Consider solving the nonlinear Schrödinger equation (NLSE) with 2D transverse dimensions. Or more simply, consider the beam propagation method (BPM).
We know that the original formulations of Feit&Fleck relied on the FFT to compute the derivatives.
In the 90s, many works about finite differences (FD) were proposed. The advantage is that the grid is more flexible and the boundary conditions are better understood and controllable (PML, TBC...). Prof. Fibich in his book suggests FD is a better approach to tackle collapses.
With the advent of GPU, FFTs with lots of points seem to be almost costless. Do you think it is not worth anymore to study finite differences? What about non-paraxial propagators?
Thank you for your suggestions.
Denys Dutykh
, thank you for your quick answer. I was wondering about how effective it is to use FFT to increase the accuracy/resolution, instead of looking for a grid improvement in FD approaches or considering a different pseudospectral algorithm. I am expert of both, but not of GPU. Are they really a game changer in this respect?
The numerical methods are all relevant and useful in the right context, i.e. to solve the right problem. The pseudo-spectral methods implemented and solved with FFT gives both accuracy and speed alltogether that is difficult to match for problems where this can be utilized. GPUs as such is not a new and not a game-changer but is a device that is designed for high-throughput for data-parallel workloads. That is, for workloads that can benefit from the high on-chip bandwidth and execute the same program (kernel) on bulks of data. So problems that fit within this computing paradigm can benefit.
Some additional comments...it is pointless to discuss FFT vs. Finite Differences as the sole criterion for choosing between numerical solvers... the fast convergence rate of spectral/peudospectral methods are exponential when solutions are smooth (algorithmic efficiency), however... the key criterion for practical use cases is... numerical efficiency (CPU time)... and it is possible to beat FFTs in terms of scalability and performance in practical implementations, e.g. when one wants to adapt to features of the solution, on large systems due to growing overhead in FFTs (the methods are globally defined) and especially the higher dimensions needed to solve a given problem, etc. Furthermore, to feed into this discussion I just mention that recently, we did some proofs using a high order finite difference method (which also can achieve convergence such as spectral methods, but have locality in the stencils) on some of the largest heterogeneous systems equipped with many GPUs, e.g. see Article A Massively Scalable Distributed Multigrid Framework for Non...
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The accuracy of the numerical results are mainly restricted to accuracy of the models, although sometimes the numerical stablilization might due to the detalied numerical methods. GPU might be treated as one possible way to decrease the CPU times in numerical simulations. Further, for a detaield numerical simulation problem, BC might be very important.
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