# Source code for pysph.sph.integrator_step

    """Integrator steps for different schemes.

    Implement as many stages as needed.
    """

    ###############################################################################
    # `IntegratorStep` class
    ###############################################################################
    class IntegratorStep(object):
        """Subclass this and implement the methods ``initialize``, ``stage1`` etc.
        Use the same conventions as the equations.
        """
        def __repr__(self):
            return '%s()'%(self.__class__.__name__)

    ###############################################################################
    # `EulerStep` class
    ###############################################################################
    class EulerStep(IntegratorStep):
        """Fast but inaccurate integrator. Use this for testing"""
        def stage1(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, d_x, d_y,
                      d_z, d_rho, d_arho, dt):
            d_u[d_idx] += dt*d_au[d_idx]
            d_v[d_idx] += dt*d_av[d_idx]
            d_w[d_idx] += dt*d_aw[d_idx]

            d_x[d_idx] += dt*d_u[d_idx]
            d_y[d_idx] += dt*d_v[d_idx]
            d_z[d_idx] += dt*d_w[d_idx]

            d_rho[d_idx] += dt*d_arho[d_idx]

    ###############################################################################
    # `WCSPHStep` class
    ###############################################################################
    class WCSPHStep(IntegratorStep):
        """Standard Predictor Corrector integrator for the WCSPH formulation

        Use this integrator for WCSPH formulations. In the predictor step,
        the particles are advanced to `t + dt/2`. The particles are then
        advanced with the new force computed at this position.

        This integrator can be used in PEC or EPEC mode.

        The same integrator can be used for other problems. Like for
        example solid mechanics (see SolidMechStep)

        """
        def initialize(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho):
            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_rho0[d_idx] = d_rho[d_idx]

        def stage1(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                       d_aw, d_ax, d_ay, d_az, d_arho, dt):
            dtb2 = 0.5*dt
            d_u[d_idx] = d_u0[d_idx] + dtb2*d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dtb2*d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dtb2*d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dtb2 * d_ax[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dtb2 * d_ay[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dtb2 * d_az[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_rho[d_idx] = d_rho0[d_idx] + dtb2 * d_arho[d_idx]

        def stage2(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                       d_aw, d_ax, d_ay, d_az, d_arho, dt):

            d_u[d_idx] = d_u0[d_idx] + dt*d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt*d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt*d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dt * d_ax[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dt * d_ay[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dt * d_az[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_rho[d_idx] = d_rho0[d_idx] + dt * d_arho[d_idx]

    ###############################################################################
    # `WCSPHTVDRK3` Integrator
    ###############################################################################
    class WCSPHTVDRK3Step(IntegratorStep):
        r"""TVD RK3 stepper for WCSPH

        This integrator requires :math:`2` stages for the storage of the
        acceleration variables.

        """
        def initialize(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho):
            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_rho0[d_idx] = d_rho[d_idx]

        def stage1(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho,
                   d_au, d_av, d_aw, d_ax, d_ay, d_az, d_arho,
                   dt):

            # update velocities
            d_u[d_idx] = d_u0[d_idx] + dt * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt * d_aw[d_idx]

            # update positions
            d_x[d_idx] = d_x0[d_idx] + dt * d_ax[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dt * d_ay[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dt * d_az[d_idx]

            # update density
            d_rho[d_idx] = d_rho0[d_idx] + dt * d_arho[d_idx]

        def stage2(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                   d_aw, d_ax, d_ay, d_az, d_arho, dt):

            # update velocities
            d_u[d_idx] = 0.75*d_u0[d_idx] + 0.25*( d_u[d_idx] + dt * d_au[d_idx] )
            d_v[d_idx] = 0.75*d_v0[d_idx] + 0.25*( d_v[d_idx] + dt * d_av[d_idx] )
            d_w[d_idx] = 0.75*d_w0[d_idx] + 0.25*( d_w[d_idx] + dt * d_aw[d_idx] )

            # update positions
            d_x[d_idx] = 0.75*d_x0[d_idx] + 0.25*( d_x[d_idx] + dt * d_ax[d_idx] )
            d_y[d_idx] = 0.75*d_y0[d_idx] + 0.25*( d_y[d_idx] + dt * d_ay[d_idx] )
            d_z[d_idx] = 0.75*d_z0[d_idx] + 0.25*( d_z[d_idx] + dt * d_az[d_idx] )

            # Update density
            d_rho[d_idx] = 0.75*d_rho0[d_idx] + 0.25*( d_rho[d_idx] + dt * d_arho[d_idx] )

        def stage3(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                   d_aw, d_ax, d_ay, d_az, d_arho, dt):

            oneby3 = 1./3.
            twoby3 = 2./3.

            # update velocities
            d_u[d_idx] = oneby3*d_u0[d_idx] + twoby3*( d_u[d_idx] + dt * d_au[d_idx] )
            d_v[d_idx] = oneby3*d_v0[d_idx] + twoby3*( d_v[d_idx] + dt * d_av[d_idx] )
            d_w[d_idx] = oneby3*d_w0[d_idx] + twoby3*( d_w[d_idx] + dt * d_aw[d_idx] )

            # update positions
            d_x[d_idx] = oneby3*d_x0[d_idx] + twoby3*( d_x[d_idx] + dt * d_ax[d_idx] )
            d_y[d_idx] = oneby3*d_y0[d_idx] + twoby3*( d_y[d_idx] + dt * d_ay[d_idx] )
            d_z[d_idx] = oneby3*d_z0[d_idx] + twoby3*( d_z[d_idx] + dt * d_az[d_idx] )

            # Update density
            d_rho[d_idx] = oneby3*d_rho0[d_idx] + twoby3*( d_rho[d_idx] + dt * d_arho[d_idx] )

    ###############################################################################
    # `SolidMechStep` class
    ###############################################################################
    class SolidMechStep(IntegratorStep):
        """Predictor corrector Integrator for solid mechanics problems"""
        def initialize(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho,
                       d_s00, d_s01, d_s02, d_s11, d_s12, d_s22,
                       d_s000, d_s010, d_s020, d_s110, d_s120, d_s220,
                       d_e0, d_e):
            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_rho0[d_idx] = d_rho[d_idx]
            d_e0[d_idx] = d_e[d_idx]

            d_s000[d_idx] = d_s00[d_idx]
            d_s010[d_idx] = d_s01[d_idx]
            d_s020[d_idx] = d_s02[d_idx]
            d_s110[d_idx] = d_s11[d_idx]
            d_s120[d_idx] = d_s12[d_idx]
            d_s220[d_idx] = d_s22[d_idx]

        def stage1(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                      d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                      d_aw, d_ax, d_ay, d_az, d_arho, d_e, d_e0, d_ae,
                      d_s00, d_s01, d_s02, d_s11, d_s12, d_s22,
                      d_s000, d_s010, d_s020, d_s110, d_s120, d_s220,
                      d_as00, d_as01, d_as02, d_as11, d_as12, d_as22,
                      dt):
            dtb2 = 0.5*dt
            d_u[d_idx] = d_u0[d_idx] + dtb2*d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dtb2*d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dtb2*d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dtb2 * d_ax[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dtb2 * d_ay[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dtb2 * d_az[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_rho[d_idx] = d_rho0[d_idx] + dtb2 * d_arho[d_idx]
            d_e[d_idx] = d_e0[d_idx] + dtb2 * d_ae[d_idx]

            # update deviatoric stress components
            d_s00[d_idx] = d_s000[d_idx] + dtb2 * d_as00[d_idx]
            d_s01[d_idx] = d_s010[d_idx] + dtb2 * d_as01[d_idx]
            d_s02[d_idx] = d_s020[d_idx] + dtb2 * d_as02[d_idx]
            d_s11[d_idx] = d_s110[d_idx] + dtb2 * d_as11[d_idx]
            d_s12[d_idx] = d_s120[d_idx] + dtb2 * d_as12[d_idx]
            d_s22[d_idx] = d_s220[d_idx] + dtb2 * d_as22[d_idx]

        def stage2(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                      d_u0, d_v0, d_w0, d_u, d_v, d_w, d_rho0, d_rho, d_au, d_av,
                      d_aw, d_ax, d_ay, d_az, d_arho, d_e, d_ae, d_e0,
                      d_s00, d_s01, d_s02, d_s11, d_s12, d_s22,
                      d_s000, d_s010, d_s020, d_s110, d_s120, d_s220,
                      d_as00, d_as01, d_as02, d_as11, d_as12, d_as22,
                      dt):

            d_u[d_idx] = d_u0[d_idx] + dt*d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt*d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt*d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dt * d_ax[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dt * d_ay[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dt * d_az[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_rho[d_idx] = d_rho0[d_idx] + dt * d_arho[d_idx]
            d_e[d_idx] = d_e0[d_idx] + dt * d_ae[d_idx]

            # update deviatoric stress components
            d_s00[d_idx] = d_s000[d_idx] + dt * d_as00[d_idx]
            d_s01[d_idx] = d_s010[d_idx] + dt * d_as01[d_idx]
            d_s02[d_idx] = d_s020[d_idx] + dt * d_as02[d_idx]
            d_s11[d_idx] = d_s110[d_idx] + dt * d_as11[d_idx]
            d_s12[d_idx] = d_s120[d_idx] + dt * d_as12[d_idx]
            d_s22[d_idx] = d_s220[d_idx] + dt * d_as22[d_idx]

    ###############################################################################
    # `TransportVelocityStep` class
    ###############################################################################
    class TransportVelocityStep(IntegratorStep):
        """Integrator defined in 'A transport velocity formulation for
        smoothed particle hydrodynamics', 2013, JCP, 241, pp 292--307

        For a predictor-corrector style of integrator, this integrator
        should operate only in PEC mode.

        """
        def initialize(self):
            pass

        def stage1(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, d_uhat, d_auhat, d_vhat,
                      d_avhat, d_what, d_awhat, d_x, d_y, d_z, dt):
            dtb2 = 0.5*dt

            # velocity update eqn (14)
            d_u[d_idx] += dtb2*d_au[d_idx]
            d_v[d_idx] += dtb2*d_av[d_idx]
            d_w[d_idx] += dtb2*d_aw[d_idx]

            # advection velocity update eqn (15)
            d_uhat[d_idx] = d_u[d_idx] + dtb2*d_auhat[d_idx]
            d_vhat[d_idx] = d_v[d_idx] + dtb2*d_avhat[d_idx]
            d_what[d_idx] = d_w[d_idx] + dtb2*d_awhat[d_idx]

            # position update eqn (16)
            d_x[d_idx] += dt*d_uhat[d_idx]
            d_y[d_idx] += dt*d_vhat[d_idx]
            d_z[d_idx] += dt*d_what[d_idx]

        def stage2(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, d_vmag2, dt):
            dtb2 = 0.5*dt

            # corrector update eqn (17)
            d_u[d_idx] += dtb2*d_au[d_idx]
            d_v[d_idx] += dtb2*d_av[d_idx]
            d_w[d_idx] += dtb2*d_aw[d_idx]

            # magnitude of velocity squared
            d_vmag2[d_idx] = (d_u[d_idx]*d_u[d_idx] + d_v[d_idx]*d_v[d_idx] +
                              d_w[d_idx]*d_w[d_idx])

    ###############################################################################
    # `AdamiVerletStep` class
    ###############################################################################
    class AdamiVerletStep(IntegratorStep):
        """Verlet time integration described in `A generalized wall
        boundary condition for smoothed particle hydrodynamics` 2012, JCP,
        231, pp 7057--7075

        This integrator can operate in either PEC mode or in EPEC mode as
        described in the paper.

        """
        def initialize(self):
            pass

        def stage1(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, d_x, d_y, d_z, dt):
            dtb2 = 0.5*dt

            # velocity predictor eqn (14)
            d_u[d_idx] += dtb2*d_au[d_idx]
            d_v[d_idx] += dtb2*d_av[d_idx]
            d_w[d_idx] += dtb2*d_aw[d_idx]

            # position predictor eqn (15)
            d_x[d_idx] += dtb2*d_u[d_idx]
            d_y[d_idx] += dtb2*d_v[d_idx]
            d_z[d_idx] += dtb2*d_w[d_idx]

        def stage2(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, d_x, d_y, d_z,
                   d_rho, d_arho, d_vmag2, dt):
            dtb2 = 0.5*dt

            # position corrector eqn (17)
            d_x[d_idx] += dtb2*d_u[d_idx]
            d_y[d_idx] += dtb2*d_v[d_idx]
            d_z[d_idx] += dtb2*d_w[d_idx]

            # velocity corrector eqn (18)
            d_u[d_idx] += dtb2*d_au[d_idx]
            d_v[d_idx] += dtb2*d_av[d_idx]
            d_w[d_idx] += dtb2*d_aw[d_idx]

            # density corrector eqn (16)
            d_rho[d_idx] += dt * d_arho[d_idx]

            # magnitude of velocity squared
            d_vmag2[d_idx] = (d_u[d_idx]*d_u[d_idx] + d_v[d_idx]*d_v[d_idx] +
                              d_w[d_idx]*d_w[d_idx])

    ###############################################################################
    # `GasDFluidStep` class
    ###############################################################################
    class GasDFluidStep(IntegratorStep):
        """Predictor Corrector integrator for Gas-dynamics"""
        def initialize(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z, d_h,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e, d_e0, d_h0,
                       d_converged, d_omega, d_rho, d_rho0, d_alpha1, d_alpha2,
                       d_alpha10, d_alpha20):
            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_e0[d_idx] = d_e[d_idx]

            d_h0[d_idx] = d_h[d_idx]
            d_rho0[d_idx] = d_rho[d_idx]

            # set the converged attribute to 0 at the beginning of a Group
            d_converged[d_idx] = 0

            # likewise, we set the default omega (grad-h) terms to 1 at
            # the beginning of this Group.
            d_omega[d_idx] = 1.0

            d_alpha10[d_idx] = d_alpha1[d_idx]
            d_alpha20[d_idx] = d_alpha2[d_idx]

        def stage1(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e0, d_e, d_au, d_av,
                   d_aw, d_ae, d_rho, d_rho0, d_arho, d_h, d_h0, d_ah,
                   d_alpha1, d_aalpha1, d_alpha10,
                   d_alpha2, d_aalpha2, d_alpha20,
                   dt):
            dtb2 = 0.5*dt

            d_u[d_idx] = d_u0[d_idx] + dtb2 * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dtb2 * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dtb2 * d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dtb2 * d_u[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dtb2 * d_v[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dtb2 * d_w[d_idx]

            # update thermal energy
            d_e[d_idx] = d_e0[d_idx] + dtb2 * d_ae[d_idx]

            # predict density and smoothing lengths for faster
            # convergence. NNPS need not be explicitly updated since it
            # will be called at the end of the predictor stage.
            d_h[d_idx] = d_h0[d_idx] + dtb2 * d_ah[d_idx]
            d_rho[d_idx] = d_rho0[d_idx] + dtb2 * d_arho[d_idx]

            # update viscosity coefficients
            d_alpha1[d_idx] = d_alpha10[d_idx] + dtb2*d_aalpha1[d_idx]
            d_alpha2[d_idx] = d_alpha20[d_idx] + dtb2*d_aalpha2[d_idx]

        def stage2(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e0, d_e, d_au, d_av,
                   d_alpha1, d_aalpha1, d_alpha10,
                   d_alpha2, d_aalpha2, d_alpha20,
                   d_aw, d_ae, dt):

            d_u[d_idx] = d_u0[d_idx] + dt * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt * d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dt * d_u[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dt * d_v[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dt * d_w[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_e[d_idx] = d_e0[d_idx] + dt * d_ae[d_idx]

            # update viscosity coefficients
            d_alpha1[d_idx] = d_alpha10[d_idx] + dt*d_aalpha1[d_idx]
            d_alpha2[d_idx] = d_alpha20[d_idx] + dt*d_aalpha2[d_idx]

    class GSPHStep(IntegratorStep):

        def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, d_e,
                   d_au, d_av, d_aw, d_ae, dt):
            dtb2 = dt*0.5
            ustar = d_u[d_idx] + dtb2*d_au[d_idx]
            vstar = d_v[d_idx] + dtb2*d_av[d_idx]
            wstar = d_w[d_idx] + dtb2*d_aw[d_idx]

            d_u[d_idx] += dt*d_au[d_idx]
            d_v[d_idx] += dt*d_av[d_idx]
            d_w[d_idx] += dt*d_aw[d_idx]
            d_e[d_idx] += dt*(d_ae[d_idx]
                              - ustar*d_au[d_idx]
                              - vstar*d_av[d_idx]
                              - wstar*d_aw[d_idx])
            d_x[d_idx] += dt*ustar
            d_y[d_idx] += dt*vstar
            d_z[d_idx] += dt*wstar

    class ADKEStep(IntegratorStep):
        """Predictor Corrector integrator for Gas-dynamics ADKE"""
        def initialize(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                       d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e, d_e0,
                       d_rho, d_rho0):
            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_e0[d_idx] = d_e[d_idx]

            d_rho0[d_idx] = d_rho[d_idx]

        def stage1(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e0, d_e, d_au, d_av,
                   d_aw, d_ae, d_rho, d_rho0, d_arho, dt):
            dtb2 = 0.5*dt

            d_u[d_idx] = d_u0[d_idx] + dtb2 * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dtb2 * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dtb2 * d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dtb2 * d_u[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dtb2 * d_v[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dtb2 * d_w[d_idx]

            # update thermal energy
            d_e[d_idx] = d_e0[d_idx] + dtb2 * d_ae[d_idx]

        def stage2(self, d_idx, d_x0, d_y0, d_z0, d_x, d_y, d_z,
                   d_u0, d_v0, d_w0, d_u, d_v, d_w, d_e0, d_e, d_au, d_av,
                   d_aw, d_ae, dt):

            d_u[d_idx] = d_u0[d_idx] + dt * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt * d_aw[d_idx]

            d_x[d_idx] = d_x0[d_idx] + dt * d_u[d_idx]
            d_y[d_idx] = d_y0[d_idx] + dt * d_v[d_idx]
            d_z[d_idx] = d_z0[d_idx] + dt * d_w[d_idx]

            # Update densities and smoothing lengths from the accelerations
            d_e[d_idx] = d_e0[d_idx] + dt * d_ae[d_idx]

    ###############################################################################
    # `TwoStageRigidBodyStep` class
    ###############################################################################
    class TwoStageRigidBodyStep(IntegratorStep):
        """Simple rigid-body motion

        At each stage of the integrator, the prescribed velocity and
        accelerations are incremented by dt/2.

        Note that the time centered velocity is used for updating the
        particle positions. This ensures exact motion for a constant
        acceleration.

        """
        def initialize(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                       d_u, d_v, d_w, d_u0, d_v0, d_w0):

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

        def stage1(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                   d_u, d_v, d_w, d_u0, d_v0, d_w0, d_au, d_av, d_aw,
                   dt):

            dtb2 = 0.5*dt

            d_u[d_idx] = d_u0[d_idx] + dtb2 * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dtb2 * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dtb2 * d_aw[d_idx]

            # positions are updated based on the time centered velocity
            d_x[d_idx] = d_x0[d_idx] + dtb2 * 0.5 * (d_u[d_idx] + d_u0[d_idx])
            d_y[d_idx] = d_y0[d_idx] + dtb2 * 0.5 * (d_v[d_idx] + d_v0[d_idx])
            d_z[d_idx] = d_z0[d_idx] + dtb2 * 0.5 * (d_w[d_idx] + d_w0[d_idx])

        def stage2(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                   d_u, d_v, d_w, d_u0, d_v0, d_w0, d_au, d_av, d_aw,
                   dt):

            d_u[d_idx] = d_u0[d_idx] + dt * d_au[d_idx]
            d_v[d_idx] = d_v0[d_idx] + dt * d_av[d_idx]
            d_w[d_idx] = d_w0[d_idx] + dt * d_aw[d_idx]

            # positions are updated based on the time centered velocity
            d_x[d_idx] = d_x0[d_idx] + dt * 0.5 * (d_u[d_idx] + d_u0[d_idx])
            d_y[d_idx] = d_y0[d_idx] + dt * 0.5 * (d_v[d_idx] + d_v0[d_idx])
            d_z[d_idx] = d_z0[d_idx] + dt * 0.5 * (d_w[d_idx] + d_w0[d_idx])

    ###############################################################################
    # `OneStageRigidBodyStep` class
    ###############################################################################
    class OneStageRigidBodyStep(IntegratorStep):
        """Simple one stage rigid-body motion """
        def initialize(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                       d_u, d_v, d_w, d_u0, d_v0, d_w0):

            d_u0[d_idx] = d_u[d_idx]
            d_v0[d_idx] = d_v[d_idx]
            d_w0[d_idx] = d_w[d_idx]

            d_x0[d_idx] = d_x[d_idx]
            d_y0[d_idx] = d_y[d_idx]
            d_z0[d_idx] = d_z[d_idx]

        def stage1(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                    d_u, d_v, d_w, d_u0, d_v0, d_w0, d_au, d_av, d_aw,
                    dt):
            pass

        def stage2(self, d_idx, d_x, d_y, d_z, d_x0, d_y0, d_z0,
                    d_u, d_v, d_w, d_u0, d_v0, d_w0, d_au, d_av, d_aw,
                    dt):

            # update velocities
            d_u[d_idx] += dt * d_au[d_idx]
            d_v[d_idx] += dt * d_av[d_idx]
            d_w[d_idx] += dt * d_aw[d_idx]

            # upadte positions using time-centered velocity
            d_x[d_idx] += dt * 0.5 * (d_u[d_idx] + d_u0[d_idx])
            d_y[d_idx] += dt * 0.5 * (d_v[d_idx] + d_v0[d_idx])
            d_z[d_idx] += dt * 0.5 * (d_w[d_idx] + d_w0[d_idx])

    ###############################################################################
    # `VerletSymplecticWCSPHStep` class
    ###############################################################################
    class VerletSymplecticWCSPHStep(IntegratorStep):
        """Symplectic second order integrator described in the review
        paper by Monaghan:

        J. Monaghan, "Smoothed Particle Hydrodynamics", Reports on
        Progress in Physics, 2005, 68, pp 1703--1759 [JM05]

        Notes:

        This integrator should run in PEC mode since in the first stage,
        the positions are updated using the current velocity. The
        accelerations are then computed to advance to the full time step
        values.

        This version of the integrator does not update the density. That
        is, the summation density is used instead of the continuity
        equation.

        """

        def initialize(self):
            pass

        def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, dt):

            dtb2 = 0.5 * dt

            # Eq. (5.39) in [JM05]
            d_x[d_idx] += dtb2 * d_u[d_idx]
            d_y[d_idx] += dtb2 * d_v[d_idx]
            d_z[d_idx] += dtb2 * d_w[d_idx]

        def stage2(self, d_idx, d_x, d_y, d_z, d_ax, d_ay, d_az,
                   d_u, d_v, d_w, d_au, d_av, d_aw, dt):

            dtb2 = 0.5 * dt

            # Eq. (5.40) in [JM05]
            d_u[d_idx] += dt * d_au[d_idx]
            d_v[d_idx] += dt * d_av[d_idx]
            d_w[d_idx] += dt * d_aw[d_idx]

            # Eq. (5.41) in [JM05] using XSPH velocity correction
            d_x[d_idx] += dtb2 * d_ax[d_idx]
            d_y[d_idx] += dtb2 * d_ay[d_idx]
            d_z[d_idx] += dtb2 * d_az[d_idx]

    ###############################################################################
    # `VelocityVerletSymplecticWCSPHStep` class
    ###############################################################################
    class VelocityVerletSymplecticWCSPHStep(IntegratorStep):
        """Another symplectic second order integrator described in the
        review paper by Monaghan:

        J. Monaghan, "Smoothed Particle Hydrodynamics", Reports on
        Progress in Physics, 2005, 68, pp 1703--1759 [JM05]

        kick--drift--kick form of the verlet integrator

        """

        def initialize(self):
            pass

        def stage1(self, d_idx, d_u, d_v, d_w, d_au, d_av, d_aw, dt):

            dtb2 = 0.5 * dt

            # Eq. (5.51) in [JM05]
            d_u[d_idx] += dtb2 * d_au[d_idx]
            d_v[d_idx] += dtb2 * d_av[d_idx]
            d_w[d_idx] += dtb2 * d_aw[d_idx]

        def stage2(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w,
                   d_au, d_av, d_aw, dt):

            dtb2 = 0.5 * dt

            # Eq. (5.52) in [JM05]
            d_x[d_idx] += dt * d_u[d_idx]
            d_y[d_idx] += dt * d_v[d_idx]
            d_z[d_idx] += dt * d_w[d_idx]

            # Eq. (5.53) in [JM05]
            d_u[d_idx] += dtb2 * d_au[d_idx]
            d_v[d_idx] += dtb2 * d_av[d_idx]
            d_w[d_idx] += dtb2 * d_aw[d_idx]

    ###############################################################################
    # `InletOutletStep` class
    ###############################################################################
    class InletOutletStep(IntegratorStep):
        """A trivial integrator for the inlet/outlet particles
        """
        def initialize(self):
            pass

        def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, dt):
            dtb2 = 0.5*dt
            d_x[d_idx] += dtb2 * d_u[d_idx]
            d_y[d_idx] += dtb2 * d_v[d_idx]
            d_z[d_idx] += dtb2 * d_w[d_idx]

        def stage2(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, dt):
            dtb2 = 0.5*dt
            d_x[d_idx] += dtb2 * d_u[d_idx]
            d_y[d_idx] += dtb2 * d_v[d_idx]
            d_z[d_idx] += dtb2 * d_w[d_idx]

    ###############################################################################
    class LeapFrogStep(IntegratorStep):

        r"""Using this stepper with XSPH as implemented in
        `pysph.base.basic_equations.XSPHCorrection` is not directly possible and
        requires a nicer implementation where the correction alone is added to ``ax,
        ay, az``.
        """

        def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, d_ax, d_ay, d_az,
                   dt):
            d_x[d_idx] += 0.5 * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += 0.5 * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += 0.5 * dt * (d_w[d_idx] + d_az[d_idx])

        def stage2(self, d_idx, d_x, d_y, d_z, d_u, d_au, d_v, d_av,
                   d_w, d_aw, d_ax, d_ay, d_az,
                   d_rho, d_arho, d_e, d_ae, dt):
            d_u[d_idx] += dt * d_au[d_idx]
            d_v[d_idx] += dt * d_av[d_idx]
            d_w[d_idx] += dt * d_aw[d_idx]

            d_rho[d_idx] += dt * d_arho[d_idx]
            d_e[d_idx] += dt * d_ae[d_idx]

            d_x[d_idx] += 0.5 * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += 0.5 * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += 0.5 * dt * (d_w[d_idx] + d_az[d_idx])

    ###############################################################################
    class PEFRLStep(IntegratorStep):

        r"""Using this stepper with XSPH as implemented in
        `pysph.base.basic_equations.XSPHCorrection` is not directly possible and
        requires a nicer implementation where the correction alone is added to ``ax,
        ay, az``.
        """

        def stage1(self, d_idx, d_x, d_y, d_z, d_u, d_v, d_w, d_ax, d_ay,
                   d_az, dt):

            xi = 0.1786178958448091

            d_x[d_idx] += xi * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += xi * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += xi * dt * (d_w[d_idx] + d_az[d_idx])

        def stage2(self, d_idx, d_x, d_y, d_z, d_u, d_au, d_v, d_av,
                   d_w, d_aw, d_ax, d_ay, d_az,
                   d_rho, d_arho, d_e, d_ae, dt=0.0):

            lamda = -0.2123418310626054
            fac = (1. - 2.*lamda) / 2.

            d_u[d_idx] += fac * dt * d_au[d_idx]
            d_v[d_idx] += fac * dt * d_av[d_idx]
            d_w[d_idx] += fac * dt * d_aw[d_idx]

            d_rho[d_idx] += fac * dt * d_arho[d_idx]
            d_e[d_idx] += fac * dt * d_ae[d_idx]

            chi = -0.06626458266981849

            d_x[d_idx] += chi * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += chi * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += chi * dt * (d_w[d_idx] + d_az[d_idx])

        def stage3(self, d_idx, d_x, d_y, d_z, d_u, d_au, d_v, d_av,
                   d_w, d_aw, d_ax, d_ay, d_az,
                   d_rho, d_arho, d_e, d_ae, dt=0.0):

            lamda = -0.2123418310626054

            d_u[d_idx] += lamda * dt * d_au[d_idx]
            d_v[d_idx] += lamda * dt * d_av[d_idx]
            d_w[d_idx] += lamda * dt * d_aw[d_idx]

            d_rho[d_idx] += lamda * dt * d_arho[d_idx]
            d_e[d_idx] += lamda * dt * d_ae[d_idx]

            xi = +0.1786178958448091
            chi = -0.06626458266981849
            fac = 1. - 2.*(xi + chi)

            d_x[d_idx] += fac * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += fac * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += fac * dt * (d_w[d_idx] + d_az[d_idx])

        def stage4(self, d_idx, d_x, d_y, d_z, d_u, d_au, d_v, d_av,
                   d_w, d_aw, d_ax, d_ay, d_az,
                   d_rho, d_arho, d_e, d_ae, dt=0.0):

            lamda = -0.2123418310626054

            d_u[d_idx] += lamda * dt * d_au[d_idx]
            d_v[d_idx] += lamda * dt * d_av[d_idx]
            d_w[d_idx] += lamda * dt * d_aw[d_idx]

            d_rho[d_idx] += lamda * dt * d_arho[d_idx]
            d_e[d_idx] += lamda * dt * d_ae[d_idx]

            chi = -0.06626458266981849

            d_x[d_idx] += chi * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += chi * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += chi * dt * (d_w[d_idx] + d_az[d_idx])

        def stage5(self, d_idx, d_x, d_y, d_z, d_u, d_au, d_v, d_av,
                   d_w, d_aw, d_ax, d_ay, d_az,
                   d_rho, d_arho, d_e, d_ae, dt=0.0):

            lamda = -0.2123418310626054
            fac = (1. - 2.*lamda) / 2.

            d_u[d_idx] += fac * dt * d_au[d_idx]
            d_v[d_idx] += fac * dt * d_av[d_idx]
            d_w[d_idx] += fac * dt * d_aw[d_idx]

            d_rho[d_idx] += fac * dt * d_arho[d_idx]
            d_e[d_idx] += fac * dt * d_ae[d_idx]

            xi = +0.1786178958448091

            d_x[d_idx] += xi * dt * (d_u[d_idx] + d_ax[d_idx])
            d_y[d_idx] += xi * dt * (d_v[d_idx] + d_ay[d_idx])
            d_z[d_idx] += xi * dt * (d_w[d_idx] + d_az[d_idx])