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I am trying to solve numerically the flow of a gas through a porous, spherically symmetric body. The non-dimensional equations read: $$ \frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\rho v\right)=g_P\\ v=-\frac{\partial \rho}{\partial r}\\ $$ where $\rho$ is the gas density, $t$ time, $r$ radius (between $0$ and outer radius $R_p$), $v$ gas (radial) velocity, $g_P$ gas production rate, and $\tau$ a non-dimensional parameter accounting for permeability among other things (actually all quantities are non-dimensional). Essentially, the first equation is the conservation of mass and the second is Darcy flow. There is also a perfect gas law hidden in the second equation. Anyway, physics is not the point here. The boundary conditions: $$ \frac{\partial\rho}{\partial r}(r=0)=0\\ \rho(r=R_p)=\rho_\infty $$ In order to solve this non-linear system, I use a Picard iteration where the second equation is solved for $v$ using finite difference, the first equation is then solved for $\rho$ also by finite difference using the calculated $v$ and an explicit time-stepping (checking for CFL condition). The process is then iterated until $\rho$ has converged, and time is then incremented.
It seems pretty straightforward, yet when it comes to running the code, oscillation rapidly appear, propagating from the outer boundary, where a pressure (i.e. density in the non-dimensional form) gradient builds up, as you can see on the figure below: <span class=$rho$ profile at $t=1$ for $\tau=0.1$, $\rho_\infty=0.1$, a time step of 1e-5 and a radial resolution of 100 grid points" /> I had the same issue with an implicit time-stepping, so I thought I'd stick to the simple version. The amplitude of the oscillations is not sensitive to the timestep, and shows little sensitivity to the radial resolution. I am using first order central FD for both equations. I thought it might be an accuracy issue and went for second order central FD for the first equation, which to some extent helped, but not enough. Also, I am not sure about how to go about implementing a second order scheme given my set of boundary conditions; what I did is switch from a central scheme to a forward/backward one when getting to the boundaries, but is doesn't feel very neat.
I can elaborate on the model if needed, I just didn't think it was required, please let me know if I should. What I need is a diagnostic really, because so far I have been trying different ways without much success! The code is provided below.
Thank you for the help!

"""
Non-dimensionalization:
r[non-dim]   = r[m]/R_out
rho[non-dim] = rho[kg/m³]/(deltaP/R/Tref)
p[non-dim]   = p[Pa]/deltaP
v[non-dim]   = v[m/s]
gP[non-dim]  = gP[kg/m³/s]/g0
t[non-dim]   = t[s]/tau_g

Timescales:
tau_g = R/M*T*deltaP/g0
tau_p = viscosity/permeability*R_out^2/deltaP

The physical parameter is tau_g/tau_p,
which quantifies the gas production
time-scale compared to the pressure
gradient-driven outgassing time-scale.

"""

import numpy as np
import matplotlib.pyplot as plt
import time
import copy

# Parameter
tau_g_tau_p = 0.1
P_out       = 0.1 # aka rho_infinite

# All quantities are non-dimensional
N        = 100               # resolution
r        = np.linspace(0,1,N) # radii
dr       = 1/(N-1)            # radius step
rho      = np.ones(N)*P_out   # density
p        = np.ones(N)*P_out   # pressure
gP       = 1.                 # gas production rate
t_max    = 1                  # final time
dt       = 1e-5               # time-step
t        = 0.                 # initial time

while t<t_max:

    rho_prev   = rho
    res = 1
    tol = 1e-5
    while res > tol:

        ##### Picard iteration #####

        ##### Solve Darcy Equation #####

        # calculate p on the staggered grid
        p_stag = (p[:-1]+p[1:])/2 # pressure on the staggered grid

        # compute Darcy flux CFD from staggered p
        v       = np.zeros(N)                  # initialize Darcy flux vector
        v[0]    = 0                            # center BC
        v[1:-1] = -(p_stag[1:]-p_stag[:-1])/dr # central finite-difference
        v[-1]   = -(P_out-p_stag[-1])/(dr/2)       # outer BC

        # CFL check
        if max(v)*dt>dr:
            raise Warning('Courant criterion not respected')

        ##### Solve Mass Conservation Equation ######
        rho_new = rho
        rho_new[1:-1] += gP*dt - tau_g_tau_p*dt/r[1:-1]**2*(r[2:]**2*rho_prev[2:]*v[2:]-r[:-2]**2*rho_prev[:-2]*v[:-2]) # central finite-difference

        # boundary conditions
        rho_new[0]  = rho_new[1] # center BC
        rho_new[-1] = P_out      # outer BC

        ##### Perfect gas law #####
        p = rho_new # non-dimensonal rho and p are the same
    
        # calculate residuals
        res = np.linalg.norm(rho-rho_new)
        rho = rho_new

    t += dt # time increment

# Plot final state
plt.figure()
plt.plot(r,p,label=r'$dt$='+str(dt)+',N='+str(N))
plt.xlabel('r [non-dim]')
plt.ylabel('rho [non-dim]')
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  • 1
    $\begingroup$ There is no reason to simmulate the Darcy flow separately. Subsitute it into the continuity equation and discretize the resulting second-order equations with finite differences. The problem you are having now is that you are applying central finite differences and you have to be careful to ensure the derivative approximation is consistent $\endgroup$
    – whpowell96
    May 31, 2023 at 21:50
  • $\begingroup$ *applying central difference twice $\endgroup$
    – whpowell96
    May 31, 2023 at 23:45

1 Answer 1

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Thank you for the tip whpowell96, I did as you suggested and it works fine. I also got inspired by this question. Here is exactly what I've done:

  • recast the two equations as one: $$\frac{\partial\rho}{\partial t}-\tau\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\rho\frac{\partial\rho}{\partial r} \right)=g_P$$ i.e., making use of $\rho\frac{\partial\rho}{\partial r}=\frac{1}{2}\frac{\partial\rho^2}{\partial r}$: $$\frac{\partial\rho}{\partial t}-\tau\left(\frac{2}{r}\frac{\partial\rho^2}{\partial r}+\frac{\partial^2\rho^2}{\partial r^2} \right)=g_P$$
  • then apply central finite difference for the first and second order derivative of $\rho^2$

It is straightforward for an explicit time-stepping, probably more complex for implicit. But explicit is fine for me. Here is the solution code:

"""
Non-dimensionalization:
r[non-dim]   = r[m]/R_out
rho[non-dim] = rho[kg/m³]/(deltaP/R/Tref)
p[non-dim]   = p[Pa]/deltaP
v[non-dim]   = v[m/s]
gP[non-dim]  = gP[kg/m³/s]/g0
t[non-dim]   = t[s]/tau_g
 
Timescales:
tau_g = R/M*T*deltaP/g0
tau_p = viscosity/permeability*R_out^2/deltaP
  
The physical parameter is tau_g/tau_p,
which quantifies the gas production
time-scale compared to the pressure
gradient-driven outgassing time-scale.

"""

import numpy as np
import matplotlib.pyplot as plt

# Parameter
tau_g_tau_p = 0.01
P_out       = 0.1 # aka rho_infinite

# All quantities are non-dimensional
N        = 20               # resolution
r        = np.linspace(0,1,N) # radii
dr       = 1/(N-1)            # radius step
rho      = np.ones(N)*P_out   # density
p        = np.ones(N)*P_out   # pressure
gP       = 1.                 # gas production rate
t_max    = 10                 # final time
dt       = 1e-2               # time-step
t        = 0.                 # initial time

start = time.time()
while t<t_max:

    # outer BC
    rho[-1] = P_out

    # FD for non-linear function rho**2
    rho[1:-1] += tau_g_tau_p*dt*((rho[2:]**2-2*rho[1:-1]**2+rho[:-2]**2)/2/dr \
                                +(rho[2:]**2-rho[:-2]**2)/2/dr/r[1:-1])       \
                                +gP*dt

    # inner BC
    rho[0] = rho[1]

    # CFL check
    v = -np.gradient(rho)
    if max(v)*dt>dr:
       raise Warning('Courant criterion not respected')

    t += dt # time increment

# Plot final state
plt.plot(r,rho,label=r'$dt$='+str(dt)+',N='+str(N))
plt.xlabel('r [non-dim]')
plt.ylabel('rho [non-dim]')
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