abhijit/MRRK Methods for the NLS Equation

Meeting Minutes

To do:

[ ] Implement step size control with ImEx+Relaxation for the NLS with a 2-soliton solution from Herbst the paper with FEM semi-discretization.

New results show FEM + ARK + Relaxation to outperform operator splitting pseudospectral on many problems (and is never worse).

Scope of paper:

Methods:
- FEM + ARK
- FEM + ARK + Relaxation (single or multiple)
- pseudospectral + Strang
- pseudospectral + AK4
- pseudospectral + OS + relaxation ? (which 2nd time discretization to use? Lie-Trotter? Strang?)
Other methods we could try:
- We could also try other IMEX methods
- Different high-order operator splitting methods (from Auzinger’s website or NLS literature)
Test problems:
- 1-, 2-, and 3-soliton solutions (Herbst & Morris)
- Focusing semiclassical problems from Markowich et. al.
Other problems we could try:
- More semiclassical problems (defocusing)
- two dimensions?

To do:

Implemented 1st-order operator splitting with exact solution for each part. Mass (\(|u|^2\)) is conserved, but the Hamiltonian is not and can’t be enforced (fsolve fails).
For the 1-soliton solution, OS gives much less accuracy than IMEX.

To do:

Find the reason for the dip in the error growth plot for 1-soliton solution for the NLS equation by plotting the error over the spatial domain.
For multi-soliton solutions, try error-based step size control time stepping to improve the error in invariants.
Try the MRRK methods for the test problems that Theodoros suggested and compare the relaxation approach with operator splitting + pseudospectral.

I was calculating the error as \(||U|-|u||\), which ignores the phase error, so I got bounded errors for the 1-soliton solution. I fixed this, and now I get a linear error growth behavior by conservative relaxation methods and quadratic error growth (eventually) by the baseline methods. Something weird happens at time \(t = 4 \pi\). The true solution is \(u(x,t) = \exp(it)*sech(x)\) and \(exp(it) = 1\) at \(t = 4 \pi\). If that dip happens for this reason, I would also expect dips in the error plot at \(t = 2n \pi\) for \(n = 1,2,3, \ldots\) I do not understand this completely.
After fixing the error calculation, I got similar error growth behavior as before for the multi-soliton solutions.

Paper of Cano & Duran shows ODE examples with bounded error growth: https://link.springer.com/article/10.1023/B:BITN.0000039391.19365.56
Results for \(q=8,18\) show expected quadratic vs. linear error growth :)

To-do:

For q =1 and 2, both semi-discretizations satisfy the time derivative of invariants up to machine accuracy. For q = 8 and 18, the time derivative of the second invariant corresponding to the pseudo-spectral semi-discretization is not small.

To-do:

[ ] Review ODE paper on relative equilibrium solutions – can we set up an ODE problem that behaves like the \(q=2\) case?
[ ] Use Thm. 3.1 of SS & Duran to show that the error should be linear/bounded for the stationary soliton solution
[X] Do more tests of \(q=8\), \(q=18\)

Both PSP and FEM semi-discretizations give the same results (also when combined with relaxation)
For \(q=2\), we see linear error growth (non-conservative) and bounded error (conservative)
For \(q=1\), PSP gives linear/quadratic (non-conservative/conservative)
For \(q=1\), FEM gives linear/1.5?? (non-conservative/conservative)
No method seems to give very good results for \(q=8\) or higher (needs more exploration)
[x] Check time derivative of invariants for PSP semi-discretization

Confirmed that time derivative of invariants is small (for HM semi-discretization)

Can we use Thm. 3.1 of SS & Duran to show that the error should be linear/bounded for the stationary soliton solution?
Verify that the different soliton solutions for the NLS equation, evaluated at grid points, satisfy the time derivative of the two nonlinear invariants.
Solve the NLS equation using the pseudospectral method in space and relaxation RK methods in time to compare the results with the numerical results from Duran and Sanz-Serna’s paper.