We are solving the unsteady heat equation
$$\text{find } u:[0,T] \to H_{0,D}^1 \quad \int_{\Omega} \partial_t u v + \int_{\Omega} \varepsilon \nabla u \nabla v + b \cdot \nabla u v = \int f v \quad \forall v \in H_{0,D}^1, \quad u(t=0) = u_0$$with a suitable advective field $b$ (the wind).
import netgen.gui
%gui tk
import tkinter
from math import pi
from ngsolve import *
from netgen.geom2d import SplineGeometry
We generate geometry $(-1,1)^2$, mesh (Dirichlet boundaries everywhere):
geo = SplineGeometry()
geo.AddRectangle( (-1, -1), (1, 1), bcs = ("bottom", "right", "top", "left"))
mesh = Mesh( geo.GenerateMesh(maxh=0.25))
fes = H1(mesh, order=3, dirichlet="bottom|right|left|top")
u = fes.TrialFunction()
v = fes.TestFunction()
time = 0.0
dt = 0.001
bcs = ('bottom', 'right', 'top', 'left')
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")
We define and assemble bilinear forms for convection-diffusion stiffness and mass matrix separately.
Note that we choose a non-symmetric memory layout for the mass matrix so that a and m have the same sparsity pattern.
a = BilinearForm(fes, symmetric=False)
a += SymbolicBFI (0.01*grad(u)*grad(v) + b*grad(u)*v)
a.Assemble()
m = BilinearForm(fes, symmetric=False)
m += SymbolicBFI (u*v)
m.Assemble()
We want to use a simple implicit Euler time stepping method, i.e.
$$ (M + \Delta t A) u^{n+1} = M u^n + f^{n+1} $$or in an incremental form:
$$ (M + \Delta t A) (u^{n+1} - u^n) = - \Delta t A u^n + f^{n+1}. $$The incremental form has the advantage that $u^{n+1} - u^n$ has homogeneous boundary conditions (unless boundary conditions are time-dependent).
For the time stepping method we want to 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 the same size and sparsity pattern as m.mat:
mstar = m.mat.CreateMatrix()
print(mstar)
Row 0: 0: 0.00520833 4: 0.00260417 31: 0.00260417 89: -0.000520833 90: 8.68056e-05 91: -0.000520833 92: 8.68056e-05 107: -0.000260417 108: -1.28749e-19 553: 0.000173611 Row 1: 1: 0.00520833 10: 0.00260417 11: 0.00260417 93: -0.000520833 94: 8.68056e-05 95: -0.000520833 96: 8.68056e-05 141: -0.000260417 142: 1.28749e-19 557: 0.000173611 Row 2: 2: 0.00520833 17: 0.00260417 18: 0.00260417 97: -0.000520833 98: 8.68056e-05 99: -0.000520833 100: 8.68056e-05 181: -0.000260417 182: 1.28749e-19 561: 0.000173611 Row 3: 3: 0.00520833 24: 0.00260417 25: 0.00260417 101: -0.000520833 102: 8.68056e-05 103: -0.000520833 104: 8.68056e-05 221: -0.000260417 222: 1.28749e-19 565: 0.000173611 Row 4: 0: 0.00260417 4: 0.0163738 5: 0.00229828 31: 0.0058886 32: 0.00558272 89: -0.000520833 90: -8.68056e-05 91: -0.000260417 92: -1.42302e-19 105: -0.000459656 106: 7.66094e-05 107: -0.00117772 108: 0.000196287 109: -0.00111654 110: 0.000186091 113: -0.000229828 114: -1.15196e-19 261: -0.000328444 262: 1.69407e-19 553: 0.000173611 554: 0.000153219 569: 0.000218962 Row 5: 4: 0.00229828 5: 0.0142215 6: 0.00235821 32: 0.00475257 44: 0.00481249 105: -0.000459656 106: -7.66094e-05 109: -0.000229828 110: -1.24514e-19 111: -0.000471641 112: 7.86068e-05 113: -0.000950513 114: 0.000158419 115: -0.000962498 116: 0.000160416 121: -0.000235821 122: -1.18585e-19 265: -0.000245428 266: 1.25361e-19 554: 0.000153219 573: 0.000157214 584: 0.000163619 Row 6: 5: 0.00235821 6: 0.0140022 7: 0.00225885 33: 0.00464288 44: 0.00474223 111: -0.000471641 112: -7.86068e-05 115: -0.000235821 116: -1.27902e-19 117: -0.000451771 118: 7.52951e-05 119: -0.000928575 120: 0.000154763 121: -0.000948446 122: 0.000158074 125: -0.000225885 126: -1.15196e-19 271: -0.000238402 272: -1.21973e-19 555: 0.00015059 573: 0.000157214 582: 0.000158935 Row 7: 6: 0.00225885 7: 0.0136816 8: 0.00228517 33: 0.00455563 45: 0.00458194 117: -0.000451771 118: -7.52951e-05 119: -0.000225885 120: -1.2282e-19 123: -0.000457033 124: 7.61722e-05 125: -0.000911126 126: 0.000151854 127: -0.000916389 128: 0.000152731 133: -0.000228517 134: -1.15196e-19 273: -0.000229678 274: 1.15196e-19 555: 0.00015059 574: 0.000152344 589: 0.000153119 Row 8: 7: 0.00228517 8: 0.0138903 9: 0.00237377 34: 0.00465998 45: 0.00457138 123: -0.000457033 124: -7.61722e-05 127: -0.000228517 128: -1.23667e-19 129: -0.000474754 130: 7.91257e-05 131: -0.000931996 132: 0.000155333 133: -0.000914276 134: 0.000152379 137: -0.000237377 138: -1.25361e-19 279: -0.000228621 280: -1.15196e-19 556: 0.000158251 574: 0.000152344 578: 0.000152414 Row 9: 8: 0.00237377 9: 0.0132999 10: 0.00223318 34: 0.00441679 46: 0.0042762 129: -0.000474754 130: -7.91257e-05 131: -0.000237377 132: -1.29596e-19 135: -0.000446635 136: 7.44392e-05 137: -0.000883358 138: 0.000147226 139: -0.00085524 140: 0.00014254 145: -0.000223318 146: -1.11808e-19 281: -0.000204302 282: 1.05032e-19 556: 0.000158251 575: 0.000148878 590: 0.000136201 Row 10: 1: 0.00260417 9: 0.00223318 10: 0.021583 11: 0.00593425 35: 0.00595416 46: 0.00485725 93: -0.000520833 94: -8.68056e-05 95: -0.000260417 96: -3.38813e-21 135: -0.000446635 136: -7.44392e-05 139: -0.000223318 140: -1.2282e-19 141: -0.00118685 142: 0.000197808 143: -0.00119083 144: 0.000198472 145: -0.000971451 146: 0.000161908 149: -0.000333008 150: -1.69407e-19 287: -0.000262408 288: -1.32137e-19 557: 0.000173611 570: 0.000222005 575: 0.000148878 609: 0.000174938 Row 11: 1: 0.00260417 10: 0.00593425 11: 0.0164725 12: 0.00230202 35: 0.00563211 93: -0.000260417 94: -1.42302e-19 95: -0.000520833 96: -8.68056e-05 141: -0.00118685 142: -0.000197808 143: -0.000333008 144: -1.82959e-19 147: -0.000460405 148: 7.67341e-05 149: -0.00112642 150: 0.000187737 153: -0.000230202 154: -1.18585e-19 557: 0.000173611 558: 0.000153468 570: 0.000222005 Row 12: 11: 0.00230202 12: 0.0143342 13: 0.00237096 35: 0.00479613 47: 0.00486507 147: -0.000460405 148: -7.67341e-05 149: -0.000230202 150: -1.23667e-19 151: -0.000474192 152: 7.9032e-05 153: -0.000959226 154: 0.000159871 155: -0.000973014 156: 0.000162169 161: -0.000237096 162: -1.18585e-19 289: -0.000249411 290: 1.28749e-19 558: 0.000153468 576: 0.000158064 602: 0.000166274 Row 13: 12: 0.00237096 13: 0.0138437 14: 0.0021948 36: 0.00455091 47: 0.00472707 151: -0.000474192 152: -7.9032e-05 155: -0.000237096 156: -1.29596e-19 157: -0.000438961 158: 7.31601e-05 159: -0.000910182 160: 0.000151697 161: -0.000945413 162: 0.000157569 165: -0.00021948 166: -1.11808e-19 293: -0.00023561 294: -1.21973e-19 559: 0.00014632 576: 0.000158064 588: 0.000157074 Row 14: 13: 0.0021948 14: 0.0130092 15: 0.00212498 36: 0.0043796 48: 0.00430978 157: -0.000438961 158: -7.31601e-05 159: -0.00021948 160: -1.19432e-19 163: -0.000424997 164: 7.08328e-05 165: -0.00087592 166: 0.000145987 167: -0.000861955 168: 0.000143659 173: -0.000212498 174: -1.0842e-19 295: -0.000218479 296: 1.15196e-19 559: 0.00014632 577: 0.000141666 591: 0.000145653 Row 15: 14: 0.00212498 15: 0.0129094 16: 0.00220475 37: 0.00432972 48: 0.00424995 163: -0.000424997 164: -7.08328e-05 167: -0.000212498 168: -1.16891e-19 169: -0.000440949 170: 7.34916e-05 171: -0.000865943 172: 0.000144324 173: -0.00084999 174: 0.000141665 177: -0.000220475 178: -1.15196e-19 301: -0.000212497 302: -1.0842e-19 560: 0.000146983 577: 0.000141666 597: 0.000141665 Row 16: 15: 0.00220475 16: 0.0125009 17: 0.0021434 37: 0.00410704 49: 0.00404569 169: -0.000440949 170: -7.34916e-05 171: -0.000220475 172: -1.19432e-19 175: -0.00042868 176: 7.14466e-05 177: -0.000821408 178: 0.000136901 179: -0.000809138 180: 0.000134856 185: -0.00021434 186: -1.11808e-19 303: -0.000190229 304: 9.82558e-20 560: 0.000146983 579: 0.000142893 592: 0.000126819 Row 17: 2: 0.00260417 16: 0.0021434 17: 0.0210621 18: 0.00585676 38: 0.00578346 49: 0.00467426 97: -0.000520833 98: -8.68056e-05 99: -0.000260417 100: -3.38813e-21 175: -0.00042868 176: -7.14466e-05 179: -0.00021434 180: -1.17738e-19 181: -0.00117135 182: 0.000195225 183: -0.00115669 184: 0.000192782 185: -0.000934852 186: 0.000155809 189: -0.00032526 190: -1.6263e-19 309: -0.000253086 310: -1.28749e-19 561: 0.000173611 571: 0.00021684 579: 0.000142893 611: 0.000168724 Row 18: 2: 0.00260417 17: 0.00585676 18: 0.0162883 19: 0.00228738 38: 0.00553997 97: -0.000260417 98: -1.42302e-19 99: -0.000520833 100: -8.68056e-05 181: -0.00117135 182: -0.000195225 183: -0.00032526 184: -1.77877e-19 187: -0.000457476 188: 7.62459e-05 189: -0.00110799 190: 0.000184666 193: -0.000228738 194: -1.18585e-19 561: 0.000173611 562: 0.000152492 571: 0.00021684 Row 19: 18: 0.00228738 19: 0.0143341 20: 0.00238991 38: 0.00477716 50: 0.00487969 187: -0.000457476 188: -7.62459e-05 189: -0.000228738 190: -1.25361e-19 191: -0.000477982 192: 7.96636e-05 193: -0.000955432 194: 0.000159239 195: -0.000975938 196: 0.000162656 201: -0.000238991 202: -1.18585e-19 311: -0.000248978 312: 1.28749e-19 562: 0.000152492 580: 0.000159327 603: 0.000165986 Row 20: 19: 0.00238991 20: 0.0143452 21: 0.00232874 39: 0.00478268 50: 0.00484385 191: -0.000477982 192: -7.96636e-05 195: -0.000238991 196: -1.29596e-19 197: -0.000465749 198: 7.76248e-05 199: -0.000956536 200: 0.000159423 201: -0.000968769 202: 0.000161462 205: -0.000232874 206: -1.21973e-19 315: -0.000245394 316: -1.28749e-19 563: 0.00015525 580: 0.000159327 598: 0.000163596 Row 21: 20: 0.00232874 21: 0.0140201 22: 0.00232374 39: 0.00468632 51: 0.00468132 197: -0.000465749 198: -7.76248e-05 199: -0.000232874 200: -1.26208e-19 203: -0.000464748 204: 7.7458e-05 205: -0.000937265 206: 0.000156211 207: -0.000936264 208: 0.000156044 213: -0.000232374 214: -1.18585e-19 317: -0.000235758 318: 1.21973e-19 563: 0.00015525 581: 0.000154916 593: 0.000157172 Row 22: 21: 0.00232374 22: 0.0139707 23: 0.00235233 40: 0.00466161 51: 0.00463302 203: -0.000464748 204: -7.7458e-05 207: -0.000232374 208: -1.27055e-19 209: -0.000470467 210: 7.84111e-05 211: -0.000932321 212: 0.000155387 213: -0.000926603 214: 0.000154434 217: -0.000235233 218: -1.21973e-19 323: -0.000230927 324: -1.18585e-19 564: 0.000156822 581: 0.000154916 599: 0.000153952 Row 23: 22: 0.00235233 23: 0.0131126 24: 0.00219508 40: 0.00436123 52: 0.00420399 209: -0.000470467 210: -7.84111e-05 211: -0.000235233 212: -1.27902e-19 215: -0.000439017 216: 7.31695e-05 217: -0.000872247 218: 0.000145374 219: -0.000840797 220: 0.000140133 225: -0.000219508 226: -1.11808e-19 325: -0.00020089 326: 1.05032e-19 564: 0.000156822 583: 0.000146339 594: 0.000133927 Row 24: 3: 0.00260417 23: 0.00219508 24: 0.0212547 25: 0.00585915 41: 0.00582808 52: 0.00476818 101: -0.000520833 102: -8.68056e-05 103: -0.000260417 104: -3.38813e-21 215: -0.000439017 216: -7.31695e-05 219: -0.000219508 220: -1.19432e-19 221: -0.00117183 222: 0.000195305 223: -0.00116562 224: 0.000194269 225: -0.000953636 226: 0.000158939 229: -0.000325499 230: -1.69407e-19 331: -0.00025731 332: -1.32137e-19 565: 0.000173611 572: 0.000216999 583: 0.000146339 619: 0.00017154 Row 25: 3: 0.00260417 24: 0.00585915 25: 0.016254 26: 0.00226785 41: 0.00552283 101: -0.000260417 102: -1.42302e-19 103: -0.000520833 104: -8.68056e-05 221: -0.00117183 222: -0.000195305 223: -0.000325499 224: -1.77877e-19 227: -0.000453569 228: 7.55949e-05 229: -0.00110457 230: 0.000184094 233: -0.000226785 234: -1.15196e-19 565: 0.000173611 566: 0.00015119 572: 0.000216999 Row 26: 25: 0.00226785 26: 0.0138853 27: 0.00228072 41: 0.00466194 53: 0.00467481 227: -0.000453569 228: -7.55949e-05 229: -0.000226785 230: -1.2282e-19 231: -0.000456144 232: 7.60239e-05 233: -0.000932388 234: 0.000155398 235: -0.000934963 236: 0.000155827 241: -0.000228072 242: -1.18585e-19 333: -0.00023941 334: 1.21973e-19 566: 0.00015119 585: 0.000152048 601: 0.000159606 Row 27: 26: 0.00228072 27: 0.0134621 28: 0.00215531 42: 0.00445032 53: 0.00457573 231: -0.000456144 232: -7.60239e-05 235: -0.000228072 236: -1.2282e-19 237: -0.000431062 238: 7.18436e-05 239: -0.000890065 240: 0.000148344 241: -0.000915147 242: 0.000152524 245: -0.000215531 246: -1.0842e-19 337: -0.000229502 338: -1.15196e-19 567: 0.000143687 585: 0.000152048 604: 0.000153001 Row 28: 27: 0.00215531 28: 0.0130849 29: 0.00218893 42: 0.00435354 54: 0.00438717 237: -0.000431062 238: -7.18436e-05 239: -0.000215531 240: -1.17738e-19 243: -0.000437786 244: 7.29644e-05 245: -0.000870709 246: 0.000145118 247: -0.000877433 248: 0.000146239 253: -0.000218893 254: -1.11808e-19 339: -0.000219824 340: 1.11808e-19 567: 0.000143687 586: 0.000145929 595: 0.000146549 Row 29: 28: 0.00218893 29: 0.0133995 30: 0.00230402 43: 0.00451081 54: 0.00439572 243: -0.000437786 244: -7.29644e-05 247: -0.000218893 248: -1.18585e-19 249: -0.000460804 250: 7.68007e-05 251: -0.000902162 252: 0.00015036 253: -0.000879144 254: 0.000146524 257: -0.000230402 258: -1.18585e-19 345: -0.000220679 346: -1.15196e-19 568: 0.000153601 586: 0.000145929 600: 0.000147119 Row 30: 29: 0.00230402 30: 0.0129629 31: 0.00218734 43: 0.00429412 55: 0.00417744 249: -0.000460804 250: -7.68007e-05 251: -0.000230402 252: -1.24514e-19 255: -0.000437468 256: 7.29113e-05 257: -0.000858824 258: 0.000143137 259: -0.000835488 260: 0.000139248 263: -0.000218734 264: -1.11808e-19 347: -0.00019901 348: 1.05032e-19 568: 0.000153601 587: 0.000145823 596: 0.000132673 Row 31: 0: 0.00260417 4: 0.0058886 30: 0.00218734 31: 0.021317 32: 0.00586698 55: 0.00476988 89: -0.000260417 90: -3.38813e-21 91: -0.000520833 92: -8.68056e-05 107: -0.00117772 108: -0.000196287 109: -0.000328444 110: -1.01644e-20 255: -0.000437468 256: -7.29113e-05 259: -0.000218734 260: -1.20279e-19 261: -0.0011734 262: 0.000195566 263: -0.000953976 264: 0.000158996 267: -0.000258254 268: -1.32137e-19 553: 0.000173611 569: 0.000218962 587: 0.000145823 627: 0.000172169 Row 32: 4: 0.00558272 5: 0.00475257 31: 0.00586698 32: 0.0316705 44: 0.00527279 55: 0.00497975 67: 0.00521572 105: -0.000229828 106: -5.0822e-21 107: -0.000328444 108: -1.81265e-19 109: -0.00111654 110: -0.000186091 113: -0.000950513 114: -0.000158419 115: -0.000245428 116: -5.0822e-21 261: -0.0011734 262: -0.000195566 263: -0.000258254 264: -1.42302e-19 265: -0.00105456 266: 0.00017576 267: -0.00099595 268: 0.000165992 269: -0.00104314 270: 0.000173857 355: -0.00028185 356: -1.42302e-19 391: -0.000239721 392: 1.25361e-19 554: 0.000153219 569: 0.000218962 584: 0.000163619 623: 0.0001879 627: 0.000172169 632: 0.000159814 Row 33: 6: 0.00464288 7: 0.00455563 33: 0.0273626 44: 0.0046814 45: 0.00448984 61: 0.00444425 66: 0.00454856 117: -0.000225885 118: -8.47033e-21 119: -0.000928575 120: -0.000154763 121: -0.000238402 122: -1.30443e-19 125: -0.000911126 126: -0.000151854 127: -0.000229678 128: -8.47033e-21 271: -0.00093628 272: 0.000156047 273: -0.000897969 274: 0.000149661 275: -0.000888851 276: 0.000148142 277: -0.000909713 278: 0.000151619 353: -0.000229738 354: 1.15196e-19 357: -0.000219307 358: -1.15196e-19 431: -0.000225119 432: -1.15196e-19 555: 0.00015059 582: 0.000158935 589: 0.000153119 612: 0.000146204 621: 0.000153158 622: 0.000150079 Row 34: 8: 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288: 0.000169635 289: -0.00107074 290: 0.000178457 291: -0.00106491 292: 0.000177486 361: -0.000246497 362: 1.21973e-19 363: -0.00028596 364: -1.49078e-19 558: 0.000153468 570: 0.000222005 602: 0.000166274 609: 0.000174938 630: 0.000164331 637: 0.00019064 Row 36: 13: 0.00455091 14: 0.0043796 36: 0.0266468 47: 0.0047088 48: 0.00424451 59: 0.00423499 68: 0.00452796 157: -0.00021948 158: -6.77626e-21 159: -0.000910182 160: -0.000151697 161: -0.00023561 162: -1.28749e-19 165: -0.00087592 166: -0.000145987 167: -0.000218479 168: -5.0822e-21 293: -0.000941759 294: 0.00015696 295: -0.000848903 296: 0.000141484 297: -0.000846999 298: 0.000141166 299: -0.000905593 300: 0.000150932 365: -0.000235269 366: 1.18585e-19 367: -0.000205972 368: -1.0842e-19 415: -0.000217527 416: -1.0842e-19 559: 0.00014632 588: 0.000157074 591: 0.000145653 608: 0.000137315 628: 0.000156846 629: 0.000145018 Row 37: 15: 0.00432972 16: 0.00410704 37: 0.0258685 48: 0.00416669 49: 0.00415419 60: 0.00466053 69: 0.00445034 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-2.15269e-05 430: 8.78797e-21 636: 1.07635e-05 Row 637: 35: 0.00019064 47: 0.00019064 57: 0.00019064 289: -2.72343e-05 290: -1.09855e-21 291: -2.72343e-05 292: -1.24937e-20 363: -2.72343e-05 364: -1.10114e-20 637: 1.36171e-05 Row 638: 50: 0.000182668 60: 0.000182668 62: 0.000182668 373: -2.60954e-05 374: -1.18585e-20 375: -2.60954e-05 376: -1.68169e-21 423: -2.60954e-05 424: 1.07997e-20 638: 1.30477e-05 Row 639: 51: 0.000156239 58: 0.000156239 70: 0.000156239 377: -2.23199e-05 378: -1.00585e-20 379: -2.23199e-05 380: -1.0968e-21 409: -2.23199e-05 410: 8.89385e-21 639: 1.116e-05 Row 640: 40: 0.000152513 51: 0.000152513 70: 0.000152513 323: -2.17876e-05 324: -9.84676e-21 329: -2.17876e-05 330: -1.23601e-21 379: -2.17876e-05 380: 9.1056e-21 640: 1.08938e-05 Row 641: 47: 0.000183468 57: 0.000183468 68: 0.000183468 363: -2.62097e-05 364: -1.19643e-20 365: -2.62097e-05 366: -1.47477e-21 403: -2.62097e-05 404: 1.07997e-20 641: 1.31048e-05 Row 642: 37: 0.000150127 49: 0.000150127 60: 0.000150127 303: -2.14467e-05 304: -1.00337e-21 305: -2.14467e-05 306: -9.635e-21 371: -2.14467e-05 372: -8.78797e-21 642: 1.07233e-05 Row 643: 37: 0.000160575 60: 0.000160575 69: 0.000160575 305: -2.29393e-05 306: -1.16082e-21 307: -2.29393e-05 308: -1.0482e-20 425: -2.29393e-05 426: -9.31736e-21 643: 1.14697e-05 Row 644: 43: 0.000148005 54: 0.000148005 71: 0.000148005 345: -2.11436e-05 346: -9.635e-21 351: -2.11436e-05 352: -9.85995e-22 389: -2.11436e-05 390: 8.99973e-21 644: 1.05718e-05 Row 645: 40: 0.000178163 65: 0.000178163 70: 0.000178163 327: -2.54519e-05 328: -1.30388e-21 329: -2.54519e-05 330: -1.14349e-20 451: -2.54519e-05 452: -1.03762e-20 645: 1.27259e-05 Row 646: 54: 0.000143445 56: 0.000143445 71: 0.000143445 387: -2.04922e-05 388: -9.42324e-21 389: -2.04922e-05 390: -1.04859e-21 395: -2.04922e-05 396: 8.78797e-21 646: 1.02461e-05 Row 647: 43: 0.000154094 55: 0.000154094 67: 0.000154094 347: -2.20134e-05 348: -1.01474e-21 349: -2.20134e-05 350: -1.01644e-20 391: -2.20134e-05 392: -9.1056e-21 647: 1.10067e-05 Row 648: 43: 0.000175565 67: 0.000175565 71: 0.000175565 349: -2.50808e-05 350: -1.07536e-21 351: -2.50808e-05 352: -1.14349e-20 461: -2.50808e-05 462: -1.05879e-20 648: 1.25404e-05 Row 649: 56: 0.000146709 71: 0.000146709 72: 0.000146709 395: -2.09585e-05 396: -1.01002e-21 397: -2.09585e-05 398: -9.42324e-21 477: -2.09585e-05 478: -8.68209e-21 649: 1.04792e-05 Row 650: 58: 0.000155324 70: 0.000155324 73: 0.000155324 409: -2.21891e-05 410: -8.99897e-22 411: -2.21891e-05 412: -1.00585e-20 473: -2.21891e-05 474: -8.89385e-21 650: 1.10946e-05 Row 651: 57: 0.000180804 63: 0.000180804 74: 0.000180804 401: -2.58292e-05 402: -1.19643e-20 405: -2.58292e-05 406: -1.59915e-21 441: -2.58292e-05 442: 1.07997e-20 651: 1.29146e-05 Row 652: 57: 0.000180757 68: 0.000180757 74: 0.000180757 403: -2.58224e-05 404: -1.27809e-21 405: -2.58224e-05 406: -1.15408e-20 465: -2.58224e-05 466: -1.05879e-20 652: 1.29112e-05 Row 653: 56: 0.000117884 64: 0.000117884 75: 0.000117884 393: -1.68406e-05 394: -7.72918e-21 399: -1.68406e-05 400: -8.72298e-22 447: -1.68406e-05 448: 6.88214e-21 653: 8.42031e-06 Row 654: 56: 0.000117663 72: 0.000117663 75: 0.000117663 397: -1.6809e-05 398: -6.42452e-22 399: -1.6809e-05 400: -7.51742e-21 481: -1.6809e-05 482: -6.98802e-21 654: 8.40451e-06 Row 655: 59: 0.0001143 69: 0.0001143 76: 0.0001143 417: -1.63286e-05 418: -7.48248e-22 419: -1.63286e-05 420: -7.30566e-21 469: -1.63286e-05 470: -6.77626e-21 655: 8.1643e-06 Row 656: 60: 0.000162153 69: 0.000162153 77: 0.000162153 425: -2.31646e-05 426: -1.03762e-20 427: -2.31646e-05 428: -1.32118e-21 471: -2.31646e-05 472: 9.95264e-21 656: 1.15823e-05 Row 657: 69: 0.000115157 76: 0.000115157 77: 0.000115157 469: -1.6451e-05 470: -7.30566e-21 471: -1.6451e-05 472: -9.59613e-22 507: -1.6451e-05 508: 6.67038e-21 657: 8.22551e-06 Row 658: 60: 0.000166875 62: 0.000166875 77: 0.000166875 423: -2.38393e-05 424: -1.03041e-21 427: -2.38393e-05 428: -1.0482e-20 437: -2.38393e-05 438: -9.74088e-21 658: 1.19197e-05 Row 659: 61: 0.000128172 66: 0.000128172 78: 0.000128172 431: -1.83103e-05 432: -8.25857e-21 433: -1.83103e-05 434: -9.77634e-22 457: -1.83103e-05 458: 7.72918e-21 659: 9.15517e-06 Row 660: 61: 0.000152314 63: 0.000152314 79: 0.000152314 429: -2.17592e-05 430: -1.08483e-21 435: -2.17592e-05 436: -9.74088e-21 443: -2.17592e-05 444: -8.68209e-21 660: 1.08796e-05 Row 661: 61: 0.000123653 78: 0.000123653 79: 0.000123653 433: -1.76648e-05 434: -7.94093e-21 435: -1.76648e-05 436: -9.80977e-22 517: -1.76648e-05 518: 7.19978e-21 661: 8.83239e-06 Row 662: 63: 0.000151977 74: 0.000151977 79: 0.000151977 441: -2.1711e-05 442: -8.39623e-22 443: -2.1711e-05 444: -9.84676e-21 497: -2.1711e-05 498: -8.89385e-21 662: 1.08555e-05 Row 663: 64: 0.000167065 65: 0.000167065 80: 0.000167065 445: -2.38664e-05 446: -1.11173e-20 449: -2.38664e-05 450: -1.25054e-21 453: -2.38664e-05 454: 1.02703e-20 663: 1.19332e-05 Row 664: 65: 0.000168922 70: 0.000168922 80: 0.000168922 451: -2.41318e-05 452: -1.09055e-20 453: -2.41318e-05 454: -1.35543e-21 475: -2.41318e-05 476: 9.74088e-21 664: 1.20659e-05 Row 665: 64: 0.000119178 75: 0.000119178 80: 0.000119178 447: -1.70254e-05 448: -8.83364e-22 449: -1.70254e-05 450: -7.51742e-21 503: -1.70254e-05 504: -6.98802e-21 665: 8.51269e-06 Row 666: 59: 0.000143906 68: 0.000143906 81: 0.000143906 415: -2.0558e-05 416: -9.31736e-21 421: -2.0558e-05 422: -1.11554e-21 467: -2.0558e-05 468: 8.57621e-21 666: 1.0279e-05 Row 667: 68: 0.000146838 74: 0.000146838 81: 0.000146838 465: -2.09769e-05 466: -9.74088e-21 467: -2.09769e-05 468: -1.18203e-21 499: -2.09769e-05 500: 8.89385e-21 667: 1.04884e-05 Row 668: 59: 0.000113315 76: 0.000113315 81: 0.000113315 419: -1.61879e-05 420: -7.50892e-22 421: -1.61879e-05 422: -7.19978e-21 509: -1.61879e-05 510: -6.56451e-21 668: 8.09394e-06 Row 669: 67: 0.000172857 71: 0.000172857 82: 0.000172857 461: -2.46939e-05 462: -1.12232e-20 463: -2.46939e-05 464: -1.17421e-21 479: -2.46939e-05 480: 1.06938e-20 669: 1.23469e-05 Row 670: 71: 0.000145027 72: 0.000145027 82: 0.000145027 477: -2.07181e-05 478: -9.52912e-21 479: -2.07181e-05 480: -1.07255e-21 483: -2.07181e-05 484: 8.47033e-21 670: 1.0359e-05 Row 671: 66: 0.000174754 67: 0.000174754 82: 0.000174754 455: -2.49649e-05 456: -1.13291e-20 459: -2.49649e-05 460: -1.39356e-21 463: -2.49649e-05 464: 1.02703e-20 671: 1.24824e-05 Row 672: 58: 0.000154839 62: 0.000154839 83: 0.000154839 407: -2.21199e-05 408: -9.84676e-21 413: -2.21199e-05 414: -1.17434e-21 439: -2.21199e-05 440: 9.31736e-21 672: 1.10599e-05 Row 673: 62: 0.000145044 77: 0.000145044 83: 0.000145044 437: -2.07206e-05 438: -9.52912e-21 439: -2.07206e-05 440: -1.09938e-21 513: -2.07206e-05 514: 8.25857e-21 673: 1.03603e-05 Row 674: 58: 0.000144717 73: 0.000144717 83: 0.000144717 411: -2.06739e-05 412: -9.27555e-22 413: -2.06739e-05 414: -9.52912e-21 491: -2.06739e-05 492: -8.57621e-21 674: 1.03369e-05 Row 675: 70: 0.000146036 73: 0.000146036 80: 0.000146036 473: -2.08622e-05 474: -9.42324e-21 475: -2.08622e-05 476: -1.20145e-21 489: -2.08622e-05 490: 8.78797e-21 675: 1.04311e-05 Row 676: 66: 0.000127291 78: 0.000127291 82: 0.000127291 457: -1.81844e-05 458: -9.55124e-22 459: -1.81844e-05 460: -8.04681e-21 519: -1.81844e-05 520: -7.30566e-21 676: 9.0922e-06 Row 677: 73: 0.000146885 80: 0.000146885 84: 0.000146885 489: -2.09836e-05 490: -9.09796e-22 493: -2.09836e-05 494: -9.31736e-21 527: -2.09836e-05 528: -8.68209e-21 677: 1.04918e-05 Row 678: 75: 0.000119572 80: 0.000119572 84: 0.000119572 503: -1.70817e-05 504: -7.72918e-21 505: -1.70817e-05 506: -1.02421e-21 527: -1.70817e-05 528: 6.98802e-21 678: 8.54083e-06 Row 679: 72: 0.00011776 75: 0.00011776 84: 0.00011776 481: -1.68229e-05 482: -7.72918e-21 485: -1.68229e-05 486: -8.6112e-22 505: -1.68229e-05 506: 6.98802e-21 679: 8.41143e-06 Row 680: 76: 0.000114288 77: 0.000114288 85: 0.000114288 507: -1.63269e-05 508: -7.10701e-22 511: -1.63269e-05 512: -7.30566e-21 515: -1.63269e-05 516: -6.35275e-21 680: 8.16344e-06 Row 681: 77: 0.000141438 83: 0.000141438 85: 0.000141438 513: -2.02054e-05 514: -9.7042e-22 515: -2.02054e-05 516: -9.31736e-21 535: -2.02054e-05 536: -8.36445e-21 681: 1.01027e-05 Row 682: 72: 0.000163837 82: 0.000163837 86: 0.000163837 483: -2.34053e-05 484: -1.0622e-21 487: -2.34053e-05 488: -1.01644e-20 533: -2.34053e-05 534: -9.31736e-21 682: 1.17027e-05 Row 683: 72: 0.000163008 84: 0.000163008 86: 0.000163008 485: -2.32869e-05 486: -1.09055e-20 487: -2.32869e-05 488: -1.14236e-21 539: -2.32869e-05 540: 9.74088e-21 683: 1.16434e-05 Row 684: 78: 0.000144071 82: 0.000144071 86: 0.000144071 519: -2.05816e-05 520: -9.42324e-21 521: -2.05816e-05 522: -1.27997e-21 533: -2.05816e-05 534: 8.47033e-21 684: 1.02908e-05 Row 685: 78: 0.000141576 79: 0.000141576 86: 0.000141576 517: -2.02252e-05 518: -9.73181e-22 521: -2.02252e-05 522: -9.1056e-21 523: -2.02252e-05 524: -8.25857e-21 685: 1.01126e-05 Row 686: 74: 0.000150894 79: 0.000150894 87: 0.000150894 497: -2.15562e-05 498: -9.74088e-21 501: -2.15562e-05 502: -1.05437e-21 525: -2.15562e-05 526: 9.1056e-21 686: 1.07781e-05 Row 687: 79: 0.000163851 86: 0.000163851 87: 0.000163851 523: -2.34072e-05 524: -1.05879e-20 525: -2.34072e-05 526: -1.1763e-21 547: -2.34072e-05 548: 1.00585e-20 687: 1.17036e-05 Row 688: 74: 0.000146983 81: 0.000146983 87: 0.000146983 499: -2.09975e-05 500: -1.06453e-21 501: -2.09975e-05 502: -9.42324e-21 531: -2.09975e-05 532: -8.78797e-21 688: 1.04988e-05 Row 689: 76: 0.000113292 81: 0.000113292 85: 0.000113292 509: -1.61846e-05 510: -7.41154e-21 511: -1.61846e-05 512: -7.95905e-22 529: -1.61846e-05 530: 6.98802e-21 689: 8.0923e-06 Row 690: 73: 0.000140563 84: 0.000140563 88: 0.000140563 493: -2.00804e-05 494: -7.87299e-22 495: -2.00804e-05 496: -8.89385e-21 541: -2.00804e-05 542: -8.04681e-21 690: 1.00402e-05 Row 691: 73: 0.00014978 83: 0.00014978 88: 0.00014978 491: -2.13971e-05 492: -9.74088e-21 495: -2.13971e-05 496: -1.02269e-21 537: -2.13971e-05 538: 8.89385e-21 691: 1.06986e-05 Row 692: 84: 0.000157217 86: 0.000157217 88: 0.000157217 539: -2.24595e-05 540: -1.07301e-21 541: -2.24595e-05 542: -1.02703e-20 549: -2.24595e-05 550: -9.31736e-21 692: 1.12298e-05 Row 693: 83: 0.00013705 85: 0.00013705 88: 0.00013705 535: -1.95786e-05 536: -8.89385e-21 537: -1.95786e-05 538: -1.04085e-21 545: -1.95786e-05 546: 8.25857e-21 693: 9.7893e-06 Row 694: 81: 0.000140737 85: 0.000140737 87: 0.000140737 529: -2.01052e-05 530: -8.72046e-22 531: -2.01052e-05 532: -9.21148e-21 543: -2.01052e-05 544: -8.25857e-21 694: 1.00526e-05 Row 695: 86: 0.000169746 87: 0.000169746 88: 0.000169746 547: -2.42494e-05 548: -1.06261e-21 549: -2.42494e-05 550: -1.10114e-20 551: -2.42494e-05 552: -1.01644e-20 695: 1.21247e-05 Row 696: 85: 0.000138883 87: 0.000138883 88: 0.000138883 543: -1.98405e-05 544: -9.1056e-21 545: -1.98405e-05 546: -1.17865e-21 551: -1.98405e-05 552: 8.04681e-21 696: 9.92025e-06
To access the entries we use the vector of nonzero-entries:
print(mstar.AsVector())
0.00520833
0.00260417
0.00260417
-0.000520833
8.68056e-05
-0.000520833
8.68056e-05
-0.000260417
-1.28749e-19
0.000173611
0.00520833
0.00260417
0.00260417
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8.68056e-05
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8.68056e-05
-0.000260417
1.28749e-19
0.000173611
0.00520833
0.00260417
0.00260417
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8.68056e-05
-0.000520833
8.68056e-05
-0.000260417
1.28749e-19
0.000173611
0.00520833
0.00260417
0.00260417
-0.000520833
8.68056e-05
-0.000520833
8.68056e-05
-0.000260417
1.28749e-19
0.000173611
0.00260417
0.0163738
0.00229828
0.0058886
0.00558272
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-8.68056e-05
-0.000260417
-1.42302e-19
-0.000459656
7.66094e-05
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0.000196287
-0.00111654
0.000186091
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-1.15196e-19
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1.69407e-19
0.000173611
0.000153219
0.000218962
0.00229828
0.0142215
0.00235821
0.00475257
0.00481249
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-0.000229828
-1.24514e-19
-0.000471641
7.86068e-05
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0.000158419
-0.000962498
0.000160416
-0.000235821
-1.18585e-19
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1.25361e-19
0.000153219
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0.000163619
0.00235821
0.0140022
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0.00464288
0.00474223
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0.000158074
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0.00015059
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0.000158935
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0.000151854
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0.000152731
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0.00015059
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0.000155333
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0.000152379
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0.000158251
0.000152344
0.000152414
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0.0132999
0.00223318
0.00441679
0.0042762
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7.44392e-05
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0.000147226
-0.00085524
0.00014254
-0.000223318
-1.11808e-19
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1.05032e-19
0.000158251
0.000148878
0.000136201
0.00260417
0.00223318
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-1.98405e-05
8.04681e-21
9.92025e-06
Using the vector we can build the linear combination of the matrices $A$ and $M$:
mstar.AsVector().data = m.mat.AsVector() + dt * a.mat.AsVector()
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")
f += SymbolicLFI(gaussp*v)
f.Assemble()
and the initial data: $u_0 = (1-y^2)x$
gfu = GridFunction(fes)
gfu.Set((1-y*y)*x)
Draw(gfu,mesh,"u")
tstep = 10 # time that we want to step over within one block-run
t_intermediate=0 # time counter within one block-run
res = gfu.vec.CreateVector()
while t_intermediate < tstep - 0.5 * dt:
res.data = dt * f.vec - dt * a.mat * gfu.vec
gfu.vec.data += invmstar * res
t_intermediate += dt
print("\r",time+t_intermediate,end="")
Redraw(blocking=True)
print("")
time+=t_intermediate
9.999999999999897
Next, we want to consider time-dependent r.h.s. data $f=f(t)$:
To this end, we introduce a parameter t representing the time.
A Parameter is a constant CoefficientFunction the value of which can be changed with the Set-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=1
tt = 0
gausspt = exp(-6*((x+sin(omega*t))*(x+sin(omega*t))+y*y))-exp(-6*((x-sin(omega*t))*(x-sin(omega*t))+y*y))
Draw(gausspt,mesh,"ft")
time = 0.0
while time < 10 - 0.5 * dt:
t.Set(time)
tt = time
#gausspt = exp(-6*((x+sin(omega*tt))*(x+sin(omega*tt))+y*y))-exp(-6*((x-sin(omega*tt))*(x-sin(omega*tt))+y*y))
Redraw(blocking=True)
time += 1e-4
Accordingly we define a different linear form which then has to be assembled in every time step.
ft = LinearForm(fes)
ft += SymbolicLFI(gausspt*v)
time = 0.0
t.Set(0.0)
gfu.Set((1-y*y)*x)
#gfu.Set(CoefficientFunction(0))
Draw(gfu,mesh,"u")
tstep = 20 # time that we want to step over within one block-run
t_intermediate=0 # time counter within one block-run
res = gfu.vec.CreateVector()
while t_intermediate < tstep - 0.5 * dt:
t.Set(time+t_intermediate+dt)
ft.Assemble()
res.data = dt * ft.vec - dt * a.mat * gfu.vec
gfu.vec.data += invmstar * res
t_intermediate += dt
print("\r",time+t_intermediate,end="")
Redraw(blocking=True)
print("")
time+=t_intermediate
20.00000000000146
Now, we want to consider changing boundary conditions $u|_{\partial \Omega} = u_D(t)$ and set $f=0$.
We use the simple implicit Euler time stepping method again for this problem. Now, we directly use the non-incremental form: $$ (M + \Delta t A) u^{n+1} = M u^n $$ We homogenize w.r.t. to the boundary conditions, i.e. we split $u^{n+1} = u^{n+1}_0 + u^{n+1}_D$ where $u^{n+1}_D$ is some function in the FESpace with the correct boundary condition and $u^{n+1}_0$ is the sought for function with zero boundary values:
$$ \underbrace{(M + \Delta t A)}_{M^\ast} u^{n+1}_0 = M u^n - \underbrace{(M + \Delta t A)}_{M^\ast} u^{n+1}_D $$uD = CoefficientFunction( [(1-x*x)*IfPos(sin(0.3*pi*t),sin(0.3*pi*t),0),0,0,0])
time = 0.0
t.Set(0.0)
gfu.Set(uD,BND)
Draw(gfu,mesh,"u")
# visualization stuff
from ngsolve.internal import *
visoptions.mminval = 0.0
visoptions.mmaxval = 0.2
visoptions.deformation = 0
visoptions.autoscale = 0
tstep = 10 # time that we want to step over within one block-run
t_intermediate=0 # time counter within one block-run
res = gfu.vec.CreateVector()
while t_intermediate < tstep - 0.5 * dt:
t.Set(time+t_intermediate+dt)
#non-incremental form
res.data = m.mat * gfu.vec
gfu.Set(uD,BND)
res.data -= mstar * gfu.vec
gfu.vec.data += invmstar * res
t_intermediate += dt
print("\r",time+t_intermediate,end="")
Redraw(blocking=True)
print("")
time+=t_intermediate
9.999999999999897
We consider more sophisticated time integration methods, SDIRK methods. To simplify presentation we set $f=0$.
SDIRK methods for the semi-discrete problem $\frac{d}{dt} u = M^{-1} F(u) = -M^{-1} \cdot A u$ are of the form: $$ u^{n+1} = u^n + \Delta t M^{-1} \sum_{i=0}^{s-1} b_i k_i $$ with $$ k_i = - A u_i \quad \text{ where $u_i$ is the solution to } \quad (M + a^\ast \Delta t A) u_i = M u^n - \Delta t \sum_{i=0}^{i-1} a_{ij} k_j, \quad i=0,..,s-1. $$ The coefficients a,b and c are stored in the butcher tableau: $$ \begin{array}{c|cccc} c_0 & a_{00} & 0 & \ddots& \\ c_1 & a_{10} & a_{11} & 0 & \ddots \\ \vdots & \vdots & \vdots & \ddots & 0\\ c_{s-1} & a_{s-1,0} & a_{s-1,1} & \dots& a_{s-1,s-1} \\ \hline & b_{0} & b_1 & \dots& b_{s-1} \\ \end{array} $$ For an SDIRK method we have $a^\ast = a_{ii},~i=0,..,s-1$.
We can use for example the 2 stage SDIRK (order 3) method $$ \begin{array}{c|cc} p & p & 0 \\ 1-p & 1-2p & p \\ \hline & 1/2 & 1/2 \\ \end{array} $$ with $p = \frac{3 - \sqrt{3}}{6}$.
class sdirk1: #order 1 (implicit Euler)
stages = 1
a = [[1]]
b = [1]
c = [1]
astar = 1
class sdirk2: #order 3
p = (3 - sqrt(3))/6
stages = 2
a = [[p, 0],
[1 - 2*p, p]]
b = [1/2, 1/2]
c = [p, 1 - p]
astar = p
class sdirk5: #order 4
stages = 5
a = [[1/4, 0, 0, 0, 0],
[1/2, 1/4, 0, 0, 0],
[17/50,-1/25, 1/4, 0, 0],
[371/1360, -137/2720, 15/544, 1/4,0],
[25/24, -49/48, 125/16, -85/12, 1/4]]
b = [25/24, -49/48, 125/16, -85/12, 1/4]
c = [1/4, 3/4, 11/20, 1/2, 1]
astar = 1/4
butchertab = sdirk2()
rhsi = gfu.vec.CreateVector()
Mu0 = gfu.vec.CreateVector()
ui = gfu.vec.CreateVector()
k = [gfu.vec.CreateVector() for i in range(butchertab.stages)]
We have to update the $M^\ast$ matrix and reset initial data
time = 0.0
t.Set(0.0)
gfu.Set(uD,BND)
Draw(gfu,mesh,"u")
# visualization stuff
from ngsolve.internal import *
visoptions.mminval = 0.0
visoptions.mmaxval = 0.2
visoptions.deformation = 0
visoptions.autoscale = 0
mstar.AsVector().data = m.mat.AsVector() + butchertab.astar * dt * a.mat.AsVector()
invmstar = mstar.Inverse(freedofs=fes.FreeDofs())
invmass = m.mat.Inverse(freedofs=fes.FreeDofs())
tstep = 10 # time that we want to step over within one block-run
t_intermediate=0 # time counter within one block-run
while t_intermediate < tstep - 0.5 * dt:
Mu0.data = m.mat * gfu.vec
for i in range(butchertab.stages):
# add up the ks as prescribed in the butcher tableau
rhsi.data = Mu0
for j in range(0,i):
rhsi.data += dt * butchertab.a[i][j] * k[j]
# Solve for ui (with homogenization for the boundary data)
t.Set(time+t_intermediate+butchertab.c[i]*dt)
gfu.Set(uD,BND)
ui.data = gfu.vec
rhsi.data -= mstar * ui
ui.data += invmstar * rhsi
# compute k[i] from ui
k[i].data = - a.mat * ui
t_intermediate += dt
t.Set(time+t_intermediate)
# Adding up the ks:
gfu.Set(uD,BND)
Mu0.data -= m.mat * gfu.vec
for i in range(0,butchertab.stages):
Mu0.data += dt * butchertab.b[i] * k[i]
gfu.vec.data += invmass * Mu0
print("\r",time+t_intermediate,end="")
Redraw(blocking=True)
print("")
time+=t_intermediate
9.999999999999897
Outputting the nice pictures to vtk (to visualize with paraview):
vtk = VTKOutput(ma=mesh,coefs=[gfu],
names=["sol"],
filename="vtk_example1",
subdivision=3)
vtk.Do()
You can also export vector fields:
vtk = VTKOutput(ma=mesh,coefs=[gfu,grad(gfu)],names=["sol","gradsol"],filename="vtk_example2",subdivision=3)
vtk.Do()
And time dependent data:
vtk = VTKOutput(ma=mesh,coefs=[gfu],names=["sol"],filename="vtk_example3",subdivision=3)
vtk.Do()
gfu.Set((1-y*y)*x)
time = 0
tstep = 1 # time that we want to step over within one block-run
t_intermediate=0 # time counter within one block-run
res = gfu.vec.CreateVector()
i = 0
while t_intermediate < tstep - 0.5 * dt:
res.data = dt * f.vec - dt * a.mat * gfu.vec
gfu.vec.data += invmstar * res
t_intermediate += dt
print("\r",time+t_intermediate,end="")
Redraw(blocking=True)
i += 1
if (i%10 == 0):
vtk.Do()
print("")
1.0000000000000007