Periodic boundary condition in EVP (D3)

[Ziyan Zhu]

Hello, 

I am a bit confused about how to impose periodic boundary conditions in Dedalus 3. 

I am interested in solving the spectrum of shallow water equations with a periodic boundary. I could get D2 and D3 to match with a no-slip boundary condition but not with periodic boundary conditions. Below is my working setup for the no-slip boundary condition. I would like to impose the fields u and v to be periodic. From the documentation and examples, it seems like the periodic boundary condition is implemented through 'integ(u) = 0' but I am having trouble applying it to my specific case. Any help is greatly appreciated! 

# Global parameters
Ny = 51 # number of points in y direction
Ly = 10*np.pi #  width the domain
kmax = np.pi # horizontal wave number extrema
kx_global = np.linspace(-kmax,kmax,100)

ky = 0.0
N = 1

def problem_builder(kx):
    # Create bases and domain
    ycoord = d3.Coordinate('y')
    dist = d3.Distributor(ycoord, dtype=np.complex128)
    ybasis = d3.Chebyshev(ycoord, size=Ny, bounds=(-Ly/2, Ly/2))
    y = dist.local_grids(ybasis) # grid

    # Fields
    u = dist.Field(name='u', bases=ybasis)
    v = dist.Field(name='v', bases=ybasis)
    h = dist.Field(name='h', bases=ybasis)
    tau_1 = dist.Field(name='tau_1')
    tau_2 = dist.Field(name='tau_2')
    omega = dist.Field(name='omega')

    # Substitution
    dy = lambda A: d3.Differentiate(A, ycoord)
    dx = lambda A: 1j*kx*A
    dt = lambda A: -1j*omega*A
    lift_basis = ybasis.derivative_basis(1)
    lift = lambda A: d3.Lift(A, lift_basis, -1)
    ky = 1

    # define non-constant coefficients
    f = dist.Field(bases=ybasis)
    f['g'] = np.sin(2*np.pi*y[0]/Ly)

    problem = d3.EVP([u, v, h, tau_1, tau_2], eigenvalue=omega, namespace=locals())

    problem.add_equation("dt(u) + dx(h) - f * v = 0")
    problem.add_equation("dt(v) + dy(h) + f * u + lift(tau_1) = 0")
    problem.add_equation("dt(h) + dx(u) + dy(v) + lift(tau_2) = 0")

#     # no-slip boundary condition
    problem.add_equation("v(y='left') = 0")
    problem.add_equation("v(y='right') = 0")

    solver = problem.build_solver()
    return solver, ybasis, dist

Best,

Zoe 

[Daniel Lecoanet]

Hi Zoe,

If you want to run a periodic simulation, I would recommend that you use a Fourier basis.

Daniel

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