# Convergence of capillary wave solution

### From Gerris

The solutions obtained using Gerris and my old marker code do not seem to converge exactly toward Prosperetti's theoretical solution (see the corresponding test case in the test suite).

The convergence is broadly OK (see figure below, wave-6, wave-7 and wave-8 are solutions obtained with Gerris on 64x64, 128x128 and 256x256 grids respectively, markers-128 is the result from the marker code, figure 1.13 p. 39 of my PhD thesis manuscript).

But not OK in the details

It looks like the numerical solutions all have a slightly larger period than the theoretical solution (as well as a slightly lower amplitude but as a second-order correction).

However all the numerical solutions seem to converge well toward one another. In particular, the solution obtained using the 128x128 grid marker code is very close to the Gerris solutions. The rate of convergence between the 64x64, 128x128 and 256x256 Gerris solutions also seems to be good and it definitely looks like the 256x256 solution is close to numerical convergence.

So, where does the difference with Prosperetti's solution come from? There are several possibilities:

- The numerical solutions have not yet reached numerical convergence and could eventually converge toward the theoretical solution for higher resolutions (or smaller integration timesteps)
- The numerical solutions have converged toward and identical but incorrect solution, due to bugs or incorrect assumptions/implementation of the integration schemes
- Prosperetti's solution is the right one but I made a mistake when computing the corresponding curve
- The assumptions made to obtain the theoretical solution are not verified by the numerical solutions (e.g. vanishing amplitude, infinite domain etc...)

Based on the figures explanation 1) is unlikely: the solutions look like being close to convergence. Furthermore, reducing the timestep does not change the results either.

Explanation 2) is not impossible, although it's hard to believe that both the marker code and Gerris which do not share any common code and are based on very different integration schemes (both for interface advection, surface tension calculation, momentum integration etc...) could both contain an error which would make them converge toward the same (incorrect) solution...

Explanation 3) would be the simplest, however I have just recomputed the solution using direct numerical inversion of the Laplace transform solution given by Prosperetti and it gives exactly the same curve as the curve I obtained during my PhD using the direct analytical formula derived by Prosperetti...

So in my opinion explanation 4) is the most likely. What is puzzling however is that some authors (e.g. Gerlach et al 2006) have reported better convergence than obtained here... Is their error calculation incorrect? Are their simulations subtly different in the way they are setup? (boundary conditions for example)

**Convergence rates**

Just to show that all the Gerris solutions seem to converge consistently, the equivalent to table 8 of the test suite, but taking the 256x256 Gerris solution as reference solution is given below

8x8 | 16x16 | 32x32 | 64x64 | 128x128 |

0.14446 | 0.02070 | 0.008811 | 0.002651 | 0.0003029 |

Up to 32x32 the results are almost identical to the results obtained using Prosperetti's solution as reference. The convergence rate is somewhat irregular (it should be computed using Richardson extrapolation rather than using the 256x256 solution as reference) but is close to two on average at all resolutions (in contrast to Table 8 which shows a lack of convergence at high resolutions).

**Larger aspect ratio**

One of the assumptions of the theoretical solution of Prosperetti which is not verified in the results above is that of an infinitely "high" domain. The test case uses a square box so boundary effects could be significant. Using a 1x3 domain instead gives the following results (using Prosperetti's solution as reference)

8x24 | 16x48 | 32x96 | 64x192 | 128x384 |

0.13953 | 0.01595 | 0.008575 | 0.001509 | 0.000545 |

which is clearly much better for higher resolutions (and is slightly better than the best results of Gerlach et al. obtained using PROST).

So the conclusion is that the domain aspect ratio is important. Gerlach et al (2006) results were probably obtained for a larger aspect ratio than used in Popinet & Zaleski (1999), although this is not mentioned in the article.