def _set_mass_mtx(self, tol):
"""set up the mass matrix"""
self.M = numpy.matrix(numpy.zeros((self.n_nodes, self.n_nodes)))
# 1st step: handle boundary modes
self.M[0, 0] = 2. / 3.
self.M[0, -1] = self.M[-1, 0] = 1. / 3.
self.M[-1, -1] = 2. / 3.
if self.n_nodes > 2:
self.M[0, 1] = self.M[1, 0] = \
jacobi_orthogonal_constant(0, 1, 1) / 8.
self.M[1, -1] = self.M[-1, 1] = \
jacobi_orthogonal_constant(0, 1, 1) / 8.
if self.n_nodes > 3:
self.M[0, 2] = self.M[2, 0] = \
- jacobi_orthogonal_constant(1, 1, 1) / 16.
self.M[2, -1] = self.M[-1, 2] = \
jacobi_orthogonal_constant(1, 1, 1) / 16.
# 2nd step: handle interior modes
for i, p in enumerate(self.expn[1:-1]):
for j in range(i + 1, min(self.n_nodes - 1, i + 2 + 2), 2):
pi = (p * self.expn[j]).integral()
self.M[i + 1, j] = self.M[j, i + 1] = pi(1) - pi(-1)
def A(i):
def Ac(i, j):
return (i + 1) * (j + 1) / 16.
a1, a2, a3, a4 = jacobi_recr_coeffs(i - 1, 0, 0)
a1 /= a3
a4 /= a3
try:
b1, b2, b3, b4 = jacobi_recr_coeffs(i - 3, 0, 0)
b1 /= b3
b4 /= b3
except JacobiOrderError:
b1 = b3 = b4 = 0
except:
raise
c1, c2, c3, c4 = jacobi_recr_coeffs(i + 1, 0, 0)
c1 /= c3
c4 /= c3
try:
A1 = Ac(i, i-2) * \
a4 * b1 * jacobi_orthogonal_constant(i-2, 0, 0)
except JacobiOrderError:
A1 = 0
except:
raise
A2 = a1 * a1 * jacobi_orthogonal_constant(i, 0, 0)
try:
A2 += a4 * a4 * jacobi_orthogonal_constant(i - 2, 0, 0)
except JacobiOrderError:
A2 += 0
except:
raise
A2 *= Ac(i, i)
A3 = Ac(i, i + 2) * a1 * c4 * jacobi_orthogonal_constant(i, 0, 0)
return A1, A2, A3
def B(i):
def Bc(i, j):
return -(i + 1) * (j - 1) / 16.
a1, a2, a3, a4 = jacobi_recr_coeffs(i - 1, 0, 0)
a1 /= a3
a4 /= a3
B1 = 0
try:
B2 = Bc(i, i) * a4 * jacobi_orthogonal_constant(i - 2, 0, 0)
except JacobiOrderError:
B2 = 0
except:
raise
B3 = Bc(i, i + 2) * a1 * jacobi_orthogonal_constant(i, 0, 0)
return B1, B2, B3
def C(i):
def Cc(i, j):
return -(j + 1) * (i - 1) / 16.
try:
a1, a2, a3, a4 = jacobi_recr_coeffs(i - 2, 0, 0)
a1 /= a3
a4 /= a3
except JacobiOrderError:
a1 = a4 = 0
except:
raise
try:
C1 = Cc(i, i - 2) * a4 * jacobi_orthogonal_constant(
i - 3, 0, 0)
except JacobiOrderError:
C1 = 0
except:
raise
C2 = Cc(i, i) * a1 * jacobi_orthogonal_constant(i - 1, 0, 0)
C3 = 0
return C1, C2, C3
def D(i):
def Dc(i, j):
return (j - 1) * (i - 1) / 16.
D1 = 0
try:
D2 = Dc(i, i) * jacobi_orthogonal_constant(i - 2, 0, 0)
except JacobiOrderError:
D2 = 0
except:
raise
D3 = 0
return D1, D2, D3
def _set_weak_laplacian(self, tol=1e-12):
"""set up the mass matrix"""
self.wL = numpy.matrix(numpy.zeros((self.n_nodes, self.n_nodes)))
# boundary-boundary
self.wL[0, 0] = self.wL[-1, -1] = 0.5
self.wL[0, -1] = self.wL[-1, 0] = -0.5
# boundary-interior
# all zeros
# interior-interior
# only diagonal entities are non-zeros
for i in range(1, self.n_nodes - 1):
self.wL[i, i] = \
i * (i + 1) * jacobi_orthogonal_constant(i-1, 1, 1) / 16.