Numerische Approximation
Approximation des Federpendels
Wir betrachten eine ideale Feder ohne Reibung. Federkonstante D und Masse m sind frei einstellbar (anfangs jeweils 1). Wähle beliebige Verfahren zum direkten Vergleich. Mit h und N bestimmst du die Schrittweite und die Anzahl Schritte: T = N · h.
Die Linien verbinden die berechneten Stützstellen. Der Fehler ist der absolute Fehler der gewählten Größe (Position in m oder Gesamtenergie in J). Die Gesamtenergie E = ½Dx² + ½mv² bleibt in der exakten Lösung konstant. Große Schritte können ungenau oder instabil sein; beim expliziten Euler wächst die Schwingungsamplitude künstlich.
Reibungsfreies FederpendelLive
| Verfahren | Fehler (m) |
|---|---|
| Euler | 1.575e+0 |
| Euler-Cromer | 5.726e-2 |
| Heun | 2.961e-2 |
| Leapfrog | 7.631e-3 |
| Runge-Kutta-4 | 1.460e-5 |
Wertetabellen zu den Diagrammen
CSV-Downloads enthalten alle Stützstellen ohne zusätzliche Rundung, mit Semikolon als Trennzeichen und Dezimalpunkt.
Die ersten 50 Stützstellen je Verfahren. Die Diagramme enthalten alle berechneten Punkte. Die Fehler werden auch bei ausgeblendeter exakter Kurve gegen die analytische Lösung berechnet.
| t | Numerisch | Exakt | Fehler |
|---|---|---|---|
| 0.00 | 1.00000 | 1.00000 | 0.000e+0 |
| 0.10 | 1.00000 | 0.995004 | 4.996e-3 |
| 0.20 | 0.990000 | 0.980067 | 9.933e-3 |
| 0.30 | 0.970000 | 0.955336 | 1.466e-2 |
| 0.40 | 0.940100 | 0.921061 | 1.904e-2 |
| 0.50 | 0.900500 | 0.877583 | 2.292e-2 |
| 0.60 | 0.851499 | 0.825336 | 2.616e-2 |
| 0.70 | 0.793493 | 0.764842 | 2.865e-2 |
| 0.80 | 0.726972 | 0.696707 | 3.027e-2 |
| 0.90 | 0.652516 | 0.621610 | 3.091e-2 |
| 1.00 | 0.570790 | 0.540302 | 3.049e-2 |
| 1.10 | 0.482540 | 0.453596 | 2.894e-2 |
| 1.20 | 0.388581 | 0.362358 | 2.622e-2 |
| 1.30 | 0.289797 | 0.267499 | 2.230e-2 |
| 1.40 | 0.187127 | 0.169967 | 1.716e-2 |
| 1.50 | 0.0815591 | 0.0707372 | 1.082e-2 |
| 1.60 | -0.0258801 | -0.0291995 | 3.319e-3 |
| 1.70 | -0.134135 | -0.128844 | 5.290e-3 |
| 1.80 | -0.242131 | -0.227202 | 1.493e-2 |
| 1.90 | -0.348785 | -0.323290 | 2.550e-2 |
| 2.00 | -0.453019 | -0.416147 | 3.687e-2 |
| 2.10 | -0.553764 | -0.504846 | 4.892e-2 |
| 2.20 | -0.649979 | -0.588501 | 6.148e-2 |
| 2.30 | -0.740657 | -0.666276 | 7.438e-2 |
| 2.40 | -0.824835 | -0.737394 | 8.744e-2 |
| 2.50 | -0.901606 | -0.801144 | 1.005e-1 |
| 2.60 | -0.970129 | -0.856889 | 1.132e-1 |
| 2.70 | -1.02964 | -0.904072 | 1.256e-1 |
| 2.80 | -1.07944 | -0.942222 | 1.372e-1 |
| 2.90 | -1.11895 | -0.970958 | 1.480e-1 |
| 3.00 | -1.14767 | -0.989992 | 1.577e-1 |
| 3.10 | -1.16519 | -0.999135 | 1.661e-1 |
| 3.20 | -1.17124 | -0.998295 | 1.729e-1 |
| 3.30 | -1.16564 | -0.987480 | 1.782e-1 |
| 3.40 | -1.14832 | -0.966798 | 1.815e-1 |
| 3.50 | -1.11935 | -0.936457 | 1.829e-1 |
| 3.60 | -1.07889 | -0.896758 | 1.821e-1 |
| 3.70 | -1.02724 | -0.848100 | 1.791e-1 |
| 3.80 | -0.964806 | -0.790968 | 1.738e-1 |
| 3.90 | -0.892096 | -0.725932 | 1.662e-1 |
| 4.00 | -0.809738 | -0.653644 | 1.561e-1 |
| 4.10 | -0.718459 | -0.574824 | 1.436e-1 |
| 4.20 | -0.619083 | -0.490261 | 1.288e-1 |
| 4.30 | -0.512522 | -0.400799 | 1.117e-1 |
| 4.40 | -0.399770 | -0.307333 | 9.244e-2 |
| 4.50 | -0.281892 | -0.210796 | 7.110e-2 |
| 4.60 | -0.160018 | -0.112153 | 4.787e-2 |
| 4.70 | -0.0353239 | -0.0123887 | 2.294e-2 |
| 4.80 | 0.0909700 | 0.0874990 | 3.471e-3 |
| 4.90 | 0.217617 | 0.186512 | 3.110e-2 |
| t | Numerisch | Exakt | Fehler |
|---|---|---|---|
| 0.00 | 1.00000 | 1.00000 | 0.000e+0 |
| 0.10 | 0.990000 | 0.995004 | 5.004e-3 |
| 0.20 | 0.970100 | 0.980067 | 9.967e-3 |
| 0.30 | 0.940499 | 0.955336 | 1.484e-2 |
| 0.40 | 0.901493 | 0.921061 | 1.957e-2 |
| 0.50 | 0.853472 | 0.877583 | 2.411e-2 |
| 0.60 | 0.796916 | 0.825336 | 2.842e-2 |
| 0.70 | 0.732392 | 0.764842 | 3.245e-2 |
| 0.80 | 0.660543 | 0.696707 | 3.616e-2 |
| 0.90 | 0.582089 | 0.621610 | 3.952e-2 |
| 1.00 | 0.497814 | 0.540302 | 4.249e-2 |
| 1.10 | 0.408561 | 0.453596 | 4.504e-2 |
| 1.20 | 0.315222 | 0.362358 | 4.714e-2 |
| 1.30 | 0.218731 | 0.267499 | 4.877e-2 |
| 1.40 | 0.120052 | 0.169967 | 4.991e-2 |
| 1.50 | 0.0201736 | 0.0707372 | 5.056e-2 |
| 1.60 | -0.0799069 | -0.0291995 | 5.071e-2 |
| 1.70 | -0.179188 | -0.128844 | 5.034e-2 |
| 1.80 | -0.276678 | -0.227202 | 4.948e-2 |
| 1.90 | -0.371401 | -0.323290 | 4.811e-2 |
| 2.00 | -0.462410 | -0.416147 | 4.626e-2 |
| 2.10 | -0.548794 | -0.504846 | 4.395e-2 |
| 2.20 | -0.629691 | -0.588501 | 4.119e-2 |
| 2.30 | -0.704291 | -0.666276 | 3.802e-2 |
| 2.40 | -0.771848 | -0.737394 | 3.445e-2 |
| 2.50 | -0.831687 | -0.801144 | 3.054e-2 |
| 2.60 | -0.883208 | -0.856889 | 2.632e-2 |
| 2.70 | -0.925898 | -0.904072 | 2.183e-2 |
| 2.80 | -0.959328 | -0.942222 | 1.711e-2 |
| 2.90 | -0.983166 | -0.970958 | 1.221e-2 |
| 3.00 | -0.997171 | -0.989992 | 7.179e-3 |
| 3.10 | -1.00121 | -0.999135 | 2.070e-3 |
| 3.20 | -0.995227 | -0.998295 | 3.068e-3 |
| 3.30 | -0.979296 | -0.987480 | 8.183e-3 |
| 3.40 | -0.953573 | -0.966798 | 1.323e-2 |
| 3.50 | -0.918314 | -0.936457 | 1.814e-2 |
| 3.60 | -0.873872 | -0.896758 | 2.289e-2 |
| 3.70 | -0.820691 | -0.848100 | 2.741e-2 |
| 3.80 | -0.759303 | -0.790968 | 3.166e-2 |
| 3.90 | -0.690322 | -0.725932 | 3.561e-2 |
| 4.00 | -0.614438 | -0.653644 | 3.921e-2 |
| 4.10 | -0.532409 | -0.574824 | 4.241e-2 |
| 4.20 | -0.445057 | -0.490261 | 4.520e-2 |
| 4.30 | -0.353254 | -0.400799 | 4.755e-2 |
| 4.40 | -0.257918 | -0.307333 | 4.941e-2 |
| 4.50 | -0.160003 | -0.210796 | 5.079e-2 |
| 4.60 | -0.0604881 | -0.112153 | 5.166e-2 |
| 4.70 | 0.0396316 | -0.0123887 | 5.202e-2 |
| 4.80 | 0.139355 | 0.0874990 | 5.186e-2 |
| 4.90 | 0.237685 | 0.186512 | 5.117e-2 |
| t | Numerisch | Exakt | Fehler |
|---|---|---|---|
| 0.00 | 1.00000 | 1.00000 | 0.000e+0 |
| 0.10 | 0.995000 | 0.995004 | 4.165e-6 |
| 0.20 | 0.980025 | 0.980067 | 4.158e-5 |
| 0.30 | 0.955225 | 0.955336 | 1.116e-4 |
| 0.40 | 0.920848 | 0.921061 | 2.130e-4 |
| 0.50 | 0.877239 | 0.877583 | 3.438e-4 |
| 0.60 | 0.824834 | 0.825336 | 5.015e-4 |
| 0.70 | 0.764159 | 0.764842 | 6.830e-4 |
| 0.80 | 0.695822 | 0.696707 | 8.846e-4 |
| 0.90 | 0.620508 | 0.621610 | 1.102e-3 |
| 1.00 | 0.538971 | 0.540302 | 1.332e-3 |
| 1.10 | 0.452029 | 0.453596 | 1.568e-3 |
| 1.20 | 0.360553 | 0.362358 | 1.805e-3 |
| 1.30 | 0.265460 | 0.267499 | 2.039e-3 |
| 1.40 | 0.167704 | 0.169967 | 2.264e-3 |
| 1.50 | 0.0682636 | 0.0707372 | 2.474e-3 |
| 1.60 | -0.0318633 | -0.0291995 | 2.664e-3 |
| 1.70 | -0.131673 | -0.128844 | 2.829e-3 |
| 1.80 | -0.230166 | -0.227202 | 2.964e-3 |
| 1.90 | -0.326353 | -0.323290 | 3.063e-3 |
| 2.00 | -0.419271 | -0.416147 | 3.124e-3 |
| 2.10 | -0.507988 | -0.504846 | 3.142e-3 |
| 2.20 | -0.591615 | -0.588501 | 3.114e-3 |
| 2.30 | -0.669313 | -0.666276 | 3.037e-3 |
| 2.40 | -0.740304 | -0.737394 | 2.910e-3 |
| 2.50 | -0.803874 | -0.801144 | 2.730e-3 |
| 2.60 | -0.859387 | -0.856889 | 2.498e-3 |
| 2.70 | -0.906286 | -0.904072 | 2.214e-3 |
| 2.80 | -0.944101 | -0.942222 | 1.879e-3 |
| 2.90 | -0.972452 | -0.970958 | 1.494e-3 |
| 3.00 | -0.991055 | -0.989992 | 1.063e-3 |
| 3.10 | -0.999723 | -0.999135 | 5.882e-4 |
| 3.20 | -0.998370 | -0.998295 | 7.476e-5 |
| 3.30 | -0.987007 | -0.987480 | 4.728e-4 |
| 3.40 | -0.965749 | -0.966798 | 1.049e-3 |
| 3.50 | -0.934810 | -0.936457 | 1.647e-3 |
| 3.60 | -0.894498 | -0.896758 | 2.261e-3 |
| 3.70 | -0.845217 | -0.848100 | 2.883e-3 |
| 3.80 | -0.787462 | -0.790968 | 3.505e-3 |
| 3.90 | -0.721812 | -0.725932 | 4.120e-3 |
| 4.00 | -0.648923 | -0.653644 | 4.720e-3 |
| 4.10 | -0.569528 | -0.574824 | 5.296e-3 |
| 4.20 | -0.484421 | -0.490261 | 5.840e-3 |
| 4.30 | -0.394455 | -0.400799 | 6.344e-3 |
| 4.40 | -0.300533 | -0.307333 | 6.800e-3 |
| 4.50 | -0.203595 | -0.210796 | 7.201e-3 |
| 4.60 | -0.104614 | -0.112153 | 7.538e-3 |
| 4.70 | -0.00458191 | -0.0123887 | 7.807e-3 |
| 4.80 | 0.0954987 | 0.0874990 | 8.000e-3 |
| 4.90 | 0.194624 | 0.186512 | 8.112e-3 |
| t | Numerisch | Exakt | Fehler |
|---|---|---|---|
| 0.00 | 1.00000 | 1.00000 | 0.000e+0 |
| 0.10 | 0.995000 | 0.995004 | 4.165e-6 |
| 0.20 | 0.980050 | 0.980067 | 1.658e-5 |
| 0.30 | 0.955299 | 0.955336 | 3.699e-5 |
| 0.40 | 0.920996 | 0.921061 | 6.499e-5 |
| 0.50 | 0.877483 | 0.877583 | 1.000e-4 |
| 0.60 | 0.825194 | 0.825336 | 1.413e-4 |
| 0.70 | 0.764654 | 0.764842 | 1.881e-4 |
| 0.80 | 0.696467 | 0.696707 | 2.394e-4 |
| 0.90 | 0.621316 | 0.621610 | 2.941e-4 |
| 1.00 | 0.539951 | 0.540302 | 3.511e-4 |
| 1.10 | 0.453187 | 0.453596 | 4.090e-4 |
| 1.20 | 0.361891 | 0.362358 | 4.666e-4 |
| 1.30 | 0.266976 | 0.267499 | 5.226e-4 |
| 1.40 | 0.169392 | 0.169967 | 5.755e-4 |
| 1.50 | 0.0701131 | 0.0707372 | 6.242e-4 |
| 1.60 | -0.0298666 | -0.0291995 | 6.671e-4 |
| 1.70 | -0.129548 | -0.128844 | 7.032e-4 |
| 1.80 | -0.227933 | -0.227202 | 7.311e-4 |
| 1.90 | -0.324039 | -0.323290 | 7.499e-4 |
| 2.00 | -0.416905 | -0.416147 | 7.585e-4 |
| 2.10 | -0.505602 | -0.504846 | 7.560e-4 |
| 2.20 | -0.589243 | -0.588501 | 7.417e-4 |
| 2.30 | -0.666991 | -0.666276 | 7.151e-4 |
| 2.40 | -0.738070 | -0.737394 | 6.759e-4 |
| 2.50 | -0.801767 | -0.801144 | 6.237e-4 |
| 2.60 | -0.857447 | -0.856889 | 5.586e-4 |
| 2.70 | -0.904553 | -0.904072 | 4.808e-4 |
| 2.80 | -0.942613 | -0.942222 | 3.906e-4 |
| 2.90 | -0.971247 | -0.970958 | 2.887e-4 |
| 3.00 | -0.990168 | -0.989992 | 1.758e-4 |
| 3.10 | -0.999188 | -0.999135 | 5.293e-5 |
| 3.20 | -0.998216 | -0.998295 | 7.881e-5 |
| 3.30 | -0.987262 | -0.987480 | 2.181e-4 |
| 3.40 | -0.966435 | -0.966798 | 3.634e-4 |
| 3.50 | -0.935944 | -0.936457 | 5.131e-4 |
| 3.60 | -0.896093 | -0.896758 | 6.655e-4 |
| 3.70 | -0.847281 | -0.848100 | 8.188e-4 |
| 3.80 | -0.789997 | -0.790968 | 9.709e-4 |
| 3.90 | -0.724812 | -0.725932 | 1.120e-3 |
| 4.00 | -0.652380 | -0.653644 | 1.264e-3 |
| 4.10 | -0.573424 | -0.574824 | 1.400e-3 |
| 4.20 | -0.488733 | -0.490261 | 1.528e-3 |
| 4.30 | -0.399155 | -0.400799 | 1.644e-3 |
| 4.40 | -0.305586 | -0.307333 | 1.747e-3 |
| 4.50 | -0.208960 | -0.210796 | 1.835e-3 |
| 4.60 | -0.110246 | -0.112153 | 1.907e-3 |
| 4.70 | -0.0104283 | -0.0123887 | 1.960e-3 |
| 4.80 | 0.0894934 | 0.0874990 | 1.994e-3 |
| 4.90 | 0.188520 | 0.186512 | 2.008e-3 |
| t | Numerisch | Exakt | Fehler |
|---|---|---|---|
| 0.00 | 1.00000 | 1.00000 | 0.000e+0 |
| 0.10 | 0.995004 | 0.995004 | 1.389e-9 |
| 0.20 | 0.980067 | 0.980067 | 1.940e-8 |
| 0.30 | 0.955337 | 0.955336 | 5.374e-8 |
| 0.40 | 0.921061 | 0.921061 | 1.038e-7 |
| 0.50 | 0.877583 | 0.877583 | 1.686e-7 |
| 0.60 | 0.825336 | 0.825336 | 2.470e-7 |
| 0.70 | 0.764843 | 0.764842 | 3.373e-7 |
| 0.80 | 0.696707 | 0.696707 | 4.379e-7 |
| 0.90 | 0.621611 | 0.621610 | 5.466e-7 |
| 1.00 | 0.540303 | 0.540302 | 6.612e-7 |
| 1.10 | 0.453597 | 0.453596 | 7.794e-7 |
| 1.20 | 0.362359 | 0.362358 | 8.986e-7 |
| 1.30 | 0.267500 | 0.267499 | 1.016e-6 |
| 1.40 | 0.169968 | 0.169967 | 1.129e-6 |
| 1.50 | 0.0707384 | 0.0707372 | 1.235e-6 |
| 1.60 | -0.0291982 | -0.0291995 | 1.331e-6 |
| 1.70 | -0.128843 | -0.128844 | 1.415e-6 |
| 1.80 | -0.227201 | -0.227202 | 1.484e-6 |
| 1.90 | -0.323288 | -0.323290 | 1.536e-6 |
| 2.00 | -0.416145 | -0.416147 | 1.568e-6 |
| 2.10 | -0.504845 | -0.504846 | 1.579e-6 |
| 2.20 | -0.588500 | -0.588501 | 1.567e-6 |
| 2.30 | -0.666274 | -0.666276 | 1.530e-6 |
| 2.40 | -0.737392 | -0.737394 | 1.469e-6 |
| 2.50 | -0.801142 | -0.801144 | 1.381e-6 |
| 2.60 | -0.856887 | -0.856889 | 1.267e-6 |
| 2.70 | -0.904071 | -0.904072 | 1.127e-6 |
| 2.80 | -0.942221 | -0.942222 | 9.618e-7 |
| 2.90 | -0.970957 | -0.970958 | 7.714e-7 |
| 3.00 | -0.989992 | -0.989992 | 5.575e-7 |
| 3.10 | -0.999135 | -0.999135 | 3.219e-7 |
| 3.20 | -0.998295 | -0.998295 | 6.646e-8 |
| 3.30 | -0.987480 | -0.987480 | 2.062e-7 |
| 3.40 | -0.966799 | -0.966798 | 4.935e-7 |
| 3.50 | -0.936457 | -0.936457 | 7.921e-7 |
| 3.60 | -0.896760 | -0.896758 | 1.099e-6 |
| 3.70 | -0.848101 | -0.848100 | 1.410e-6 |
| 3.80 | -0.790969 | -0.790968 | 1.722e-6 |
| 3.90 | -0.725934 | -0.725932 | 2.031e-6 |
| 4.00 | -0.653646 | -0.653644 | 2.332e-6 |
| 4.10 | -0.574827 | -0.574824 | 2.622e-6 |
| 4.20 | -0.490264 | -0.490261 | 2.897e-6 |
| 4.30 | -0.400802 | -0.400799 | 3.152e-6 |
| 4.40 | -0.307336 | -0.307333 | 3.383e-6 |
| 4.50 | -0.210799 | -0.210796 | 3.587e-6 |
| 4.60 | -0.112156 | -0.112153 | 3.760e-6 |
| 4.70 | -0.0123926 | -0.0123887 | 3.898e-6 |
| 4.80 | 0.0874950 | 0.0874990 | 4.000e-6 |
| 4.90 | 0.186508 | 0.186512 | 4.061e-6 |