Computing time vs order of accuracyΒΆ

Using a higher order implies using smaller time steps and using more basis functions. The expected increase in computed time can then be estimated as follow:

Decrease of the time steps width by a factor: \((2N+1)/(2n+1)\)
Increase of the number of basis functions by a factor: \((N+1)(N+2)(N+3)/((n+1)(n+2)(n+3))\)
with:
n: initial order of accuracy -1
N: new order of accuracy -1

Here is an example of the theoretical increase of the compute time relative to order 3

3 4 5 6 7 8 9 10
Increase due to time steps 1 1.4 1.8 2.2 2.6 3 3.4 3.8
Increase due to number of basis functions 1 2 3.5 5.6 8.4 12 16.5 22
theoretical increase relative to order 3 1 2.8 6.3 12.32 21.84 36 56.1 83.6
observed increase (on SM2) 1 2.0 4.6 11.5        

The last line shows the observed time increase on SuperMUC Phase 2 on a small run with dynamic rupture and LTS-DR. Low order calculation (up to order 5 included) are memory bounds, and are then less efficient. As a consequence, the higher-order simulations cost less than expected in comparison with order 3.