has the degree of the polynomial p_T that is used to discretize the temperature in it:

∆t  ≤ min_K  C  h_K  / ( p_T  || u ||_{L_∞} (K) )

where C is the CFL number and K is the cell index. My understanding (guess) is that this came about because in the (2012) paper it was written that C defined by

CFL_K =   ∆t_n ||u||_{∞,K}  /  h_K   ≤  C

and experimentally it was found that good values of C were

C = 1 / (5.91 p)

in 2D and

C = 1 / (43.61 p)

in 3-D, where p is the polynomial degree with which the temperature variable is discretized.

Is this correct?

Is there any more insight into why the degree of the polynomial p or p_T is needed to maintain stability?


Think of h/p as the distance between nodes of the mesh. That’s the characteristic lengthscale of a discretization that corresponds to the ‘Delta x’ you would find in the typical finite difference stability analysis.