Research Article
Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation
Table 1
The AEs of Example (
1) using fourth-order Runge–Kutta (RK4) methods.
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| | Case I: classical RK4 method | | 1.1153e-07 | 7.1708e-09 | 4.5427e-10 | 2.8580e-11 | 1.7923e-12 | 1.1258e-13 | 7.3275e-15 | | 2.6297e-07 | 1.6906e-08 | 1.0710e-09 | 6.7377e-11 | 4.2253e-12 | 2.6446e-13 | 1.7097e-14 | | 4.6838e-07 | 3.0109e-08 | 1.9073e-09 | 1.1999e-10 | 7.5240e-12 | 4.7051e-13 | 3.0642e-14 | | 7.4674e-07 | 4.8000e-08 | 3.0404e-09 | 1.9127e-10 | 1.1994e-11 | 7.4984e-13 | 4.9738e-14 | | 1.1237e-06 | 7.2225e-08 | 4.5748e-09 | 2.8779e-10 | 1.8045e-11 | 1.1273e-12 | 7.5051e-14 | | 1.6338e-06 | 1.0501e-07 | 6.6511e-09 | 4.1841e-10 | 2.6235e-11 | 1.6427e-12 | 1.1013e-13 | | 2.3239e-06 | 1.4935e-07 | 9.4596e-09 | 5.9507e-10 | 3.7311e-11 | 2.3390e-12 | 1.5632e-13 | | 3.2569e-06 | 2.0931e-07 | 1.3257e-08 | 8.3395e-10 | 5.2287e-11 | 3.2805e-12 | 2.1760e-13 | | 4.5182e-06 | 2.9036e-07 | 1.8390e-08 | 1.1568e-09 | 7.2531e-11 | 4.5524e-12 | 2.9932e-13 | | 6.2225e-06 | 3.9989e-07 | 2.5327e-08 | 1.5932e-09 | 9.9895e-11 | 6.2537e-12 | 3.8369e-13 |
| | Case II: England version of RK4 method | | 3.6077e-09 | 2.2755e-10 | 1.4254e-11 | 8.9129e-13 | 5.5733e-14 | 2.8866e-15 | 1.3323e-15 | | 7.5843e-09 | 4.7887e-10 | 3.0005e-11 | 1.8767e-12 | 1.1746e-13 | 7.7716e-15 | 2.4425e-15 | | 1.1965e-08 | 7.5640e-10 | 4.7408e-11 | 2.9652e-12 | 1.8630e-13 | 1.3101e-14 | 1.9984e-15 | | 1.6788e-08 | 1.0628e-09 | 6.6638e-11 | 4.1682e-12 | 2.6201e-13 | 1.8208e-14 | 2.2204e-16 | | 2.2093e-08 | 1.4011e-09 | 8.7883e-11 | 5.4976e-12 | 3.4595e-13 | 2.3981e-14 | 1.5543e-15 | | 2.7922e-08 | 1.7744e-09 | 1.1135e-10 | 6.9678e-12 | 4.3876e-13 | 2.7534e-14 | 4.4409e-15 | | 3.4318e-08 | 2.1862e-09 | 1.3728e-10 | 8.5918e-12 | 5.4046e-13 | 3.1086e-14 | 7.1054e-15 | | 4.1327e-08 | 2.6404e-09 | 1.6592e-10 | 1.0387e-11 | 6.5414e-13 | 3.6415e-14 | 1.0214e-14 | | 4.8992e-08 | 3.1410e-09 | 1.9756e-10 | 1.2370e-11 | 7.7893e-13 | 4.1744e-14 | 1.4655e-14 | | 5.7356e-08 | 3.6925e-09 | 2.3249e-10 | 1.4556e-11 | 9.0861e-13 | 6.2617e-14 | 8.4377e-15 |
| | Case III: a new formula of RK4 method | | 3.4181e-07 | 2.1968e-08 | 1.3913e-09 | 8.7523e-11 | 5.4883e-12 | 3.4306e-13 | 2.1316e-14 | | 8.0409e-07 | 5.1676e-08 | 3.2729e-09 | 2.0588e-10 | 1.2910e-11 | 8.0780e-13 | 4.9294e-14 | | 1.4291e-06 | 9.1841e-08 | 5.8166e-09 | 3.6590e-10 | 2.2943e-11 | 1.4355e-12 | 8.9262e-14 | | 2.2738e-06 | 1.4612e-07 | 9.2546e-09 | 5.8216e-10 | 3.6501e-11 | 2.2840e-12 | 1.4455e-13 | | 3.4152e-06 | 2.1948e-07 | 1.3900e-08 | 8.7438e-10 | 5.4823e-11 | 3.4304e-12 | 2.1871e-13 | | 4.9573e-06 | 3.1857e-07 | 2.0176e-08 | 1.2692e-09 | 7.9574e-11 | 4.9818e-12 | 3.1841e-13 | | 7.0403e-06 | 4.5243e-07 | 2.8653e-08 | 1.8024e-09 | 1.1301e-10 | 7.0774e-12 | 4.5297e-13 | | 9.8535e-06 | 6.3322e-07 | 4.0103e-08 | 2.5226e-09 | 1.5816e-10 | 9.9059e-12 | 6.3283e-13 | | 1.3653e-05 | 8.7737e-07 | 5.5565e-08 | 3.4952e-09 | 2.1914e-10 | 1.3727e-11 | 8.7397e-13 | | 1.8782e-05 | 1.2071e-06 | 7.6445e-08 | 4.8086e-09 | 3.0150e-10 | 1.8870e-11 | 1.1742e-12 |
| | Case IV: a second new formula of RK4 method | | 3.4772e-08 | 2.2386e-09 | 1.4192e-10 | 8.9324e-12 | 5.6022e-13 | 3.4861e-14 | 1.3323e-15 | | 8.2602e-08 | 5.3161e-09 | 3.3699e-10 | 2.1208e-11 | 1.3303e-12 | 8.3045e-14 | 3.5527e-15 | | 1.4815e-07 | 9.5322e-09 | 6.0415e-10 | 3.8020e-11 | 2.3841e-12 | 1.4855e-13 | 8.6597e-15 | | 2.3772e-07 | 1.5291e-08 | 9.6905e-10 | 6.0980e-11 | 3.8232e-12 | 2.3848e-13 | 1.5321e-14 | | 3.5983e-07 | 2.3141e-08 | 1.4663e-09 | 9.2267e-11 | 5.7845e-12 | 3.6016e-13 | 2.5757e-14 | | 5.2599e-07 | 3.3820e-08 | 2.1428e-09 | 1.3482e-10 | 8.4523e-12 | 5.2913e-13 | 3.8414e-14 | | 7.5175e-07 | 4.8327e-08 | 3.0617e-09 | 1.9263e-10 | 1.2077e-11 | 7.5939e-13 | 5.6399e-14 | | 1.0581e-06 | 6.8011e-08 | 4.3084e-09 | 2.7106e-10 | 1.6994e-11 | 1.0703e-12 | 7.8604e-14 | | 1.4734e-06 | 9.4694e-08 | 5.9983e-09 | 3.7736e-10 | 2.3660e-11 | 1.4921e-12 | 1.0791e-13 | | 2.0359e-06 | 1.3084e-07 | 8.2872e-09 | 5.2135e-10 | 3.2694e-11 | 2.0446e-12 | 1.2035e-13 |
| | Case V: a third new formula of RK4 method | | 2.6635e-08 | 1.7072e-09 | 1.0796e-10 | 6.7855e-12 | 4.2544e-13 | 2.7089e-14 | 2.6645e-15 | | 6.1696e-08 | 3.9559e-09 | 2.5020e-10 | 1.5727e-11 | 9.8588e-13 | 6.2617e-14 | 5.5511e-15 | | 1.0803e-07 | 6.9295e-09 | 4.3835e-10 | 2.7556e-11 | 1.7273e-12 | 1.0969e-13 | 7.7716e-15 | | 1.6949e-07 | 1.0875e-08 | 6.8805e-10 | 4.3256e-11 | 2.7112e-12 | 1.7164e-13 | 9.1038e-15 | | 2.5125e-07 | 1.6126e-08 | 1.0204e-09 | 6.4155e-11 | 4.0221e-12 | 2.5424e-13 | 1.2434e-14 | | 3.6027e-07 | 2.3131e-08 | 1.4638e-09 | 9.2041e-11 | 5.7707e-12 | 3.6171e-13 | 1.7097e-14 | | 5.0596e-07 | 3.2494e-08 | 2.0567e-09 | 1.2932e-10 | 8.1086e-12 | 5.0493e-13 | 2.3981e-14 | | 7.0098e-07 | 4.5031e-08 | 2.8505e-09 | 1.7925e-10 | 1.1239e-11 | 6.9855e-13 | 3.4639e-14 | | 9.6242e-07 | 6.1842e-08 | 3.9151e-09 | 2.4621e-10 | 1.5439e-11 | 9.5746e-13 | 4.9294e-14 | | 1.3133e-06 | 8.4410e-08 | 5.3443e-09 | 3.3610e-10 | 2.1069e-11 | 1.3225e-12 | 9.4591e-14 |
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