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.. _sphx_glr_auto_tutorials_errors_g_error_dealiasing.py:


Implementing an anti-aliasing technique with a model
====================================================


When performing nonlinear computations on discrete grids, it is important to
choose the time window and the number of sample points large enough to prevent
aliasing errors [B2001]_.  This example demonstrates how to
implement an anti-aliasing technique directly with a model.

For a nonlinearity of order three, as for the cubic Kerr nonlinearity, an easy
anti-aliasing procedure is to extend the spectrum by a factor of two and to
proceed by zero-padding [HCL2008]_ [FCGK2005]_.  Therefore, in each step, after
the nonlinear term is evaluated in the time domain and transformed to the
Fourier domain, the upper half of the spectrum is set to zero.

Here, the symmetric split-step Fourier method is used and the propagation of a
fourth-order soliton is considered.

.. codeauthor:: Oliver Melchert <melchert@iqo.uni-hannover.de>

.. GENERATED FROM PYTHON SOURCE LINES 24-26

We first import the functionality needed to perform the sequence of numerical
experiments: 

.. GENERATED FROM PYTHON SOURCE LINES 26-36

.. code-block:: default


    import sys; sys.path.append('../../')
    import numpy as np
    import numpy.fft as nfft
    from fmas.models import ModelBaseClass
    from fmas.config import FTFREQ, FT, IFT, C0
    from fmas.solver import  SySSM
    from fmas.grid import Grid
    from fmas.tools import plot_evolution








.. GENERATED FROM PYTHON SOURCE LINES 37-49

Next, we implement a model for the nonlinear Schrödinger equation.  In
particular, we here consider the standard nonlinear Schrödinger equation,
given by

.. math::
   \partial_z u = -i \frac{\beta_2}{2}\partial_t^2 u + i\gamma |u|^2 u,

wherein :math:`u = u(z, t)` represents the slowly varying pulse envelope,
:math:`\beta_2=-1` is the second order dispersion parameter, and
:math:`\gamma=1` is the nonlinear parameter. 
As discussed above, we here implement a simple technique allowing
to compute the nonlinear term free of aliasing errors.

.. GENERATED FROM PYTHON SOURCE LINES 49-67

.. code-block:: default


    class NSE(ModelBaseClass):

        def __init__(self, w, b2 = -1.0, gamma = 1.):
            super().__init__(w, 0.5*b2*w*w)
            self.gamma = gamma
            # ANTI-ALIASING FILTER SETTING UPPER HALF OF SPECTRUM TO ZERO
            self._de_alias = lambda uw: np.where(np.abs(w) < 0.5 * w.max(), uw, 0j)

        @property
        def Lw(self):
            return 1j*self.beta_w

        def Nw(self, uw):
            ut = IFT(uw)
            return self._de_alias(1j*self.gamma*FT(np.abs(ut)**2*ut))









.. GENERATED FROM PYTHON SOURCE LINES 68-74

Next, we initialize the computational domain and use a symmetric split-step
Fourier method to propagate a single third-order soliton for six soliton
periods.
For this numerical experiment, the extend of the time domain and the number 
of sample points is chosen large enough to allow for a zero padding 
anti-aliasing technique without cropping important parts of the spectrum.

.. GENERATED FROM PYTHON SOURCE LINES 74-89

.. code-block:: default


    grid = Grid( t_max = 34., t_num = 2**12)
    t, w = grid.t, grid.w
    model = NSE(w, b2=-1., gamma=1.)
    u_0t = 4./np.cosh(t)

    solver = SySSM(model.Lw, model.Nw)
    solver.set_initial_condition(w, FT(u_0t))
    solver.propagate(z_range = 3*np.pi/2, n_steps = 10000, n_skip = 50)
    z, utz = solver.z_, solver.utz

    plot_evolution( solver.z, grid.t, solver.utz,
        t_lim = (-5,5), w_lim = (-60,60), DO_T_LOG=False)





.. image:: /auto_tutorials/errors/images/sphx_glr_g_error_dealiasing_001.png
    :alt: $|u|^2/{\rm{max}}\left(|u|^2\right)$, $|u_\omega|^2/{\rm{max}}\left(|u_\omega|^2\right)$
    :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 90-103

**References:**

.. [B2001] J.P. Boyd, Chebychev and Fourier Spectral Methods, Dover, New York (2001)

.. [HCL2008] H. Holmas, D. Clamond, H.P. Langtangen, A pseudospectral Fourier
           method for a 1D incompressible two-fluid model, Int. J. Numer.
           Meth. Fluids 58 (2008) 639, https://doi.org/10.1002/fld.1772

.. [FCGK2005] D. Fuctus, D. Clamond, J. Grue, O. Kristiansen, An efficient 
           model for three-dimensional surface wave simulations Part I: Free
           space problems, J. Comp. Phys. 205 (2005) 665,
           https://doi.org/10.1016/j.jcp.2004.11.027 



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