def pde_fem_hDt(u0, T, a, Nref, kappa, neref, L, epsilon, fhandle):
"""
A3.12 Page 123
"""
ne = neref // L
h = a / ne
nvtx = ne + 1
dtref = T / Nref
Dt = kappa * dtref
t = np.linspace(0, T, Nref // kappa)
p = epsilon
q = 1
f = 1
uh, A, b, K, M = ch2.oned_linear_FEM(ne, p, q, f)
EE = M + Dt * K
ZM = 0
# set initial condition
ut = np.zeros((nvtx, Nref // kappa + 1))
ut[:, 1] = u0
u = np.copy(u0)
EEinv = sparse.linalg.factorized(EE)
for n in range(Nref // kappa): # time loop
fu = fhandle(u)
b = oned_linear_FEM_b(ne, h, fu)
u_new = EEinv(M.dot(u[1:-1]) + Dt * b)
u = np.hstack([ZM, u_new, ZM])
ut[:, n + 1] = u
return t, ut
def spde_fem_MhDt(u0,T,a,Nref,kappa,neref,L,
epsilon,fhandle,ghandle,r,M,s0):
"""
Alg 10.12 Page 473
"""
ne=neref // L; assert(ne%1 == 0)
h=(a / ne); nvtx=ne + 1
dtref=T / Nref; Dt=kappa * dtref
t=np.linspace(0,T,Nref//kappa+1)
p=epsilon * np.ones(ne); q=np.ones(ne); f=np.ones(ne)
uh,A,b,KK,MM=ch2.oned_linear_FEM(ne,p,q,f)
EE=MM + Dt * KK
#
ZM=np.zeros((M,1))
#
bj=get_onedD_bj(dtref,neref,a,r); bj[ne:-1]=0
iFspace=0
#
u=np.matlib.repmat(u0,M,1)
ut=np.zeros((nvtx,Nref // kappa + 1)); ut[:,0]=u[0,:]
b=np.zeros(ne-1); gdw=np.copy(b)
EEinv=sparse.linalg.factorized(EE)
for k in range(Nref // kappa):
dWJ=get_onedD_dW(bj,kappa,iFspace,M,s0)
dWL=(np.hstack([ZM,dWJ,ZM])); dWL=dWL[:,::L]
gdW=ghandle(u)*dWL
fu=fhandle(u)
for m in range(M):
b=ch3.oned_linear_FEM_b(ne,h,fu[m,:])
gdw=ch3.oned_linear_FEM_b(ne,h,gdW[m,:])
u1=EEinv(MM.dot(u[m,1:-1]) + Dt * b+gdw)
u[m,:]=np.hstack([0,u1,0])
ut[:,k + 1]=u[-1,:]
return t,u,ut
def pde_fem(u0, T, a, N, ne, epsilon, fhandle):
"""
A3.11 Page 122
"""
h = a / ne
nvtx = ne + 1
Dt = T / N
t = np.linspace(0, T, N + 1)
p = epsilon
q = 1
f = 1
uh, A, b, KK, MM = ch2.oned_linear_FEM(ne, p, q, f)
EE = (MM + Dt * KK)
ZM = 0
# set initial condition
ut = np.zeros((nvtx, N + 1))
ut[:, 0] = u0
u = np.copy(u0)
EEinv = sparse.linalg.factorized(EE)
for n in range(N): # time loop
fu = fhandle(u)
b = oned_linear_FEM_b(ne, h, fu)
u_new = EEinv(MM.dot(u[1:-1]) + Dt * b)
u = np.hstack([ZM, u_new, ZM])
ut[:, n + 1] = u
return t, ut