Neumann BCs and Singular Error

[Jake Bowhay]

Hi All,

I am trying to solve the following system of equations:

\frac{\partial u}{\partial t} &= \nabla^2u + a - (b + 1)u + u^2v,\\
 \frac{\partial v}{\partial t} &= D\nabla^2v + bu - u^2v

with the boundary conditions \mathbf{n}\cdot\nabla\mathbf{u}=0

I have introduced tau variables however I am getting "Factor is exactly singular". I am not sure if this is because dx(u)(x=0) is singular or because I need a Gauge Conditions?

Thanks in advance,

Jake

import numpy as np
import matplotlib.pyplot as plt
import dedalus.public as d3

# equation parameters
a = 1
b = 2
D = 1

# define domain and discretisation
coords = d3.CartesianCoordinates('x', 'y')
dist = d3.Distributor(coords, dtype=np.float64)
xbasis = d3.Chebyshev(coords['x'], size=32, bounds=(0,1), dealias=3/2)
ybasis = d3.Chebyshev(coords['y'], size=32, bounds=(0,1), dealias=3/2)

# Field quantities
u = dist.Field(name='u', bases=(xbasis, ybasis))
v = dist.Field(name='v', bases=(xbasis, ybasis))
tau1 = dist.VectorField(coords, name='tau1', bases=(xbasis))
tau2 = dist.VectorField(coords, name='tau2', bases=(ybasis))
tau3 = dist.VectorField(coords, name='tau3', bases=(xbasis))
tau4 = dist.VectorField(coords, name='tau4', bases=(ybasis))
problem = d3.IVP([u, v, tau1, tau2, tau3, tau4], namespace=locals())

# Substitutions
dx = lambda A: d3.Differentiate(A, coords['x'])
dy = lambda A: d3.Differentiate(A, coords['y'])

# Equations
problem.add_equation("dt(u) = lap(u) + a - (b + 1)*u + u*u*v")
problem.add_equation("dt(v) = D*lap(v) + b*u - u*u*v")

# Boundary Condition
problem.add_equation("dx(u)(x=0) = 0")
problem.add_equation("dx(v)(x=0) = 0")
problem.add_equation("dx(u)(x=1) = 0")
problem.add_equation("dx(v)(x=1) = 0")
problem.add_equation("dy(u)(y=0) = 0")
problem.add_equation("dy(v)(y=0) = 0")
problem.add_equation("dy(u)(y=1) = 0")
problem.add_equation("dy(v)(y=1) = 0")

# Initial Conditions
x = dist.local_grid(xbasis)
u['g'] = x
v['g'] = 1 - x

# Setup solver
solver = problem.build_solver(d3.SBDF3)
solver.stop_sim_time = 10

u.change_scales(1)
u_list = [np.copy(u['g'])]
t_list = [solver.sim_time]

# Main loop
timestep = 0.05
while solver.proceed:
    solver.step(timestep)
    if solver.iteration % 10 == 0:
        u.change_scales(1)
        u_list.append(np.copy(u['g']))
        t_list.append(solver.sim_time)

[Jeffrey S. Oishi]

Hi Jake,

Double Chebyshev problems, while technically possible in Dedalus, are not supported right now, partly because the taus are very non-trivial and an active research problem. You can read about some of the issues in this paper:

https://arxiv.org/pdf/2211.17259.pdf

We welcome community contributions to the understanding of double Chebyshev taus, but we currently cannot offer specific support on them.

Jeff

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[Jake Bowhay]

Hi Jeff,

Many thanks for the quick reply.

Regards,

Jake

[Shuai Dong]

Hi Jeff,

Is there any development in the double-chebyshev  expansion in Dedalus project? any examples?

best regards,

Simon