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


#1

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?


#2

Gerry,
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.

Best
Wolfgang