Research Article
Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation
Table 2
The AEs of Example (
2) using fourth-order Runge–Kutta (RK4) methods.
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| | Case I: classical RK4 method | | 2.2551e-06 | 1.5184e-07 | 9.8491e-09 | 6.2708e-10 | 3.9557e-11 | 2.4840e-12 | 1.5543e-13 | | 4.7763e-06 | 3.1936e-07 | 2.0641e-08 | 1.3119e-09 | 8.2681e-11 | 5.1891e-12 | 3.2457e-13 | | 7.3083e-06 | 4.8505e-07 | 3.1235e-08 | 1.9814e-09 | 1.2476e-10 | 7.8259e-12 | 4.9027e-13 | | 9.5635e-06 | 6.3023e-07 | 4.0441e-08 | 2.5610e-09 | 1.6111e-10 | 1.0101e-11 | 6.3527e-13 | | 1.1301e-05 | 7.4039e-07 | 4.7374e-08 | 2.9957e-09 | 1.8833e-10 | 1.1803e-11 | 7.4463e-13 | | 1.2408e-05 | 8.0984e-07 | 5.1724e-08 | 3.2678e-09 | 2.0534e-10 | 1.2869e-11 | 8.1413e-13 | | 1.2940e-05 | 8.4328e-07 | 5.3815e-08 | 3.3985e-09 | 2.1351e-10 | 1.3382e-11 | 8.4910e-13 | | 1.3100e-05 | 8.5316e-07 | 5.4419e-08 | 3.4357e-09 | 2.1581e-10 | 1.3529e-11 | 8.6020e-13 | | 1.3141e-05 | 8.5375e-07 | 5.4381e-08 | 3.4308e-09 | 2.1543e-10 | 1.3503e-11 | 8.5909e-13 | | 1.3245e-05 | 8.5473e-07 | 5.4260e-08 | 3.4174e-09 | 2.1441e-10 | 1.3427e-11 | 8.3977e-13 |
| | Case II: England version of RK4 method | | 1.8238e-06 | 1.2286e-07 | 7.9738e-09 | 5.0789e-10 | 3.2046e-11 | 2.0125e-12 | 1.2590e-13 | | 3.7840e-06 | 2.5339e-07 | 1.6397e-08 | 1.0428e-09 | 6.5747e-11 | 4.1272e-12 | 2.5807e-13 | | 5.7022e-06 | 3.7952e-07 | 2.4483e-08 | 1.5547e-09 | 9.7944e-11 | 6.1453e-12 | 3.8519e-13 | | 7.4121e-06 | 4.9044e-07 | 3.1547e-08 | 2.0004e-09 | 1.2593e-10 | 7.8977e-12 | 4.9760e-13 | | 8.7887e-06 | 5.7860e-07 | 3.7126e-08 | 2.3512e-09 | 1.4792e-10 | 9.2731e-12 | 5.8653e-13 | | 9.7606e-06 | 6.4011e-07 | 4.0995e-08 | 2.5937e-09 | 1.6310e-10 | 1.0225e-11 | 6.4893e-13 | | 1.0321e-05 | 6.7501e-07 | 4.3170e-08 | 2.7294e-09 | 1.7158e-10 | 1.0757e-11 | 6.8523e-13 | | 1.0545e-05 | 6.8794e-07 | 4.3938e-08 | 2.7761e-09 | 1.7445e-10 | 1.0938e-11 | 6.9811e-13 | | 1.0585e-05 | 6.8771e-07 | 4.3828e-08 | 2.7662e-09 | 1.7373e-10 | 1.0892e-11 | 6.9567e-13 | | 1.0608e-05 | 6.8394e-07 | 4.3423e-08 | 2.7354e-09 | 1.7164e-10 | 1.0749e-11 | 6.7191e-13 |
| | Case III: a new formula of RK4 method | | 3.1177e-06 | 2.0981e-07 | 1.3600e-08 | 8.6546e-10 | 5.4580e-11 | 3.4268e-12 | 2.1448e-13 | | 6.7609e-06 | 4.5128e-07 | 2.9131e-08 | 1.8500e-09 | 1.1655e-10 | 7.3131e-12 | 4.5755e-13 | | 1.0520e-05 | 6.9611e-07 | 4.4738e-08 | 2.8350e-09 | 1.7840e-10 | 1.1188e-11 | 7.0077e-13 | | 1.3866e-05 | 9.0980e-07 | 5.8229e-08 | 3.6821e-09 | 2.3147e-10 | 1.4508e-11 | 9.1094e-13 | | 1.6326e-05 | 1.0640e-06 | 6.7872e-08 | 4.2849e-09 | 2.6914e-10 | 1.6861e-11 | 1.0610e-12 | | 1.7701e-05 | 1.1493e-06 | 7.3182e-08 | 4.6159e-09 | 2.8981e-10 | 1.8155e-11 | 1.1446e-12 | | 1.8177e-05 | 1.1798e-06 | 7.5107e-08 | 4.7367e-09 | 2.9737e-10 | 1.8630e-11 | 1.1773e-12 | | 1.8211e-05 | 1.1836e-06 | 7.5383e-08 | 4.7551e-09 | 2.9854e-10 | 1.8707e-11 | 1.1846e-12 | | 1.8255e-05 | 1.1858e-06 | 7.5485e-08 | 4.7600e-09 | 2.9881e-10 | 1.8723e-11 | 1.1871e-12 | | 1.8519e-05 | 1.1963e-06 | 7.5933e-08 | 4.7814e-09 | 2.9994e-10 | 1.8780e-11 | 1.1759e-12 |
| | Case IV: a second new formula of RK4 method | | 1.9675e-06 | 1.3252e-07 | 8.5989e-09 | 5.4762e-10 | 3.4549e-11 | 2.1696e-12 | 1.3577e-13 | | 4.1148e-06 | 2.7538e-07 | 1.7812e-08 | 1.1325e-09 | 7.1392e-11 | 4.4811e-12 | 2.8008e-13 | | 6.2376e-06 | 4.1469e-07 | 2.6734e-08 | 1.6969e-09 | 1.0688e-10 | 6.7053e-12 | 4.2016e-13 | | 8.1292e-06 | 5.3704e-07 | 3.4512e-08 | 2.1872e-09 | 1.3766e-10 | 8.6320e-12 | 5.4312e-13 | | 9.6262e-06 | 6.3253e-07 | 4.0542e-08 | 2.5660e-09 | 1.6139e-10 | 1.0116e-11 | 6.3882e-13 | | 1.0643e-05 | 6.9669e-07 | 4.4571e-08 | 2.8184e-09 | 1.7718e-10 | 1.1107e-11 | 7.0366e-13 | | 1.1194e-05 | 7.3110e-07 | 4.6718e-08 | 2.9525e-09 | 1.8555e-10 | 1.1633e-11 | 7.3919e-13 | | 1.1397e-05 | 7.4302e-07 | 4.7432e-08 | 2.9960e-09 | 1.8824e-10 | 1.1801e-11 | 7.5162e-13 | | 1.1437e-05 | 7.4305e-07 | 4.7346e-08 | 2.9877e-09 | 1.8763e-10 | 1.1762e-11 | 7.4984e-13 | | 1.1487e-05 | 7.4087e-07 | 4.7035e-08 | 2.9628e-09 | 1.8590e-10 | 1.1640e-11 | 7.2764e-13 |
| | Case V: a third new formula of RK4 method | | 1.7375e-06 | 1.1706e-07 | 7.5988e-09 | 4.8405e-10 | 3.0543e-11 | 1.9182e-12 | 1.1996e-13 | | 3.5855e-06 | 2.4020e-07 | 1.5548e-08 | 9.8900e-10 | 6.2360e-11 | 3.9146e-12 | 2.4469e-13 | | 5.3810e-06 | 3.5841e-07 | 2.3133e-08 | 1.4693e-09 | 9.2579e-11 | 5.8089e-12 | 3.6421e-13 | | 6.9818e-06 | 4.6248e-07 | 2.9769e-08 | 1.8882e-09 | 1.1889e-10 | 7.4570e-12 | 4.7018e-13 | | 8.2862e-06 | 5.4624e-07 | 3.5076e-08 | 2.2222e-09 | 1.3984e-10 | 8.7679e-12 | 5.5511e-13 | | 9.2312e-06 | 6.0616e-07 | 3.8849e-08 | 2.4589e-09 | 1.5466e-10 | 9.6974e-12 | 6.1595e-13 | | 9.7971e-06 | 6.4135e-07 | 4.1041e-08 | 2.5956e-09 | 1.6319e-10 | 1.0233e-11 | 6.5215e-13 | | 8.2862e-06 | 6.5490e-07 | 4.1841e-08 | 2.6442e-09 | 1.6618e-10 | 1.0421e-11 | 6.6547e-13 | | 9.2312e-06 | 6.5450e-07 | 4.1718e-08 | 2.6333e-09 | 1.6539e-10 | 1.0371e-11 | 6.6280e-13 | | 1.0081e-05 | 6.4978e-07 | 4.1255e-08 | 2.5990e-09 | 1.6309e-10 | 1.0214e-11 | 6.3838e-13 |
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