# Compare integration time-stepping schemes for KS equation.
# Conclusions: 'ETD-RK4' is superior to 'SI-RK3',
# - Superiority deteriorates when adding noise to ICs.
# - Superiority fairly independent of N.
# - The naive methods (rk4, rk1) are quite terrible.
import time
import numpy as np
from matplotlib import pyplot as plt
from dapper.mods.KS import Model
model = Model()
print("THERE WILL BE WARNINGS GENERATED.")
print("This is normal, because some schemes cause NaNs for some dt's.")
# Experiment range
minexpo = -8 # use -11 to observe saturation due to num. prec.
hh = 2.0**np.arange(minexpo, 0) # list of dt
# Params
T = 30
N = 4
# IC -- NB: integration accuracy depends on noise level
E0 = model.x0_Kassam + 1e-3*np.random.randn(N, model.Nx)
# Allocate stats
mref = 'step_ETD_RK4'
def test_KS():
from dapper.mods.KS import Model
KS = Model()
assert _allclose(KS.dxdt, KS.d2x_dtdx, KS.x0)
import numpy as np
import dapper as dpr
from dapper.mods.KS import Model, Tplot
from dapper.mods.Lorenz96 import LPs
from dapper.tools.localization import nd_Id_localization
from dapper.tools.math import Id_Obs
KS = Model(dt=0.5)
Nx = KS.Nx
# nRepeat=10
t = dpr.Chronology(KS.dt,
dkObs=2,
KObs=2 * 10**4,
BurnIn=2 * 10**3,
Tplot=Tplot)
Dyn = {'M': Nx, 'model': KS.step, 'linear': KS.dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(mu=KS.x0, C=0.001)
Obs = Id_Obs(Nx)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (4, ))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(np.arange(Nx))
####################
if mod == "LUV":
from dapper.mods.LorenzUV.lorenz96 import LUV
ii = np.arange(LUV.nU)
step = with_rk4(LUV.dxdt, autonom=True)
Nx = LUV.M
T = 1e2
dt = 0.005
LUV.F = 10
x0 = 0.01 * randn(LUV.M)
eps = 0.001
N = 66 # Don't need all Nx for a good approximation of upper spectrum.
# Lyapunov exponents: [ 0.08 0.07 0.06 ... -37.9 -39.09 -41.55]
if mod == "KS":
from dapper.mods.KS import Model
KS = Model()
step = KS.step
x0 = KS.x0
dt = KS.dt
N = KS.Nx
Nx = len(x0)
T = 1e3
eps = 0.0002
# n0 ≈ 140
if mod == "QG":
from dapper.mods.QG import model_config, sample_filename, shape
# NB: There may arise an ipython/multiprocessing bug/issue.
# Ref https://stackoverflow.com/a/45720872 . If so, set mp=False,
# or run outside of ipython. However, I did not encounter lately.