Project: s41467-024-51560-x | Function: _rk546m (python)

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**DESCRIPTION:** Alternative Dormand-Prince implementation for solving nonlinear optical propagation equations without interaction picture, providing flexibility in numerical methods for simulating the frequency conversion processes in periodically poled lithium niobate waveguides

**KEY TERMS:** {
  "field": "optical field amplitude A(z)",
  "dz": "propagation step Δz along waveguide",
  "k_1 through k_6": "Runge-Kutta stage derivatives",
  "sol_5, sol_4": "5th and 4th order accuracy solutions",
  "dispersion_step": "dispersion propagation operator",
  "nonlinear_term": "χ^(2) and χ^(3) nonlinear interactions"
}

**CONTEXT CODE:**         """
        if method is None:
            if hasattr(self.model, '_recommended_solver'):
                return self.model._recommended_solver
            else:
                return 'DP5IP'
        else:
            return method
        
    def _compute_error(self, field_1, field_2):
        """
        Error estimation for adaptive step size

        Parameters
        ----------
        field_1 : array
            Reference field.
        field_2 : array
            Test field.

        Returns
        -------
        float
            Error estimate between the two fields.
        """
        raise NotImplementedError("Function implementation has been masked")
    
    def _dormand_prince_step(self, field, dz):
        """
        Dormand Prince order 5(4) adaptive scheme in the interaction picture. This
        new stepper uses the interaction picture in an embedded scheme of order 5.
        Tests have shown it to be up to four times faster than RK4IP.

        Parameters
        ----------
        field : array
            Current field.
        dz : float
            Current step size.

        Returns
        -------
        array
            New field estimate.
        
        float
            Error estimate.
        """
        raise NotImplementedError("Function implementation has been masked")
    
    def _rk546m(self, field, dz):
        """
        Some other Dormand-Prince method that has one less step but does not
        leverage the interaction picture and does not have the FSAL property.
        Usually worse than the other dormand-prince method.
        References:
            DOI: 10.1016/0771-050X(80)90013-3

        Parameters
        ----------
        field : array
            Current field.
        dz : float
            Current step size.

        Returns
        -------
        array
            New field estimate.
        
        float
            Error estimate.
        """
        raise NotImplementedError("Function implementation has been masked")
    
    def _adams_bashforth54_step(self, field, dz):
        """To be implemented"""
        pass

    def _rk4ip_single_step(self, field, dz):
        """
        Runge-Kutta order 4 step in the interaction picture (RK4IP)

        Parameters
        ----------
        field : array
            Current field.
        dz : float
            Current step size.

        Returns
        -------
        array
            New field estimate.
        """
        dispersion_op = numbaexp(-1j*self.model.waveguide.k*dz/2)
        a_int = dispersion_op*field
        k_1 = dz*dispersion_op*self.model.nonlinear_term(field)
        k_2 = dz*self.model.nonlinear_term(a_int + k_1/2)
        k_3 = dz*self.model.nonlinear_term(a_int + k_2/2)
        k_4 = dz*self.model.nonlinear_term(dispersion_op*(a_int + k_3))
        return dispersion_op*(a_int + k_1/6 + k_2/3 + k_3/3) + k_4/6
    
    def _rk4ip_step(self, field, dz):
        """
        Runge-Kutta order 4 adaptive scheme in the interaction picture (RK4IP)
        with step doubling and combining. Provided for comparison but slower
        than the dormand-prince method.
        References:
            DOI: 10.1109/JLT.2007.909373
            DOI: 10.1109/JLT.2009.2021538

        Parameters
        ----------
        field : array
            Current field.
        dz : float
            Current step size.

        Returns
        -------
        array
            New field estimate.
        
        float
            Error estimate.
        """
        sol_single = self._rk4ip_single_step(field, dz)
        sol_double = self._rk4ip_single_step(field, dz/2)
        sol_double = self._rk4ip_single_step(sol_double, dz/2)
        error = self._compute_error(sol_double, sol_single)
        sol_5 = (sol_double*16 - sol_single)/15
        return sol_5, error
    
    def _erk4ip_step(self, field, dz):
        """
        Embedded Runge-Kutta order 4(3) adaptive scheme in the interaction
        picture (ERK4IP). Can be faster than DP5IP if accuracy is irrelevant,
        for example in SPM only cases. Otherwise, DP5IP performs better.
        References:
            DOI: 10.1016/j.cpc.2012.12.020

        Parameters
        ----------
        field : array
            Current field.
        dz : float
            Current step size.

        Returns


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