3.4. Time dependant problems#
In this unit we want to turn to instationary problems. And we will start with a very basic setup: implicit Euler time stepping for the convection diffusion equation. After the main part of this tutorial we have supplementary material to extend the case in several regards.
We are solving the unsteady heat equation
\[\text{find } u:[0,T] \to H_{0,D}^1 \quad \int_{\Omega} \partial_t u v + \int_{\Omega} \nabla u \nabla v + b \cdot \nabla u v = \int f v \quad \forall v \in H_{0,D}^1, \quad u(t=0) = u_0\]
with a suitable advective field \(b\) (the wind).
#imports
from ngsolve import *
#from netgen.geom2d import SplineGeometry
from ngsolve.webgui import Draw
Geometry: \((-1,1)^2\)
Dirichlet boundaries everywhere
Mesh
from netgen.occ import *
from netgen.webgui import Draw as DrawGeo
shape = Rectangle(2,2).Face().Move((-1,-1,0))
shape.edges.Min(X).name="left"
shape.edges.Max(X).name="right"
shape.edges.Min(Y).name="bottom"
shape.edges.Max(Y).name="top"
mesh = Mesh(OCCGeometry(shape, dim=2).GenerateMesh(maxh=0.25))
fes = H1(mesh, order=3, dirichlet="bottom|right|left|top")
u,v = fes.TnT()
time = 0.0
dt = 0.001
We define the field \(b\) (the wind) as
\[b(x,y) := (2y(1-x^2),-2x(1-y^2)).\]
b = CoefficientFunction((2*y*(1-x*x),-2*x*(1-y*y)))
Draw(b,mesh,"wind", vectors={"grid_size": 32}, order=3)
BaseWebGuiScene
bilinear forms for
convection-diffusion stiffness and
mass matrix separately.
non-symmetric memory layout for the mass matrix so that a and m have the same sparsity pattern.
a = BilinearForm(fes, symmetric=False)
a += 0.01*grad(u)*grad(v)*dx + b*grad(u)*v*dx
a.Assemble()
m = BilinearForm(fes, symmetric=False)
m += u*v*dx
m.Assemble()
<ngsolve.comp.BilinearForm at 0x7fce4788c630>
3.4.1. Implicit Euler method#
\[
\underbrace{(M + \Delta t A)}_{M^\ast} u^{n+1} = M u^n + \Delta tf^{n+1}
\]
or in an incremental form:
\[
M^\ast (u^{n+1} - u^n) = - \Delta t A u^n + \Delta tf^{n+1}.
\]
Incremental form: \(u^{n+1} - u^n\) has homogeneous boundary conditions (unless boundary conditions are time-dependent).
For the time stepping method: set up linear combinations of matrices.
(Only) if the sparsity pattern of the matrices agree we can copy the pattern and sum up the entries.
First, we create a matrix of same format as m.mat and compare number of non-zero entries and sparsity pattern:
mstar = m.mat.CreateMatrix()
print(f"m.mat.nze = {m.mat.nze}, a.mat.nze={a.mat.nze}, mstar.nze={mstar.nze}")
m.mat.nze = 11257, a.mat.nze=11257, mstar.nze=11257
Using the vector we can build the linear combination of the a and the m matrix:
mstar.AsVector().data = m.mat.AsVector() + dt * a.mat.AsVector()
# We cannot do this:
# mstar.data = m.mat + dt * a.mat
# unsupported operand type(s) for *: 'NoneType' and 'ngsolve.la.DynamicVectorExpression'
# corresponds to M* = M + dt * A
invmstar = mstar.Inverse(freedofs=fes.FreeDofs())
We set the r.h.s. \(f = exp(-6 ((x+\frac12)^2+y^2)) - exp(-6 ((x-\frac12)^2+y^2))\)
f = LinearForm(fes)
gaussp = exp(-6*((x+0.5)*(x+0.5)+y*y))-exp(-6*((x-0.5)*(x-0.5)+y*y))
Draw(gaussp,mesh,"f", deformation=True)
f += gaussp*v*dx
f.Assemble()
<ngsolve.comp.LinearForm at 0x7fce271da8b0>
and the initial data: \(u_0 = (1-y^2)x\)
gfu = GridFunction(fes)
gfu.Set((1-y*y)*x) # note that boundary conditions remain
scene = Draw(gfu,mesh,"u")
we define a simple time loop including some visualization sampling:
def TimeStepping(invmstar, initial_cond = None, t0 = 0, tend = 2,
nsamples = 10):
if initial_cond:
gfu.Set(initial_cond)
cnt = 0; time = t0
sample_int = int(floor(tend / dt / nsamples)+1)
gfut = GridFunction(gfu.space,multidim=0)
gfut.AddMultiDimComponent(gfu.vec)
while time < tend - 0.5 * dt:
res = dt * f.vec - dt * a.mat * gfu.vec
gfu.vec.data += invmstar * res
print("\r",time,end="")
scene.Redraw()
if cnt % sample_int == 0:
gfut.AddMultiDimComponent(gfu.vec)
cnt += 1; time = cnt * dt
return gfut
%%time
gfut = TimeStepping(invmstar, (1-y*y)*x)
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1.999CPU times: user 7.57 s, sys: 586 ms, total: 8.16 s
Wall time: 5.24 s
1.999CPU times: user 7.05 s, sys: 630 ms, total: 7.68 s Wall time: 6.57 s
Draw(gfut, mesh, interpolate_multidim=True, animate=True)
BaseWebGuiScene
3.4.2. Using iterative solvers#
For a factorization of \(M^\ast\) (and \({M^\ast}^{-1}\)) we required a sparse matrix \(M^\ast\)
To store \(M^\ast\) as a sparse matrix requires new storage (and same memory layout of \(A\) and \(M\))
For iterative solvers we only require the matrix (and preconditioner) applications and it suffices to have \(M^\ast\) available as a linear operator
mstar_alt = m.mat + dt * a.mathas no storage but defines the operator action: sum of two matrix-vector multiplications
iterative solver version (with Jacobi preconditining):
** does not perform well for this problem, just for explanation **
mstar_alt = m.mat + dt * a.mat
precond = preconditioners.Local
preflags ={"GS":False}
prea = precond(a, **preflags)
prem = precond(m, **preflags)
premstar_alt = prem.mat + dt * prea.mat
print(premstar_alt)
Sum of
Scale a = 1
Print base-matrix
Scale b = 1
Scale with 0.001:
Print base-matrix
Now, replace the action of the inverse Matrix from the previous script with a solution of a conjugate gradient method:
from ngsolve.krylovspace import CGSolver
invmstar_alt = CGSolver(mstar_alt, premstar_alt, tol=1e-8, plotrates=False, maxiter=200)
Now, we can repeat the time stepping. As in 2D direct solvers are quite efficient, this simple preconditioned solver can hardly compete. We hence only do a few time steps:
%%time
gfut_a1 = TimeStepping(invmstar_alt, (1-y*y)*x )
0
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0.63
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0.64
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0.67
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0.71
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0.718
0.719
0.72
0.721
0.722
0.723
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0.729
0.73
0.731
0.732
0.733
0.734
0.735
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0.737
0.738
0.739
0.74
0.741
0.742
0.743
0.744
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0.746
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0.748
0.749
0.75
0.751
0.752
0.753
0.754
0.755
0.756
0.757
0.758
0.759
0.76
0.761
0.762
0.763
0.764
0.765
0.766
0.767
0.768
0.769
0.77
0.771
0.772
0.773
0.774
0.775
0.776
0.777
0.778
0.779
0.78
0.781
0.782
0.783
0.784
0.785
0.786
0.787
0.788
0.789
0.79
0.791
0.792
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0.794
0.795
0.796
0.797
0.798
0.799
0.8
0.801
0.802
0.803
0.804
0.805
0.806
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0.808
0.809
0.81
0.811
0.812
0.8130000000000001
0.8140000000000001
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0.8320000000000001
0.833
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0.84
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0.845
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0.85
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0.895
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0.899
0.9
0.901
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0.904
0.905
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0.91
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0.913
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0.916
0.917
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0.919
0.92
0.921
0.922
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0.926
0.927
0.928
0.929
0.93
0.931
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0.933
0.934
0.935
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0.937
0.9380000000000001
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0.9590000000000001
0.96
0.961
0.962
0.963
0.964
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0.97
0.971
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0.978
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0.98
0.981
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0.99
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1.0
1.0010000000000001
1.002
1.0030000000000001
1.004
1.0050000000000001
1.006
1.0070000000000001
1.008
1.0090000000000001
1.01
1.0110000000000001
1.012
1.0130000000000001
1.014
1.0150000000000001
1.016
1.0170000000000001
1.018
1.0190000000000001
1.02
1.0210000000000001
1.022
1.0230000000000001
1.024
1.025
1.026
1.027
1.028
1.029
1.03
1.031
1.032
1.033
1.034
1.035
1.036
1.037
1.038
1.039
1.04
1.041
1.042
1.043
1.044
1.045
1.046
1.047
1.048
1.049
1.05
1.051
1.052
1.053
1.054
1.055
1.056
1.057
1.058
1.059
1.06
1.061
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1.066
1.067
1.068
1.069
1.07
1.071
1.072
1.073
1.074
1.075
1.076
1.077
1.078
1.079
1.08
1.081
1.082
1.083
1.084
1.085
1.086
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1.088
1.089
1.09
1.091
1.092
1.093
1.094
1.095
1.096
1.097
1.098
1.099
1.1
1.101
1.102
1.103
1.104
1.105
1.106
1.107
1.108
1.109
1.11
1.111
1.112
1.113
1.114
1.115
1.116
1.117
1.118
1.119
1.12
1.121
1.122
1.123
1.124
1.125
1.1260000000000001
1.127
1.1280000000000001
1.129
1.1300000000000001
1.131
1.1320000000000001
1.133
1.1340000000000001
1.135
1.1360000000000001
1.137
1.1380000000000001
1.139
1.1400000000000001
1.141
1.1420000000000001
1.143
1.1440000000000001
1.145
1.1460000000000001
1.147
1.1480000000000001
1.149
1.1500000000000001
1.151
1.1520000000000001
1.153
1.154
1.155
1.156
1.157
1.158
1.159
1.16
1.161
1.162
1.163
1.164
1.165
1.166
1.167
1.168
1.169
1.17
1.171
1.172
1.173
1.174
1.175
1.176
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1.18
1.181
1.182
1.183
1.184
1.185
1.186
1.187
1.188
1.189
1.19
1.191
1.192
1.193
1.194
1.195
1.196
1.197
1.198
1.199
1.2
1.201
1.202
1.203
1.204
1.205
1.206
1.207
1.208
1.209
1.21
1.211
1.212
1.213
1.214
1.215
1.216
1.217
1.218
1.219
1.22
1.221
1.222
1.223
1.224
1.225
1.226
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1.228
1.229
1.23
1.231
1.232
1.233
1.234
1.235
1.236
1.237
1.238
1.239
1.24
1.241
1.242
1.243
1.244
1.245
1.246
1.247
1.248
1.249
1.25
1.2510000000000001
1.252
1.2530000000000001
1.254
1.2550000000000001
1.256
1.2570000000000001
1.258
1.2590000000000001
1.26
1.2610000000000001
1.262
1.2630000000000001
1.264
1.2650000000000001
1.266
1.2670000000000001
1.268
1.2690000000000001
1.27
1.2710000000000001
1.272
1.2730000000000001
1.274
1.2750000000000001
1.276
1.2770000000000001
1.278
1.2790000000000001
1.28
1.281
1.282
1.283
1.284
1.285
1.286
1.287
1.288
1.289
1.29
1.291
1.292
1.293
1.294
1.295
1.296
1.297
1.298
1.299
1.3
1.301
1.302
1.303
1.304
1.305
1.306
1.307
1.308
1.309
1.31
1.311
1.312
1.313
1.314
1.315
1.316
1.317
1.318
1.319
1.32
1.321
1.322
1.323
1.324
1.325
1.326
1.327
1.328
1.329
1.33
1.331
1.332
1.333
1.334
1.335
1.336
1.337
1.338
1.339
1.34
1.341
1.342
1.343
1.344
1.345
1.346
1.347
1.348
1.349
1.35
1.351
1.352
1.353
1.354
1.355
1.356
1.357
1.358
1.359
1.36
1.361
1.362
1.363
1.364
1.365
1.366
1.367
1.368
1.369
1.37
1.371
1.372
1.373
1.374
1.375
1.3760000000000001
1.377
1.3780000000000001
1.379
1.3800000000000001
1.381
1.3820000000000001
1.383
1.3840000000000001
1.385
1.3860000000000001
1.387
1.3880000000000001
1.389
1.3900000000000001
1.391
1.3920000000000001
1.393
1.3940000000000001
1.395
1.3960000000000001
1.397
1.3980000000000001
1.399
1.4000000000000001
1.401
1.4020000000000001
1.403
1.4040000000000001
1.405
1.4060000000000001
1.407
1.408
1.409
1.41
1.411
1.412
1.413
1.414
1.415
1.416
1.417
1.418
1.419
1.42
1.421
1.422
1.423
1.424
1.425
1.426
1.427
1.428
1.429
1.43
1.431
1.432
1.433
1.434
1.435
1.436
1.437
1.438
1.439
1.44
1.441
1.442
1.443
1.444
1.445
1.446
1.447
1.448
1.449
1.45
1.451
1.452
1.453
1.454
1.455
1.456
1.457
1.458
1.459
1.46
1.461
1.462
1.463
1.464
1.465
1.466
1.467
1.468
1.469
1.47
1.471
1.472
1.473
1.474
1.475
1.476
1.477
1.478
1.479
1.48
1.481
1.482
1.483
1.484
1.485
1.486
1.487
1.488
1.489
1.49
1.491
1.492
1.493
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1.495
1.496
1.497
1.498
1.499
1.5
1.5010000000000001
1.502
1.5030000000000001
1.504
1.5050000000000001
1.506
1.5070000000000001
1.508
1.5090000000000001
1.51
1.5110000000000001
1.512
1.5130000000000001
1.514
1.5150000000000001
1.516
1.5170000000000001
1.518
1.5190000000000001
1.52
1.5210000000000001
1.522
1.5230000000000001
1.524
1.5250000000000001
1.526
1.5270000000000001
1.528
1.5290000000000001
1.53
1.5310000000000001
1.532
1.5330000000000001
1.534
1.5350000000000001
1.536
1.537
1.538
1.539
1.54
1.541
1.542
1.543
1.544
1.545
1.546
1.547
1.548
1.549
1.55
1.551
1.552
1.553
1.554
1.555
1.556
1.557
1.558
1.559
1.56
1.561
1.562
1.563
1.564
1.565
1.566
1.567
1.568
1.569
1.57
1.571
1.572
1.573
1.574
1.575
1.576
1.577
1.578
1.579
1.58
1.581
1.582
1.583
1.584
1.585
1.586
1.587
1.588
1.589
1.59
1.591
1.592
1.593
1.594
1.595
1.596
1.597
1.598
1.599
1.6
1.601
1.602
1.603
1.604
1.605
1.606
1.607
1.608
1.609
1.61
1.611
1.612
1.613
1.614
1.615
1.616
1.617
1.618
1.619
1.62
1.621
1.622
1.623
1.624
1.625
1.6260000000000001
1.627
1.6280000000000001
1.629
1.6300000000000001
1.631
1.6320000000000001
1.633
1.6340000000000001
1.635
1.6360000000000001
1.637
1.6380000000000001
1.639
1.6400000000000001
1.641
1.6420000000000001
1.643
1.6440000000000001
1.645
1.6460000000000001
1.647
1.6480000000000001
1.649
1.6500000000000001
1.651
1.6520000000000001
1.653
1.6540000000000001
1.655
1.6560000000000001
1.657
1.6580000000000001
1.659
1.6600000000000001
1.661
1.6620000000000001
1.663
1.6640000000000001
1.665
1.666
1.667
1.668
1.669
1.67
1.671
1.672
1.673
1.674
1.675
1.676
1.677
1.678
1.679
1.68
1.681
1.682
1.683
1.684
1.685
1.686
1.687
1.688
1.689
1.69
1.691
1.692
1.693
1.694
1.695
1.696
1.697
1.698
1.699
1.7
1.701
1.702
1.703
1.704
1.705
1.706
1.707
1.708
1.709
1.71
1.711
1.712
1.713
1.714
1.715
1.716
1.717
1.718
1.719
1.72
1.721
1.722
1.723
1.724
1.725
1.726
1.727
1.728
1.729
1.73
1.731
1.732
1.733
1.734
1.735
1.736
1.737
1.738
1.739
1.74
1.741
1.742
1.743
1.744
1.745
1.746
1.747
1.748
1.749
1.75
1.7510000000000001
1.752
1.7530000000000001
1.754
1.7550000000000001
1.756
1.7570000000000001
1.758
1.7590000000000001
1.76
1.7610000000000001
1.762
1.7630000000000001
1.764
1.7650000000000001
1.766
1.7670000000000001
1.768
1.7690000000000001
1.77
1.7710000000000001
1.772
1.7730000000000001
1.774
1.7750000000000001
1.776
1.7770000000000001
1.778
1.7790000000000001
1.78
1.7810000000000001
1.782
1.7830000000000001
1.784
1.7850000000000001
1.786
1.7870000000000001
1.788
1.7890000000000001
1.79
1.7910000000000001
1.792
1.793
1.794
1.795
1.796
1.797
1.798
1.799
1.8
1.801
1.802
1.803
1.804
1.805
1.806
1.807
1.808
1.809
1.81
1.811
1.812
1.813
1.814
1.815
1.816
1.817
1.818
1.819
1.82
1.821
1.822
1.823
1.824
1.825
1.826
1.827
1.828
1.829
1.83
1.831
1.832
1.833
1.834
1.835
1.836
1.837
1.838
1.839
1.84
1.841
1.842
1.843
1.844
1.845
1.846
1.847
1.848
1.849
1.85
1.851
1.852
1.853
1.854
1.855
1.856
1.857
1.858
1.859
1.86
1.861
1.862
1.863
1.864
1.865
1.866
1.867
1.868
1.869
1.87
1.871
1.872
1.873
1.874
1.875
1.8760000000000001
1.877
1.8780000000000001
1.879
1.8800000000000001
1.881
1.8820000000000001
1.883
1.8840000000000001
1.885
1.8860000000000001
1.887
1.8880000000000001
1.889
1.8900000000000001
1.891
1.8920000000000001
1.893
1.8940000000000001
1.895
1.8960000000000001
1.897
1.8980000000000001
1.899
1.9000000000000001
1.901
1.9020000000000001
1.903
1.9040000000000001
1.905
1.9060000000000001
1.907
1.9080000000000001
1.909
1.9100000000000001
1.911
1.9120000000000001
1.913
1.9140000000000001
1.915
1.9160000000000001
1.917
1.9180000000000001
1.919
1.92
1.921
1.922
1.923
1.924
1.925
1.926
1.927
1.928
1.929
1.93
1.931
1.932
1.933
1.934
1.935
1.936
1.937
1.938
1.939
1.94
1.941
1.942
1.943
1.944
1.945
1.946
1.947
1.948
1.949
1.95
1.951
1.952
1.953
1.954
1.955
1.956
1.957
1.958
1.959
1.96
1.961
1.962
1.963
1.964
1.965
1.966
1.967
1.968
1.969
1.97
1.971
1.972
1.973
1.974
1.975
1.976
1.977
1.978
1.979
1.98
1.981
1.982
1.983
1.984
1.985
1.986
1.987
1.988
1.989
1.99
1.991
1.992
1.993
1.994
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1.997
1.998
1.999CPU times: user 23.5 s, sys: 400 ms, total: 23.9 s
Wall time: 23.3 s
Draw(gfut_a1, mesh, interpolate_multidim=True, animate=True);
3.4.3. time-dependent r.h.s. data#
Next: time-dependent r.h.s. data \(f=f(t)\):
Use
Parametert representing the time.A
Parameteris a constantCoefficientFunctionthe value of which can be changed with theSet-function.
t = Parameter(0.0)
An example of a time-dependent coefficient that we want to use as r.h.s. in the following is
omega=pi
gausspt = exp(-10*((x-0.5)**2 + (y-0.5)**2) )*(sin(omega*t))
gff = GridFunction(L2(mesh,order=gfu.space.globalorder+1))
gfft = GridFunction(gff.space,multidim=0)
time = 0.0
for i in range(7):
t.Set(3*i/6)
gff.Set(gausspt)
gfft.AddMultiDimComponent(gff.vec)
Draw(gfft, mesh, interpolate_multidim=True, animate=True, min=-1, max=1, autoscale=False);
Accordingly, we define a different linear form which then has to be assembled in every time step.
ft = LinearForm(fes)
ft += gausspt*v*dx
def TimeStepping_app2(invmstar, initial_cond = None, t0 = 0, tend = 2,
nsamples = 20):
if initial_cond:
gfu.Set(initial_cond)
cnt = 0; time = t0
sample_int = int(floor(tend / dt / nsamples)+1)
gfut = GridFunction(gfu.space,multidim=0)
gfut.AddMultiDimComponent(gfu.vec)
while time < tend - 0.5 * dt:
t.Set(time)
ft.Assemble()
res = dt * ft.vec - dt * a.mat * gfu.vec
gfu.vec.data += invmstar * res
print("\r",time,end="")
if cnt % sample_int == 0:
gfut.AddMultiDimComponent(gfu.vec)
cnt += 1; time = cnt * dt
return gfut
%%time
gfut_a2 = TimeStepping_app2(invmstar, initial_cond=CF(0),tend=5)
Draw(gfut_a2, mesh, interpolate_multidim=True, animate=True,
min=-0.25,max=0.25,autoscale=False);
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CPU times: user 1.85 s, sys: 19.7 ms, total: 1.87 s
Wall time: 1.85 s
3.4.4. Time dependent boundary conditions#
\(u|_{\partial \Omega} = u_D(t)\), \(f=0\)
implicit Euler time stepping method, non-incremental form:
\[ M^\ast u^{n+1} = (M + \Delta t A) u^{n+1} = M u^n \]Homogenize w.r.t. to boundary conditions, i.e. we split
\[ u^{n+1} = u^{n+1}_0 + u^{n+1}_D \]where \(u^{n+1}_D\) is a (discrete) function with correct boundary condition:
\[
{M^\ast} u^{n+1}_0 = M u^n - {M^\ast} u^{n+1}_D
\]
uD = CoefficientFunction( (cos(5*(x+t))*y ))
time = 0.0
t.Set(0.0)
gfu.Set(uD,BND)
Draw(gfu,mesh,"uD");
def TimeStepping_app3(invmstar, initial_cond = None, t0 = 0, tend = 2,
nsamples = 10):
if initial_cond:
gfu.Set(initial_cond)
cnt = 0; time = t0
sample_int = int(floor(tend / dt / nsamples)+1)
gfuD = GridFunction(gfu.space)
gfut = GridFunction(gfu.space,multidim=0)
gfut.AddMultiDimComponent(gfu.vec)
while time < tend - 0.5 * dt:
t.Set(time)
gfuD.Set(uD,BND)
res = m.mat * gfu.vec - mstar *gfuD.vec
gfu.vec.data = gfuD.vec + invmstar * res
print("\r",time,end="")
if cnt % sample_int == 0:
gfut.AddMultiDimComponent(gfu.vec)
cnt += 1; time = cnt * dt
return gfut
%%time
gfut_a3 = TimeStepping_app3(invmstar, initial_cond=CF(gaussp),tend=2)
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1.0
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1.3
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1.31
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1.999CPU times: user 533 ms, sys: 0 ns, total: 533 ms
Wall time: 531 ms
Draw(gfut_a3, mesh, interpolate_multidim=True, animate=True,
#settings = {"subdivision" : 10},
deformation = False, min = 0, max = 1, autoscale = True);