def stencil_diff2(nr, pos, coeffs = None):
"""Create stencil matrix for a zero 2nd derivative"""
assert(coeffs.get('a', None) is not None)
assert(coeffs.get('b', None) is not None)
assert(coeffs.get('c', None) is None)
ns = np.arange(0,nr,1)
if pos == 0:
offsets = [-2, 0]
# Generate subdiagonal
def d_1(n):
return -(n - 3.0)*(n - 2.0)**2/(n**2*(n + 1.0))
else:
offsets = [-1, 0]
# Generate subdiagonal
def d_1(n):
return -pos*(n - 2.0)*(n - 1.0)/(n*(n + 1.0))
# Generate diagonal
def d0(n):
return np.ones(n.shape)
ds = [d_1, d0]
diags = utils.build_diagonals(ns, -1, ds, offsets, None, False)
diags[-1] = diags[-1][0:nr+offsets[0]]
return spsp.diags(diags, offsets, (nr,nr+offsets[0]))
def stencil_value(nr, pos, coeffs = None):
"""Create stencil matrix for a zero boundary value"""
assert(coeffs.get('a', None) is not None)
assert(coeffs.get('b', None) is not None)
assert(coeffs.get('c', None) is None)
ns = np.arange(0,nr,1)
if pos == 0:
offsets = [-2, 0]
sgn = -1.0
else:
offsets = [-1, 0]
sgn = -pos
# Generate subdiagonal
def d_1(n):
return galerkin_c(n+offsets[0])*sgn
# Generate diagonal
def d0(n):
return np.ones(n.shape)
ds = [d_1, d0]
diags = utils.build_diagonals(ns, -1, ds, offsets, None, False)
diags[-1] = diags[-1][0:nr+offsets[0]]
return spsp.diags(diags, offsets, (nr,nr+offsets[0]))
def stencil_insulating(nr, pos, coeffs = None):
"""Create stencil matrix for an insulating boundary"""
assert(coeffs.get('a', None) is not None)
assert(coeffs.get('b', None) is not None)
assert(coeffs.get('c', None) is None)
assert(coeffs.get('l', None) is not None)
assert(pos == 0)
a = coeffs['a']
b = coeffs['b']
l = coeffs['l']
ns = np.arange(0,nr,1)
offsets = [-2, -1, 0]
# Generate 2nd subdiagonal
def d_2(n):
val_num = a**2*(2.0*l*(l+1.0)-2.0*(n-3.0)*n*((n-3.0)*n+5.0)-13.0)+a*b*(2.0*l+1.0)*(2.0*(n-3.0)*n+5.0)+2.0*b**2*(n**2-3.0*n+2.0)**2
val_den = a**2*(2.0*l*(l+1.0)-2.0*(n-1.0)*n*((n-1.0)*n+1.0)-1.0)+a*b*(2.0*l+1.0)*(2.0*(n-1.0)*n+1.0)+2.0*b**2*(n-1.0)**2.0*n**2
val = -val_num/val_den
for i,j in enumerate(n):
if j == 2:
corr_num = a*(a*(2.0*l**2+2.0*l-1.0)+2.0*b*l+b)
corr_den = 2.0*(a**2*(2.0*l**2+2.0*l-13.0)+5.0*a*(2.0*b*l+b)+8.0*b**2)
val[i] = -corr_num/corr_den
if j > 2:
break
return val
# Generate 1st subdiagonal
def d_1(n):
val_num = 4.0*a*n*((2.0*l+1.0)*a - b)
val_den = a**2*(2.0*l*(l+1.0)-2.0*n*(n+1.0)*(n**2+n+1.0)-1.0)+a*b*(2.0*l+1.0)*(2.0*n*(n+1.0)+1.0)+2.0*b**2*n**2*(n+1.0)**2
val = val_num/val_den
for i,j in enumerate(n):
if j == 1:
corr_num = 2.0*a*(2.0*a*l+a-b)
corr_den = a**2*(2.0*l**2+2.0*l-13.0)+5.0*a*(2.0*b*l+b)+8.0*b**2
val[i] = corr_num/corr_den
if j > 1:
break
return val
# Generate diagonal
def d0(n):
return np.ones(n.shape)
ds = [d_2, d_1, d0]
diags = utils.build_diagonals(ns, -1, ds, offsets, None, False)
diags[-1] = diags[-1][0:nr+offsets[0]]
return spsp.diags(diags, offsets, (nr,nr+offsets[0]))
def stencil_rdiffdivr(nr, pos, coeffs = None):
"""Create stencil matrix for a zero r D 1/r derivative"""
assert(coeffs.get('a', None) is not None)
assert(coeffs.get('b', None) is not None)
assert(coeffs.get('c', None) is None)
assert(pos == 0)
a = coeffs['a']
b = coeffs['b']
ns = np.arange(0,nr,1)
offsets = [-2, -1, 0]
# Generate 2nd subdiagonal
def d_2(n):
#val_num = a**2*((n - 3.0)**2*n**2 - 2.0) - b**2*(n**2 - 3.0*n + 2.0)**2
#val_den = -a**2*((n - 2.0)*n - 1.0)*(n**2 - 2.0) + b**2*(n - 1.0)**2*n**2
val_num = (n - 2.0)*(b**2*(n - 2.0)*(n - 1.0) - a**2*(n - 3.0)*n)
val_den = n*(a**2*(n - 2.0)*(n + 1.0) - b**2*(n - 1.0)*n)
val = val_num/val_den
for i,j in enumerate(n):
if j == 2:
#val[i] = a**2/(2.0*a**2 + 4.0*b**2)
val[i] = 0
if j > 2:
break
return val
# Generate 1st subdiagonal
def d_1(n):
#val_num = -8.0*a*b*n
#val_den = a**2*(n**2 - 2.0)*(n*(n + 2.0) - 1.0) - b**2*n**2*(n + 1.0)**2
val_num = -4.0*a*b
val_den = (n + 1.0)*(a**2*(n**2 + n - 2.0) - b**2*n*(n + 1.0))
val = val_num/val_den
for i,j in enumerate(n):
if j == 1:
#val[i] = 2.0*a*b/(a**2 + 2.0*b**2)
val[i] = a/(2.0*b)
if j > 1:
break
return val
# Generate diagonal
def d0(n):
return np.ones(n.shape)
ds = [d_2, d_1, d0]
diags = utils.build_diagonals(ns, -1, ds, offsets, None, False)
diags[-1] = diags[-1][0:nr+offsets[0]]
return spsp.diags(diags, offsets, (nr,nr+offsets[0]))
def stencil_value_diff2(nr, pos, coeffs = None):
"""Create stencil matrix for a zero boundary value and a zero 2nd derivative"""
assert(coeffs.get('a', None) is not None)
assert(coeffs.get('b', None) is not None)
assert(coeffs.get('c', None) is None)
assert(pos == 0)
ns = np.arange(0,nr,1)
offsets = [-4, -2, 0]
# Generate 2nd subdiagonal
def d_2(n):
val_num = (n - 3.0)*(2.0*n**2 - 12.0*n + 19.0)
val_den = (n - 1.0)*(2.0*n**2 - 4.0*n + 3.0)
val = val_num/val_den
for i,j in enumerate(n):
if j == 4:
val[i] = 1.0/38.0
if j > 4:
break
return val
# Generate 1st subdiagonal
def d_1(n):
val_num = -2.0*n*(2.0*n**2 + 7.0)
val_den = (n + 1.0)*(2.0*n**2 + 4.0*n + 3.0)
val = val_num/val_den
for i,j in enumerate(n):
if j == 2:
val[i] = -10.0/19.0
if j > 2:
break
return val
# Generate diagonal
def d0(n):
return np.ones(n.shape)
ds = [d_2, d_1, d0]
diags = utils.build_diagonals(ns, -1, ds, offsets, None, False)
diags[-1] = diags[-1][0:nr+offsets[0]]
return spsp.diags(diags, offsets, (nr,nr+offsets[0]))