def test_grad_P_varying_qn():
NX = 32 #const.NX
xmin = 0.0 #const.xmin
xmax = 2*np.pi #const.xmax
dx = xmax / NX
x = np.arange(xmin, xmax, dx/100.) # Simulation domain space 0,NX (normalized to grid)
k = 2.0
# Physical location of nodes
E_nodes = (np.arange(NX + 3) - 0.5) * dx
B_nodes = (np.arange(NX + 3) - 1.0) * dx
# Set analytic solutions (input/output)
qn_input = const.q*(np.sin(k*E_nodes) + 1) # Analytic input at node points (number density varying)
te_input = np.ones(NX + 3)*const.Te0
pe_anal = const.kB * const.Te0 * k*np.cos(k*B_nodes) # Analytic solution at nodes
grad_P_anal= const.kB * const.Te0 * k*np.cos(k*x) # Highly sampled output (derivative)
# Finite differences
grad_PB, grad_P = fields.get_grad_P(qn_input, te_input, DX=dx)
r2 = r_squared(grad_P[1:-2], pe_anal[1:-2])
## PLOT ##
plot = False
if plot == True:
plt.figure(figsize=(15, 15))
marker_size = None
plt.plot(x, grad_P_anal, linestyle=':', c='b', label='Analytic Solution')
plt.scatter(B_nodes, pe_anal, marker='o', c='k', s=marker_size, label='Node Solution')
plt.scatter(B_nodes, grad_PB, marker='x', c='b', s=marker_size, label='Finite Difference (on B)')
plt.scatter(E_nodes, grad_P, marker='x', c='r', s=marker_size, label='Finite Difference (on E)')
plt.title(r'Test of $\nabla p_e$')
for kk in range(NX + 3):
plt.axvline(E_nodes[kk], linestyle='--', c='r', alpha=0.2)
plt.axvline(B_nodes[kk], linestyle='--', c='b', alpha=0.2)
plt.axvline(xmin, linestyle='-', c='k', alpha=0.2)
plt.axvline(xmax, linestyle='-', c='k', alpha=0.2)
plt.gcf().text(0.15, 0.93, '$R^2 = %.4f$' % r2)
plt.xlim(xmin - 1.5*dx, xmax + 2*dx)
plt.legend()
return
def set_equilibrium_te0(q_dens, Te0):
'''
Modifies the initial Te array to allow grad(P_e) = grad(nkT) = 0
Iterative? Analytic?
NOTE: Removed factors 2dx from qdens_gradient and m_arr, since they cancel out
Note: Could probably calculate Te0 from some sort of density/temperature relationship
with the ions, rather than defining it a priori, for now it should be ok (set via beta
same as cold ions)
'''
qdens_gradient = np.zeros(NC , dtype=np.float64)
# Get density gradient: Central differencing, internal points
for ii in nb.prange(1, NC - 1):
qdens_gradient[ii] = (q_dens[ii + 1] - q_dens[ii - 1])
# Forwards/Backwards difference at physical boundaries (In this case, the gradient will be zero)
qdens_gradient[0] = 0
qdens_gradient[NC - 1] = 0
m_arr = (qdens_gradient/q_dens)
# Construct solution array to work out Te
soln_array = np.zeros((NC, NC), dtype=np.float64)
ans_arr = np.zeros(NC, dtype=np.float64)
# Construct central points (Centered finite difference with constant)
for ii in range(1, NC-1):
soln_array[ii, ii - 1] = -1.0
soln_array[ii, ii ] = 1.0 * m_arr[ii]
soln_array[ii, ii + 1] = 1.0
# Enter boundary points : Neumann boundary conditions
soln_array[0, 0] = 1.0
soln_array[NC - 1, NC - 1] = 1.0
ans_arr[0] = Te0_scalar
ans_arr[NC - 1] = Te0_scalar
Te0[:] = np.dot(np.linalg.inv(soln_array), ans_arr)
if False:
# Test: This should be zero if working
grad_P = np.zeros(NC , dtype=np.float64)
temp = np.zeros(NC + 1, dtype=np.float64)
fields.get_grad_P(q_dens, Te0, grad_P, temp)
import sys
fig, axes = plt.subplots(4, sharex=True, figsize=(15, 10))
axes[0].set_title('Initial Temp/Dens with zero derivative at ND-NX interface')
axes[0].plot(q_dens / (q*ne))
axes[0].set_ylabel('ne / ne0')
axes[1].plot(qdens_gradient)
axes[1].set_ylabel('dne/dx')
axes[2].plot(Te0)
axes[2].set_ylabel('Te')
axes[3].plot(grad_P)
axes[3].set_ylabel('grad(P)')
axes[3].set_xlabel('Cell number')
sys.exit()
return