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averaged_sw_diff.py
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averaged_sw_diff.py
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from cheby_exp import *
from firedrake import *
from firedrake.petsc import PETSc
from math import pi
from math import ceil
from timestepping_methods import *
from latlon import *
import numpy as np
import argparse
#get command arguments
parser = argparse.ArgumentParser(description='Williamson 5 testcase for averaged propagator.')
parser.add_argument('--ref_level', type=int, default=4, help='Refinement level of icosahedral grid. Default 4.')
parser.add_argument('--space_parallel', type=int, default=4, help='Default 4.')
parser.add_argument('--tmax', type=float, default=1200, help='Final time in hours. Default 24x50=1200.')
parser.add_argument('--dumpt', type=float, default=24, help='Dump time in hours. Default 24.')
parser.add_argument('--normt', type=float, default=6, help='Collect norms every normt (in hours). Default 6.')
parser.add_argument('--dt', type=float, default=0.5, help='Timestep for the averaged model in hours. Default 0.5.')
parser.add_argument('--dts', type=float, default=0.0125, help='Timestep for the standard model in hours. Default 0.0125.')
parser.add_argument('--rho', type=float, default=1, help='Averaging window width as a multiple of dt. Default 1.')
parser.add_argument('--linear', action='store_false', dest='nonlinear', help='Run linear model if present, otherwise run nonlinear model')
parser.add_argument('--Mbar', action='store_true', dest='get_Mbar', help='Compute suitable Mbar, print it and exit.')
parser.add_argument('--filter', action='store_true', help='Use a filter in the averaging exponential')
parser.add_argument('--filter2', action='store_true', help='Use a filter for cheby2')
parser.add_argument('--filter_val', type=float, default=0.75, help='Cut-off for filter')
parser.add_argument('--filter_freq', action='store_true', help='Cut-off frequency for filter')
parser.add_argument('--ppp', type=float, default=4, help='Points per time-period for averaging.')
parser.add_argument('--timestepping', type=str, default='rk4', choices=['rk2', 'rk4', 'heuns', 'ssprk3', 'leapfrog'], help='Choose a time steeping method. Default SSPRK3.')
parser.add_argument('--asselin', type=float, default=0.3, help='Asselin Filter coefficient. Default 0.3.')
parser.add_argument('--filename', type=str, default='control')
parser.add_argument('--pickup', action='store_true', help='Pickup the result from the checkpoint.')
args = parser.parse_known_args()
args = args[0]
filter = args.filter
filter2 = args.filter2
filter_val = args.filter_val
filter_freq = args.filter_freq
timestepping = args.timestepping
asselin = args.asselin
ref_level = args.ref_level
filename = args.filename
space_parallel = args.space_parallel
print(args)
if filter_freq and filter:
print("To many filters are on")
import sys; sys.exit()
#ensemble communicator
ensemble = Ensemble(COMM_WORLD, space_parallel)
#parameters
R0 = 6371220.
R = Constant(R0)
H = Constant(5960.)
Omega = Constant(7.292e-5) # rotation rate
g = Constant(9.8) # Gravitational constant
mesh = IcosahedralSphereMesh(radius=R0,
refinement_level=ref_level, degree=3,
comm = ensemble.comm)
x = SpatialCoordinate(mesh)
global_normal = as_vector([x[0], x[1], x[2]])
mesh.init_cell_orientations(global_normal)
outward_normals = CellNormal(mesh)
perp = lambda u: cross(outward_normals, u)
V1 = FunctionSpace(mesh, "BDM", 2)
V2 = FunctionSpace(mesh, "DG", 1)
W = MixedFunctionSpace((V1, V2))
f_expr = 2 * Omega * x[2] / R
Vf = FunctionSpace(mesh, "CG", 3)
f = Function(Vf).interpolate(f_expr) # Coriolis frequency
u_0 = 20.0 # maximum amplitude of the zonal wind [m/s]
u_max = Constant(u_0)
u_expr = as_vector([-u_max*x[1]/R, u_max*x[0]/R, 0.0])
eta_expr = -((R*Omega*u_max+u_max*u_max/2.0)*(x[2]*x[2]/(R*R)))/g
h_expr = eta_expr + H
#topography (D = H + eta - b)
rl = pi/9.0
lambda_x = atan_2(x[1]/R0, x[0]/R0)
lambda_c = -pi/2.0
phi_x = asin(x[2]/R0)
phi_c = pi/6.0
minarg = Min(pow(rl, 2), pow(phi_x - phi_c, 2) + pow(lambda_x - lambda_c, 2))
bexpr = 2000.0*(1 - sqrt(minarg)/rl)
b = Function(V2, name="Topography")
b.interpolate(bexpr)
#checking cheby parameters based on ref_level
eigs = [0.003465, 0.007274, 0.014955] #maximum frequency
min_time_period = 2*pi/eigs[ref_level-3]
hours = args.dt
dt = 60*60*hours
dts = 60*60*args.dts
rho = args.rho #averaging window is rho*dt
L = eigs[ref_level-3]*dt*rho
ppp = args.ppp #points per (minimum) time period
#rho*dt/min_time_period = number of min_time_periods that fit in rho*dt
# we want at least ppp times this number of sample points
Mbar = ceil(ppp*rho*dt*eigs[ref_level-3]/2/pi)
if args.get_Mbar:
print("Mbar="+str(Mbar))
import sys; sys.exit()
svals = np.arange(0.5, Mbar)/Mbar #tvals goes from -rho*dt/2 to rho*dt/2
weights = np.exp(-1.0/svals/(1.0-svals))
weights = weights/np.sum(weights)
print(weights)
svals -= 0.5
#parameters for timestepping
t = 0.
tmax = 60.*60.*args.tmax
dumpt = args.dumpt*60.*60.
normt = args.normt*60.*60.
tdump = 0.
tnorm = 0.
#print out settings
print = PETSc.Sys.Print
assert Mbar*space_parallel==COMM_WORLD.size, str(Mbar)+' '+str(COMM_WORLD.size)
print('averaging window', rho*dt, 'sample width', rho*dt/Mbar)
print('Mbar', Mbar, 'samples per min time period', min_time_period/(rho*dt/Mbar))
print(args)
#pickup the result
if args.pickup:
chkfile = DumbCheckpoint(filename, mode=FILE_READ, comm = ensemble.comm)
un = Function(V1, name="Velocity")
etan = Function(V2, name="Elevation")
urn = Function(V1, name="VelocityR")
hn = Function(V2, name="Depth")
chkfile.load(un, name="Velocity")
chkfile.load(etan, name="Elevation")
chkfile.load(urn, name="VelocityR")
chkfile.load(hn, name="Depth")
t = chkfile.read_attribute("/", "time")
tdump = chkfile.read_attribute("/", "tdump")
tnorm = chkfile.read_attribute("/", "tnorm")
eta_norm_ar = chkfile.read_attribute("/", "eta_norm")
u_norm_Hdiv_ar = chkfile.read_attribute("/", "u_norm_Hdiv")
u_norm_L2_ar = chkfile.read_attribute("/", "u_norm_L2")
eta_norm = eta_norm_ar.tolist()
u_norm_Hdiv = u_norm_Hdiv_ar.tolist()
u_norm_L2 = u_norm_L2_ar.tolist()
print("eta_norm = ", eta_norm)
print("u_norm_Hdiv = ", u_norm_Hdiv)
print("u_norm_L2 = ", u_norm_L2)
chkfile.close()
else:
un = Function(V1, name="Velocity").project(u_expr)
etan = Function(V2, name="Elevation").interpolate(eta_expr)
urn = Function(V1, name="VelocityR").assign(un)
hn = Function(V2, name="Depth").interpolate(h_expr)
hn -= b
##############################################################################
# Set up the exponential operator
##############################################################################
operator_in = Function(W)
u_in, eta_in = split(operator_in)
u, eta = TrialFunctions(W)
v, phi = TestFunctions(W)
F = (
- inner(f*perp(u_in),v)*dx
+g*eta_in*div(v)*dx
- H*div(u_in)*phi*dx
)
a = inner(v,u)*dx + phi*eta*dx
operator_out = Function(W)
params = {
'ksp_type': 'preonly',
'pc_type': 'fieldsplit',
'fieldsplit_0_ksp_type':'cg',
'fieldsplit_0_pc_type':'bjacobi',
'fieldsplit_0_sub_pc_type':'ilu',
'fieldsplit_1_ksp_type':'preonly',
'fieldsplit_1_pc_type':'bjacobi',
'fieldsplit_1_sub_pc_type':'ilu'
}
Prob = LinearVariationalProblem(a, F, operator_out)
OperatorSolver = LinearVariationalSolver(Prob, solver_parameters=params)
ncheb = 10000
cheby = cheby_exp(OperatorSolver, operator_in, operator_out,
ncheb, tol=1.0e-8, L=L, filter=filter, filter_val=filter_val, filter_freq=filter_freq)
cheby2 = cheby_exp(OperatorSolver, operator_in, operator_out,
ncheb, tol=1.0e-8, L=L, filter=filter2, filter_val=filter_val)
##############################################################################
# Set up solvers for the slow part
##############################################################################
USlow_in = Function(W) #value at previous timestep
USlow_out = Function(W) #value at RK stage
u0, eta0 = split(USlow_in)
#RHS for Forward Euler step
gradperp = lambda f: perp(grad(f))
n = FacetNormal(mesh)
Upwind = 0.5 * (sign(dot(u0, n)) + 1)
both = lambda u: 2*avg(u)
K = 0.5*inner(u0, u0)
uup = 0.5 * (dot(u0, n) + abs(dot(u0, n)))
dT = Constant(dt)
vector_invariant = True
if vector_invariant:
L = (
dT*inner(perp(grad(inner(v, perp(u0)))), u0)*dx
- dT*inner(both(perp(n)*inner(v, perp(u0))),
both(Upwind*u0))*dS
+ dT*div(v)*K*dx
+ dT*inner(grad(phi), u0*(eta0-b))*dx
- dT*jump(phi)*(uup('+')*(eta0('+')-b('+'))
- uup('-')*(eta0('-') - b('-')))*dS
)
else:
L = (
dT*inner(div(outer(u0, v)), u0)*dx
- dT*inner(both(inner(n, u0)*v), both(Upwind*u0))*dS
+ dT*inner(grad(phi), u0*(eta0-b))*dx
- dT*jump(phi)*(uup('+')*(eta0('+')-b('+'))
- uup('-')*(eta0('-') - b('-')))*dS
)
SlowProb = LinearVariationalProblem(a, L, USlow_out)
SlowSolver = LinearVariationalSolver(SlowProb,
solver_parameters = params)
##############################################################################
# Set up depth advection solver (DG upwinded scheme)
##############################################################################
up = Function(V1)
hp = Function(V2)
hps = Function(V2)
h = TrialFunction(V2)
phi = TestFunction(V2)
hh = 0.5 * (hn + h)
uh = 0.5 * (urn + up)
n = FacetNormal(mesh)
uup = 0.5 * (dot(uh, n) + abs(dot(uh, n)))
Heqn = ((h - hn)*phi*dx - dts*inner(grad(phi), uh*hh)*dx
+ dts*jump(phi)*(uup('+')*hh('+')-uup('-')*hh('-'))*dS)
Hproblem = LinearVariationalProblem(lhs(Heqn), rhs(Heqn), hps)
lu_params = {'ksp_type': 'preonly',
'pc_type': 'lu',
'pc_factor_mat_solver_type': 'mumps'}
Hsolver = LinearVariationalSolver(Hproblem,
solver_parameters=lu_params,
options_prefix="H-advection")
##############################################################################
# Velocity advection (Natale et. al (2016) extended to SWE)
##############################################################################
ups = Function(V1)
u = TrialFunction(V1)
v = TestFunction(V1)
hh = 0.5 * (hn + hp)
ubar = 0.5 * (urn + up)
uup = 0.5 * (dot(ubar, n) + abs(dot(ubar, n)))
uh = 0.5 * (urn + u)
Upwind = 0.5 * (sign(dot(ubar, n)) + 1)
K = 0.5 * (inner(0.5 * (urn + up), 0.5 * (urn + up)))
both = lambda u: 2*avg(u)
outward_normals = CellNormal(mesh)
perp = lambda arg: cross(outward_normals, arg)
Ueqn = (inner(u - urn, v)*dx + dts*inner(perp(uh)*f, v)*dx
- dts*inner(perp(grad(inner(v, perp(ubar)))), uh)*dx
+ dts*inner(both(perp(n)*inner(v, perp(ubar))),
both(Upwind*uh))*dS
- dts*div(v)*(g*(hh + b) + K)*dx)
Uproblem = LinearVariationalProblem(lhs(Ueqn), rhs(Ueqn), ups)
Usolver = LinearVariationalSolver(Uproblem,
solver_parameters=lu_params,
options_prefix="U-advection")
##############################################################################
# Linear solver for incremental updates
##############################################################################
HU = Function(W)
deltaU, deltaH = HU.split()
w, phi = TestFunctions(W)
du, dh = TrialFunctions(W)
alpha = 0.5
HUlhs = (inner(w, du + alpha*dts*f*perp(du))*dx
- alpha*dts*div(w)*g*dh*dx
+ phi*(dh + alpha*dts*H*div(du))*dx)
HUrhs = -inner(w, up - ups)*dx - phi*(hp - hps)*dx
HUproblem = LinearVariationalProblem(HUlhs, HUrhs, HU)
params = {'ksp_type': 'preonly',
'mat_type': 'aij',
'pc_type': 'lu',
'pc_factor_mat_solver_type': 'mumps'}
HUsolver = LinearVariationalSolver(HUproblem,
solver_parameters=params,
options_prefix="impl-solve")
##############################################################################
# Time loop
##############################################################################
U = Function(W)
DU = Function(W)
U1 = Function(W)
U2 = Function(W)
U3 = Function(W)
X1 = Function(W)
X2 = Function(W)
X3 = Function(W)
V = Function(W)
U_u, U_eta = U.split()
U_u.assign(un)
U_eta.assign(etan)
k_max = 4 # Maximum number of Picard iterations
iter_max = int(dt/dts)
print("dt, dts, iter_max =", dt, dts, iter_max)
u_out = Function(V1, name="Velocity").assign(urn)
eta_out = Function(V2, name="Elevation").assign(hn + b - H)
u_diff = Function(V1, name="Velocity Difference").assign(un - u_out)
eta_diff = Function(V2, name="Elevation Difference").assign(etan - eta_out)
#set weights
rank = ensemble.ensemble_comm.rank
expt = rho*dt*svals[rank]
wt = weights[rank]
print(wt, "weight", expt)
print("svals", svals)
etanorm = 0.
unorm_Hdiv = 0.
unorm_L2 = 0.
if rank==0:
#calculate norms at the initial state
etanorm = errornorm(etan, eta_out)/norm(eta_out)
unorm_Hdiv = errornorm(un, u_out, norm_type="Hdiv")/norm(u_out, norm_type="Hdiv")
unorm_L2 = errornorm(un, u_out)/norm(u_out)
print('etanorm', etanorm, 'unorm_Hdiv', unorm_Hdiv, 'unorm_L2', unorm_L2)
if not args.pickup:
eta_norm = []
u_norm_Hdiv = []
u_norm_L2 = []
eta_norm.append(etanorm)
u_norm_Hdiv.append(unorm_Hdiv)
u_norm_L2.append(unorm_L2)
print('etanorm =', eta_norm)
print('unorm_Hdiv =', u_norm_Hdiv)
print('unorm_L2 =', u_norm_L2)
#setup PV solver
PV = Function(Vf, name="PotentialVorticity")
gamma = TestFunction(Vf)
q = TrialFunction(Vf)
D = etan + H - b
a = q*gamma*D*dx
L = (- inner(perp(grad(gamma)), un))*dx + gamma*f*dx
PVproblem = LinearVariationalProblem(a, L, PV)
PVsolver = LinearVariationalSolver(PVproblem, solver_parameters={"ksp_type": "cg"})
PVsolver.solve()
PVout = Function(Vf, name="PotentialVorticity")
Dout = eta_out + H - b
a = q*gamma*Dout*dx
L = (- inner(perp(grad(gamma)), u_out))*dx + gamma*f*dx
PVoutproblem = LinearVariationalProblem(a, L, PVout)
PVoutsolver = LinearVariationalSolver(PVoutproblem, solver_parameters={"ksp_type": "cg"})
PVoutsolver.solve()
PVdiff = Function(Vf, name="PV Difference").assign(PV- PVout)
#write out initial fields
mesh_ll = get_latlon_mesh(mesh)
file_sw = File(filename+'_avg.pvd', comm=ensemble.comm, mode="a")
file_r = File(filename+'_serial.pvd', comm=ensemble.comm, mode="a")
file_d = File(filename+'_diff.pvd', comm=ensemble.comm, mode="a")
field_un = Function(
functionspaceimpl.WithGeometry(un.function_space(), mesh_ll),
val=un.topological)
field_etan = Function(
functionspaceimpl.WithGeometry(etan.function_space(), mesh_ll),
val=etan.topological)
field_PV = Function(
functionspaceimpl.WithGeometry(PV.function_space(), mesh_ll),
val=PV.topological)
field_b = Function(
functionspaceimpl.WithGeometry(b.function_space(), mesh_ll),
val=b.topological)
field_uout = Function(
functionspaceimpl.WithGeometry(u_out.function_space(), mesh_ll),
val=u_out.topological)
field_etaout = Function(
functionspaceimpl.WithGeometry(eta_out.function_space(), mesh_ll),
val=eta_out.topological)
field_PVout = Function(
functionspaceimpl.WithGeometry(PVout.function_space(), mesh_ll),
val=PVout.topological)
field_udiff = Function(
functionspaceimpl.WithGeometry(u_diff.function_space(), mesh_ll),
val=u_diff.topological)
field_etadiff = Function(
functionspaceimpl.WithGeometry(eta_diff.function_space(), mesh_ll),
val=eta_diff.topological)
field_PVdiff = Function(
functionspaceimpl.WithGeometry(PVdiff.function_space(), mesh_ll),
val=PVdiff.topological)
if not args.pickup:
file_sw.write(field_un, field_etan, field_PV, field_b)
file_r.write(field_uout, field_etaout, field_PVout, field_b)
file_d.write(field_udiff, field_etadiff, field_PVdiff, field_b)
#create checkpointing file
print("create checkpointing file at rank =", rank)
chk = DumbCheckpoint(filename, mode=FILE_CREATE, comm = ensemble.comm)
#start time loop
print('tmax', tmax, 'dt', dt, 'dts', dts)
while t < tmax - 0.5*dt:
print(t)
t += dt
tdump += dt
tnorm += dt
if t < dt*1.5 and timestepping == 'leapfrog':
U_old = Function(W)
U_new = Function(W)
U_old.assign(U)
rk2(U, USlow_in, USlow_out, DU, V, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt)
else:
if timestepping == 'leapfrog':
leapfrog(U, USlow_in, USlow_out, U_old, U_new, DU, U1, U2, V, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt, asselin)
elif timestepping == 'ssprk3':
ssprk3(U, USlow_in, USlow_out, DU, U1, U2, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt)
elif timestepping == 'rk2':
rk2(U, USlow_in, USlow_out, DU, V, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt)
elif timestepping == 'rk4':
rk4(U, USlow_in, USlow_out, DU, U1, U2, U3, X1, X2, X3, V, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt)
elif timestepping == 'heuns':
heuns(U, USlow_in, USlow_out, DU, U1, U2, W,
expt, ensemble, cheby, cheby2, SlowSolver, wt, dt)
if rank == 0:
#run the serial solver
for iter in range(iter_max):
up.assign(urn)
hp.assign(hn)
#start picard cycle
for i in range(k_max):
#advect to get candidates
Hsolver.solve()
Usolver.solve()
#linear solve for updates
HUsolver.solve()
#increment updates
up += deltaU
hp += deltaH
#update fields for next time step
urn.assign(up)
hn.assign(hp)
#calculate norms
un.assign(U_u)
etan.assign(U_eta)
u_out.assign(urn)
eta_out.assign(hn + b - H)
etanorm = errornorm(etan, eta_out)/norm(eta_out)
unorm_Hdiv = errornorm(un, u_out, norm_type="Hdiv")/norm(u_out, norm_type="Hdiv")
unorm_L2 = errornorm(un, u_out)/norm(u_out)
print('etanorm', etanorm, 'unorm_Hdiv', unorm_Hdiv, 'unorm_L2', unorm_L2)
#dumping results
if tdump > dumpt - dt*0.5:
#dump averaged results
PVsolver.solve()
file_sw.write(field_un, field_etan, field_PV, field_b)
#dump non averaged results
PVoutsolver.solve()
file_r.write(field_uout, field_etaout, field_PVout, field_b)
#dump differences
u_diff.assign(un - u_out)
eta_diff.assign(etan - eta_out)
PVdiff.assign(PV- PVout)
file_d.write(field_udiff, field_etadiff, field_PVdiff, field_b)
#update dumpt
print("dumped at t =", t)
tdump -= dumpt
#append norms to array every normt
if tnorm > normt - dt*0.5:
eta_norm.append(etanorm)
u_norm_Hdiv.append(unorm_Hdiv)
u_norm_L2.append(unorm_L2)
print('etanorm =', eta_norm)
print('unorm_Hdiv =', u_norm_Hdiv)
print('unorm_L2 =', u_norm_L2)
tnorm -= normt
if rank == 0:
#checkpointing every time step
chk.store(un)
chk.store(etan)
chk.store(urn)
chk.store(hn)
chk.write_attribute("/", "time", t)
chk.write_attribute("/", "tdump", tdump)
chk.write_attribute("/", "tnorm", tnorm)
chk.write_attribute("/", "eta_norm", eta_norm)
chk.write_attribute("/", "u_norm_Hdiv", u_norm_Hdiv)
chk.write_attribute("/", "u_norm_L2", u_norm_L2)
chk.close()
#check if dumbcheckpoint is working
und = Function(V1)
etand = Function(V2)
urnd = Function(V1)
hnd = Function(V2)
chkfile = DumbCheckpoint(filename, mode=FILE_READ, comm = ensemble.comm)
chkfile.load(und, name="Velocity")
chkfile.load(etand, name="Elevation")
chkfile.load(urnd, name="VelocityR")
chkfile.load(hnd, name="Depth")
t = chkfile.read_attribute("/", "time")
tdump = chkfile.read_attribute("/", "tdump")
tnorm = chkfile.read_attribute("/", "tnorm")
eta_norm = chkfile.read_attribute("/", "eta_norm")
u_norm_Hdiv = chkfile.read_attribute("/", "u_norm_Hdiv")
u_norm_L2 = chkfile.read_attribute("/", "u_norm_L2")
valf = assemble(etan*dx)
valg = assemble(etand*dx)
valh = assemble(hn*dx)
vald = assemble(hnd*dx)
print("t = ", t)
print("tdump = ", tdump)
print("tnorm = ", tnorm)
print("eta_norm = ", eta_norm)
print("u_norm_Hdiv = ", u_norm_Hdiv)
print("u_norm_L2 = ", u_norm_L2)
ud_out = Function(V1).assign(urnd)
etad_out = Function(V2).assign(hnd + b - H)
etanorm = errornorm(etand, etad_out)/norm(etad_out)
unorm_Hdiv = errornorm(und, ud_out, norm_type="Hdiv")/norm(ud_out, norm_type="Hdiv")
unorm_L2 = errornorm(und, ud_out)/norm(ud_out)
print('etanorm', etanorm, 'unorm_Hdiv', unorm_Hdiv, 'unorm_L2', unorm_L2)
print("valf = ", valf)
print("valg = ", valg)
assert(valf == valg)
print("valh = ", valh)
print("vald = ", vald)
assert(valh == vald)
chkfile.close()