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"AccuracyGoal" and NDSolve

Posted 9 years ago
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Hi!

I would like to know how "AccuracyGoal" works. I know that "AccuracyGoal" is an option for various numerical operations which specifies how many effective digits of accuracy should be sought in the final result" so i tought that if I choose a bigger value the results should be more accurate, but if I put, for example AccuracyGoal->2 I obtain results that are completly different from the results I obtein with AccuracyGoal->4... It seems that a bigger value brings to a worse result and if I reach AccuracyGoal->10 the results disappear!

I also have another point: I plot the evolution of a function (displacement) in the time. I plot the displacement of a point in the x direction and y direction. In both the directions the situation is the same but the solicitation in direction y is 30% of that in x. So I expect that the displacements follow the same evolution but one is simply smaller than the other. This is true and I can see it in the output but just until a certain time (about 20 seconds, while the time history lasts more than 40 seconds). So, I don't understand...what happen from second 20?!

(*Record*)

dT = 0.02;

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   0.0835, -0.0427, -0.0301, 0.048, 0.0143, 0.0125, 0.0377, 0.0116, 
   0.0142, -0.0219, -0.0211, 0.0569, 0.0324, -0.0022, -0.0349, -0.0066, 
   0.2009, 0.0705, -0.2713, -0.2289, -0.0056, 0.0274, 0.0007, 
   0.0277, -0.0239, -0.0671, 0.0309, 0.0559, -0.0114, -0.0092, 0.0599, 
   0.0445, -0.0615, -0.0462, -0.0335, -0.0078, 
   0.0069, -0.0168, -0.0675, -0.0361, 0.0563, 0.1029, 
   0.0713, -0.032, -0.1001, -0.0048, 0.1375, 
   0.1713, -0.0174, -0.2902, -0.2388, -0.1289, -0.1109, -0.0805, 0.0152, 
   0.0294, -0.0821, -0.1841, -0.1935, -0.0633, -0.0376, 0.0176, 0.1738, 
   0.2059, 0.1181, -0.0075, -0.0048, 0.0453, 0.036, -0.0228, -0.077, -0.0463, 
   0.032, 0.037, -0.0419, -0.0616, 0.0408, 
   0.0488, -0.0412, -0.0802, -0.1253, -0.1554, -0.0747, 0.1319, 
   0.1144, -0.0087, 0.0306, 0.0349, 0.0539, 0.0728, 0.1098, 0.1196, 0.0885, 
   0.0782, 0.0463, -0.0031, -0.0259, 0.0055, 0.0414, 0.0739, 0.1158, 
   0.0768, -0.0657, -0.1661, -0.1212, 0.0084, 0.0208, 0.0041, 0.065, 0.0918, 
   0.0676, 0.0035, -0.0739, -0.1074, -0.0524, -0.0186, 0.0478, 0.1711, 0.2268,
    0.1728, 0.0242, -0.0729, -0.1506, -0.0918, 0.0612, 
   0.024, -0.0857, -0.0656, 0.0263, 
   0.0366, -0.0201, -0.0145, -0.0391, -0.0776, -0.0525, -0.0976, -0.0682, 
   0.0213, -0.007, -0.0871, -0.0249, 0.0993, 0.1438, 
   0.0401, -0.2518, -0.332, -0.2115, -0.1179, -0.0065, 0.0788, 0.112, 0.171, 
   0.1447, 0.0003, -0.1357, -0.1396, -0.037, -0.0072, 0.0406, 0.0051, 0.0104, 
   0.071, 0.0497, 0.0548, 0.0749, -0.0083, -0.1041, -0.0016, 0.0607, 0.0394, 
   0.0292, 0.0021, -0.0814, -0.0733, 0.0041, -0.0277, -0.0298, 0.0302, 0.0407,
    0.0027, -0.0144, -0.0371, -0.0425, 0.0197, 0.0958, 
   0.0866, -0.0267, -0.1093, -0.0693, 0.0367, 0.0922, 
   0.0586, -0.0221, -0.0759, -0.0386, 0.0597, 
   0.0594, -0.0063, -0.0638, -0.117, -0.0846, 0.0278, 0.121, 0.1123, 
   0.025, -0.0363, 0.0042, 0.0868, 0.1026, 0.0817, 
   0.0208, -0.0082, -0.0074, -0.0477, -0.0227, 0.0759, 0.0946, 
   0.0143, -0.0609, -0.0576, -0.0004, -0.0036, -0.1016, -0.097, 0.02, 0.0472, 
   0.0313, -0.0246, -0.0466, -0.0194, -0.0451, -0.0763, -0.1171, -0.0968, 
-0.0221, 0.0412, 0.1433, 0.1995, 0.0717, -0.1602, -0.1421, 0.0641, 0.1797, 
   0.1541, 0.0208, -0.0657, -0.112, -0.0591, 0.1236, 
   0.0702, -0.1791, -0.2918, -0.1892, -0.0228, -0.0116, -0.0544, -0.0561, 
   0.066, 0.1739, 0.1398, 0.059, 0.0039, 0.0004, 0.074, 0.1403, 0.1916, 
   0.1674, 0.0762, 0.0027, -0.0426, 0.0223, 0.1027, 0.1614, 
   0.0867, -0.1322, -0.3131, -0.3139, -0.1517, -0.0406, 0.0133, 
   0.0396, -0.0301, -0.201, -0.2275, -0.0505, 0.0577, 0.1915, 0.3301, 0.2343, 
   0.1235, 0.0858, 0.1044, 0.1797, 0.195, 0.1886, 0.0196, -0.3086, -0.0761, 
   0.0692, -0.3487, -0.3312, -0.1134, 0.0917, 0.2933, 0.0735, -0.0173, 0.0624,
    0.0476, 0.0941, -0.0325, 0.1203, 0.2663, 0.1367, -0.1176, -0.185, -0.0253,
    0.0894, 0.1611, 0.0943, -0.141, -0.1633, 0.0378, 0.1362, 0.146, 
   0.0526, -0.0261, -0.0434, -0.0248, -0.0485, -0.0891, -0.1162, -0.1998, \
-0.2119, -0.1698, -0.0786, 0.0481, 0.1385, 0.248, 0.2966, 0.284, 0.203, 
   0.0184, -0.0996, -0.1382, -0.074, -0.0669, -0.0763, 0.0417, 0.1416, 0.2211,
    0.1903, -0.0322, -0.1971, -0.2447, -0.2224, -0.1597, -0.142, -0.0905, \
-0.043, 0.003, 0.0536, 0.0562, 0.0782, 
   0.0735, -0.049, -0.119, -0.162, -0.181, -0.183, -0.1389, 0.0577, 0.1729, 
   0.0468, -0.2103, -0.2416, -0.0804, 0.0464, 0.1305, 
   0.0958, -0.0787, -0.0861, 0.0637, 0.0313, 0.0345, 0.0833, 
   0.0377, -0.0292, -0.1031, -0.1808, -0.1751, -0.0636, 0.0283, 0.1021, 
   0.0812, -0.0564, -0.1066, -0.0212, 0.0104, 0.0192, 
   0.0299, -0.0505, -0.0708, -0.0067, 0.0281, 0.044, 0.0749, 0.0654, 
   0.0082, -0.025, -0.0453, -0.029, 0.0256, 0.0103, -0.0592, -0.0304, 0.0623, 
   0.0886, 0.0691, -0.0023, -0.0709, -0.0306, 0.0737, 0.0915, 0.0636, 0.0577, 
   0.0069, -0.0572, -0.0434, 0.0381, 
   0.0183, -0.0349, -0.045, -0.1226, -0.0972, -0.0117, 0.0196, 0.076, 
   0.0149, -0.0341, 0.0021, -0.0338, -0.11, -0.1043, -0.0244, 0.0402, 0.0632, 
   0.0937, 0.0956, 0.0416, 0.0163, 0.0201, 0.061, 0.0726, 0.0492, 0.0688, 
   0.0776, 0.0792, 0.0981, 0.129, 
   0.0402, -0.1363, -0.2369, -0.2509, -0.1321, -0.0457, -0.0018, 0.0499, 
   0.0665, 0.0156, -0.0182, 0.0638, 0.1661, 0.1383, 0.0126, -0.002, 
   0.036, -0.0119, -0.0661, -0.0454, -0.0229, -0.0482, -0.0438, -0.0728, 
-0.0127, 0.1641, 0.2442, 0.1557, -0.0216, -0.1065, -0.1029, -0.0245, 0.0465, 
   0.0964, 0.102, 0.0438, 0.0119, 0.0934, 0.1466, 0.0467, -0.029, -0.0497, 
   0.001, 0.0253, 0.0102, 0.0653, 0.1294, 0.1468, 
   0.0707, -0.0229, -0.0906, -0.1235, -0.1419, -0.1384, -0.0215, 0.0858, 
   0.1563, 0.1094, -0.051, -0.1494, -0.1197, 0.0192, 
   0.0329, -0.0384, -0.0544, -0.041, 0.0618, 0.1508, 
   0.1165, -0.0003, -0.1077, -0.1241, -0.0141, 0.0577, 0.0655, 
   0.0286, -0.0454, -0.0806, -0.0451, 0.0143, 0.0301, 0.0336, 0.0895, 
   0.0335, -0.1382, -0.1466, 0.0016, 0.0605, -0.0131, -0.0982, -0.0781, 
   0.0136, 0.074, 0.0639, -0.0522, -0.0731, 0.0422, 0.118, 0.1015, 
   0.0339, -0.023, -0.096, -0.1142, -0.1102, -0.126, -0.1351, -0.0968, 0.0461,
    0.1305, 0.0775, -0.043, -0.1205, -0.1306, -0.0922, -0.0573, -0.0484, 
-0.051, -0.0249, 0.0155, 0.0626, 0.1029, 0.0873, 
   0.0041, -0.0564, -0.074, -0.0759, 0.0116, 0.0696, 0.0687, 0.0501, 0.0657, 
   0.0801, 0.0341, 0.0279, 0.0505, -0.0131, -0.0513, 0.0086, 
   0.0051, -0.0389, -0.0071, 0.0352, 0.05, 0.0454, 0.084, 0.1018, 
   0.0298, -0.0338, -0.0464, 0.0151, 0.02, -0.0475, -0.076, -0.0573, 0.0258, 
   0.0617, -0.0251, -0.0794, -0.028, -0.0006, -0.0438, -0.0406, 0.0503, 
   0.1284, 0.0992, -0.017, -0.0658, 0.0087, 0.081, 
   0.0566, -0.0046, -0.0684, -0.0876, -0.0618, -0.0608, -0.035, -0.0344, 
-0.0295, 0.0103, 0.0464, 0.0321, -0.0507, -0.0476, 0.0283, 0.0539, 0.0286, 
   0.0395, 0.0528, 0.0189, 0.0461, 0.0642, 0.0073, -0.0149, 0.0454, 
   0.0642, -0.0015, -0.0926, -0.1003, -0.0517, -0.0025, 0.0787, 0.154, 0.1576,
    0.0478, -0.0144, -0.0547, -0.0719, -0.021, -0.0717, -0.0917, 0.0204, 
   0.1141, 0.0925, -0.0516, -0.1962, -0.1679, 0.0102, 0.1063, 0.1206, 
   0.0481, -0.1055, -0.094, 0.0355, 0.1127, 0.1146, 
   0.0063, -0.1312, -0.1669, -0.0411, 0.0274, 0.0688, 
   0.0535, -0.0507, -0.1143, -0.0981, -0.025, -0.019, 0.0236, 0.088, 
   0.0757, -0.0005, -0.0643, -0.0345, 0.0298, 0.0328, 0.0374, 0.0598, 0.0759, 
   0.007, -0.0096, 0.1016, 0.1252, 0.0045, -0.1357, -0.1571, -0.0613, 0.0031, 
   0.0383, 0.059, 0.024, -0.0912, -0.1004, 0.0711, 0.1007, -0.0042, -0.0404, 
   0.008, 0.0361, 0.0329, 
   0.0008, -0.0346, -0.0292, -0.0523, -0.069, -0.0321, -0.0239, -0.0367, 
-0.057, -0.0688, -0.0139, 0.0211, 0.0082, -0.0609, -0.0537, 0.0507, 
   0.047, -0.0405, -0.0483, 0.0211, 0.0539, 0.0361, 0.0143, 0.0224, 
   0.0247, -0.0146, -0.0061, 0.0256, 0.0177, -0.0024, 0.0333, 0.0438, 0.039, 
   0.0165, -0.0987, -0.1258, -0.0493, -0.0044, 0.0354, 0.0224, -0.0306, 0.006,
    0.0539, 0.0708, 0.1089, 0.1015, -0.0159, -0.0984, -0.0732, 0.0249, 0.1018,
    0.1338, 0.1166, -0.0069, -0.0617, -0.0031, -0.0146, -0.012, 0.0895, 
   0.0974, -0.0407, -0.0827, -0.0336, -0.0405, -0.0307, 
   0.0035, -0.0196, -0.0857, -0.1136, -0.0982, -0.0403, 0.0115, 0.0514, 
   0.0316, -0.1217, -0.1382, 0.0584, 0.1369, 0.0837, 0.0393, -0.0575, -0.0834,
    0.0247, 0.0787, 0.0886, 0.0613, 0.0198, -0.0069, -0.0389, -0.0392, -0.013,
    0.0177, 0.0712, 0.0939, 0.0392, -0.0643, -0.0942, -0.0238, 0.0012, 0.0154,
    0.0163, -0.0098, -0.0217, -0.0074, -0.0319, -0.0485, -0.0119, 0.0148, 
   0.0548, 0.0698, 0.0403, -0.0081, -0.0512, 0.0008, 0.104, 0.14, 
   0.0792, -0.0361, -0.0785, 0.0016, 0.0833, 0.1106, 
   0.0613, -0.027, -0.074, -0.0421, -0.0034, 0.0265, 
   0.0303, -0.0346, -0.0555, -0.0026, 0.0362, 0.0405, -0.0187, -0.0441, 
   0.0339, 0.0445, -0.011, 0.0014, 0.0367, -0.0166, -0.0451, 0.0082, 0.0519, 
   0.0345, -0.0639, -0.1447, -0.1122, 0.0216, 0.0888, 
   0.0691, -0.0185, -0.1361, -0.1281, -0.0137, 0.0331, 0.0196, 
   0.0294, -0.0276, -0.078, -0.0361, 0.0059, 
   0.0132, -0.0285, -0.0475, -0.0026, 0.031, 0.0308, 
   0.0123, -0.0012, -0.0104, -0.0101, 0.0015, 0.008, 0.0071, 0.0027, -0.0002, 
   0.0004, 0.002, 0.0023, 0.0017, 0.0012, 0.001, 0.001, 0.001, 0.0009, 0.0009,
    0.0009, 0.0008, 0.0008, 0.0008, 0.0008, 0.0008, 0.0007, 0.0007, 0.0007, 
   0.0007, 0.0007, 0.0007, 0.0007, 0.0007, 0.0007, 0.0007, 0};

ListLinePlot[mList, 
 PlotRange -> {{0, Length[mList]}, {Min[mList], Max[mList]}}]

ListLinePlot[(0.3*mList), 
 PlotRange -> {{0, Length[mList]}, {0.3*Min[mList], 0.3*Max[mList]}}]

GetIndex[t_] := IntegerPart[t/dT] + 1;

mList[[GetIndex[0.04]]]

GetGroundAcceleration[t_] := 
  mList[[GetIndex[
     t]]] + (mList[[GetIndex[t] + 1]] - mList[[GetIndex[t]]]) FractionalPart[
     t/dT];

GetGroundAcceleration[40]

( Force-Displacement)

\[Mu] = 0.033;

g = 9.81;

R = 3.7;

\[Omega] = Sqrt[g/R];

YP = 0.001;

W = 726992.03;

\[Gamma] = 0.5;

tTotal = 45;

n = 5;

ParametricPlot[
 Evaluate[{x1[t], (1/R)*x1[t] + \[Mu]*x3[t]} /.
   Quiet@NDSolve[
     {x1'[t] == x5[t],
      x2'[t] == x6[t],
      x3'[t] == 
       x5[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x5[t]* x3[t]] + (1 - \[Gamma]))),
      x4'[t] == 
       x6[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x6[t]* x4[t]] + (1 - \[Gamma]))),
      x5'[t] == -(\[Omega]^2)*x1[t] - \[Mu]*9.81*x3[t] - 
        9.81*GetGroundAcceleration[t],
      x6'[t] == -(\[Omega]^2)*x2[t] - \[Mu]*9.81*x4[t] - 
        9.81*0.3*GetGroundAcceleration[t],
      x1[0] == 0,
      x2[0] == 0,
      x3[0] == 0,
      x4[0] == 0,
      x5[0] == 0,
      x6[0] == 0},
     {x1[t], x2[t], x3[t], x4[t], x5[t], x6[t]},
     {t, 0, tTotal}, AccuracyGoal -> 2]],
 {t, 0, tTotal},
 ImageSize -> {500, 500}, PlotRange -> {{-0.5, 0.5}, {-0.2, 0.2}}, 
 AspectRatio -> 1/1, AxesLabel -> {"ux", "F"}]

ParametricPlot[
 Evaluate[{x2[t], (1/R)*x2[t] + \[Mu]*x4[t]} /.
   Quiet@NDSolve[
     {x1'[t] == x5[t],
      x2'[t] == x6[t],
      x3'[t] == 
       x5[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x5[t]* x3[t]] + (1 - \[Gamma]))),
      x4'[t] == 
       x6[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x6[t]* x4[t]] + (1 - \[Gamma]))),
      x5'[t] == -(\[Omega]^2)*x1[t] - \[Mu]*9.81*x3[t] - 
        9.81*GetGroundAcceleration[t],
      x6'[t] == -(\[Omega]^2)*x2[t] - \[Mu]*9.81*x4[t] - 
        9.81*0.3*GetGroundAcceleration[t],
      x1[0] == 0,
      x2[0] == 0,
      x3[0] == 0,
      x4[0] == 0,
      x5[0] == 0,
      x6[0] == 0},
     {x1[t], x2[t], x3[t], x4[t], x5[t], x6[t]},
     {t, 0, tTotal}, AccuracyGoal -> 5]],
 {t, 0, tTotal},
 ImageSize -> {500, 500}, PlotRange -> {{-0.1, 0.1}, {-0.05, 0.05}}, 
 AspectRatio -> 1/1, AxesLabel -> {"uy", "F"}]

(* Displacement (time)*)

ParametricPlot[
 Evaluate[{t, x1[t]} /.
   Quiet@NDSolve[
     {x1'[t] == x5[t],
      x2'[t] == x6[t],
      x3'[t] == 
       x5[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x5[t]* x3[t]] + (1 - \[Gamma]))),
      x4'[t] == 
       x6[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x6[t]* x4[t]] + (1 - \[Gamma]))),
      x5'[t] == -(\[Omega]^2)*x1[t] - \[Mu]*9.81*x3[t] - 
        9.81*GetGroundAcceleration[t],
      x6'[t] == -(\[Omega]^2)*x2[t] - \[Mu]*9.81*x4[t] - 
        9.81*0.3*GetGroundAcceleration[t],
      x1[0] == 0,
      x2[0] == 0,
      x3[0] == 0,
      x4[0] == 0,
      x5[0] == 0,
      x6[0] == 0},
     {x1[t], x2[t], x3[t], x4[t], x5[t], x6[t]},
     {t, 0, tTotal}, AccuracyGoal -> 2]],
 {t, 0, tTotal},
 ImageSize -> {500, 500}, PlotRange -> {{0, tTotal}, {-0.4, 0.4}}, 
 AspectRatio -> 1/1, AxesLabel -> {"t", "ux"}]

ParametricPlot[
 Evaluate[{t, x2[t]} /.
   Quiet@NDSolve[
     {x1'[t] == x5[t],
      x2'[t] == x6[t],
      x3'[t] == 
       x5[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x5[t]* x3[t]] + (1 - \[Gamma]))),
      x4'[t] == 
       x6[t]/YP (1 - 
          Abs[x3[t]]^n* (\[Gamma] *Sign[x6[t]* x4[t]] + (1 - \[Gamma]))),
      x5'[t] == -(\[Omega]^2)*x1[t] - \[Mu]*9.81*x3[t] - 
        9.81*GetGroundAcceleration[t],
      x6'[t] == -(\[Omega]^2)*x2[t] - \[Mu]*9.81*x4[t] - 
        9.81*0.3*GetGroundAcceleration[t],
      x1[0] == 0,
      x2[0] == 0,
      x3[0] == 0,
      x4[0] == 0,
      x5[0] == 0,
      x6[0] == 0},
     {x1[t], x2[t], x3[t], x4[t], x5[t], x6[t]},
     {t, 0, tTotal}, AccuracyGoal -> 2]],
 {t, 0, tTotal},
 ImageSize -> {500, 500}, PlotRange -> {{0, tTotal}, {-0.3, 0.3}}, 
 AspectRatio -> 1/1, AxesLabel -> {"t", "uy"}]

Thanks!

POSTED BY: Richard Dir

AccuracyGoal is fairly complicated. For NDSolve, the documentation states that it affects the step size of the integration.

AccuracyGoal effectively specifies the absolute local error allowed at each step in finding a solution, while PrecisionGoal specifies the relative local error.

I'd imagine how it reduces the step size could depend on the numerical method. If no numerical method is specified, then I can imagine it's possible that AccuracyGoal might affect which numerical method is automatically selected.

AccuracyGoal doesn't gurantee that entire solution is within a certain accuracy. It looks like it instead tries to get each step within a certain accuracy by reducing step sizes.

POSTED BY: Sean Clarke
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