Research Article
Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation
Table 3
The AEs of Example (
3) using fourth-order Runge–Kutta (RK4) methods.
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| | Case I: classical RK4 method | | 3.8296e-07 | 2.1510e-08 | 1.2712e-09 | 7.7213e-11 | 4.7566e-12 | 2.9510e-13 | 1.8319e-14 | | 5.7951e-07 | 3.2725e-08 | 1.9396e-09 | 1.1798e-10 | 7.2735e-12 | 4.5164e-13 | 2.7978e-14 | | 6.8133e-07 | 3.8623e-08 | 2.2939e-09 | 1.3968e-10 | 8.6161e-12 | 5.3524e-13 | 3.2752e-14 | | 7.3394e-07 | 4.1722e-08 | 2.4816e-09 | 1.5123e-10 | 9.3322e-12 | 5.7987e-13 | 3.4639e-14 | | 7.6091e-07 | 4.3344e-08 | 2.5808e-09 | 1.5737e-10 | 9.7137e-12 | 6.0407e-13 | 3.5416e-14 | | 7.7483e-07 | 4.4202e-08 | 2.6340e-09 | 1.6067e-10 | 9.9198e-12 | 6.1640e-13 | 3.5860e-14 | | 7.8257e-07 | 4.4692e-08 | 2.6648e-09 | 1.6260e-10 | 1.0040e-11 | 6.2339e-13 | 3.5638e-14 | | 7.8799e-07 | 4.5037e-08 | 2.6865e-09 | 1.6396e-10 | 1.0126e-11 | 6.2883e-13 | 3.5638e-14 | | 7.9326e-07 | 4.5365e-08 | 2.7069e-09 | 1.6523e-10 | 1.0205e-11 | 6.3338e-13 | 3.5194e-14 | | 7.9961e-07 | 4.5748e-08 | 2.7304e-09 | 1.6669e-10 | 1.0295e-11 | 6.4021e-13 | 3.7692e-14 |
| | Case II: England version of RK4 method | | 2.1401e-08 | 8.5002e-10 | 3.8238e-11 | 1.9329e-12 | 1.0647e-13 | 6.2172e-15 | 4.4409e-16 | | 3.7105e-08 | 1.6162e-09 | 7.9370e-11 | 4.2936e-12 | 2.4736e-13 | 1.4988e-14 | 1.1102e-15 | | 4.6667e-08 | 2.1219e-09 | 1.0802e-10 | 5.9908e-12 | 3.5072e-13 | 2.1538e-14 | 1.1102e-15 | | 5.1839e-08 | 2.4094e-09 | 1.2480e-10 | 7.0011e-12 | 4.1289e-13 | 2.5757e-14 | 7.7716e-16 | | 5.4295e-08 | 2.5521e-09 | 1.3332e-10 | 7.5204e-12 | 4.4509e-13 | 2.7978e-14 | 2.2204e-16 | | 5.5228e-08 | 2.6096e-09 | 1.3685e-10 | 7.7387e-12 | 4.5852e-13 | 2.8200e-14 | 1.1102e-16 | | 5.5412e-08 | 2.6231e-09 | 1.3773e-10 | 7.7948e-12 | 4.6207e-13 | 2.8089e-14 | 6.6613e-16 | | 5.5315e-08 | 2.6185e-09 | 1.3748e-10 | 7.7798e-12 | 4.6108e-13 | 2.7534e-14 | 1.1102e-15 | | 5.5215e-08 | 2.6116e-09 | 1.3702e-10 | 7.7501e-12 | 4.5897e-13 | 2.7200e-14 | 1.8874e-15 | | 5.5268e-08 | 2.6114e-09 | 1.3689e-10 | 7.7386e-12 | 4.5747e-13 | 2.8089e-14 | 2.7756e-16 |
| | Case III: a new formula of RK4 method | | 1.1061e-06 | 6.2829e-08 | 3.7371e-09 | 2.2777e-10 | 1.4057e-11 | 8.7286e-13 | 5.4290e-14 | | 1.6643e-06 | 9.4942e-08 | 5.6599e-09 | 3.4536e-10 | 2.1325e-11 | 1.3248e-12 | 8.2934e-14 | | 1.9506e-06 | 1.1163e-07 | 6.6656e-09 | 4.0706e-10 | 2.5147e-11 | 1.5626e-12 | 9.6922e-14 | | 2.0981e-06 | 1.2035e-07 | 7.1953e-09 | 4.3969e-10 | 2.7171e-11 | 1.6886e-12 | 1.0425e-13 | | 2.1741e-06 | 1.2493e-07 | 7.4759e-09 | 4.5705e-10 | 2.8251e-11 | 1.7564e-12 | 1.0802e-13 | | 2.2140e-06 | 1.2739e-07 | 7.6283e-09 | 4.6654e-10 | 2.8842e-11 | 1.7926e-12 | 1.0925e-13 | | 2.2369e-06 | 1.2883e-07 | 7.7188e-09 | 4.7220e-10 | 2.9197e-11 | 1.8144e-12 | 1.0991e-13 | | 2.2533e-06 | 1.2987e-07 | 7.7845e-09 | 4.7632e-10 | 2.9454e-11 | 1.8304e-12 | 1.1025e-13 | | 2.2693e-06 | 1.3087e-07 | 7.8467e-09 | 4.8020e-10 | 2.9697e-11 | 1.8451e-12 | 1.1058e-13 | | 2.2883e-06 | 1.3202e-07 | 7.9175e-09 | 4.8459e-10 | 2.9970e-11 | 1.8632e-12 | 1.1374e-13 |
| | Case IV: a second new formula of RK4 method | | 1.4192e-07 | 7.7366e-09 | 4.4923e-10 | 2.7026e-11 | 1.6567e-12 | 1.0247e-13 | 6.7724e-15 | | 2.1791e-07 | 1.1986e-08 | 6.9943e-10 | 4.2190e-11 | 2.5897e-12 | 1.6087e-13 | 1.0547e-14 | | 2.5822e-07 | 1.4289e-08 | 8.3663e-10 | 5.0555e-11 | 3.1062e-12 | 1.9318e-13 | 1.1879e-14 | | 2.7921e-07 | 1.5514e-08 | 9.1041e-10 | 5.5078e-11 | 3.3863e-12 | 2.1050e-13 | 1.2212e-14 | | 2.8983e-07 | 1.6149e-08 | 9.4916e-10 | 5.7469e-11 | 3.5351e-12 | 2.2005e-13 | 1.1990e-14 | | 2.9509e-07 | 1.6474e-08 | 9.6923e-10 | 5.8717e-11 | 3.6128e-12 | 2.2415e-13 | 1.1879e-14 | | 2.9780e-07 | 1.6646e-08 | 9.8007e-10 | 5.9396e-11 | 3.6553e-12 | 2.2615e-13 | 1.1435e-14 | | 2.9954e-07 | 1.6758e-08 | 9.8715e-10 | 5.9840e-11 | 3.6833e-12 | 2.2760e-13 | 1.1102e-14 | | 3.0123e-07 | 1.6863e-08 | 9.9365e-10 | 6.0244e-11 | 3.7086e-12 | 2.2859e-13 | 1.0547e-14 | | 3.0338e-07 | 1.6990e-08 | 1.0014e-09 | 6.0722e-11 | 3.7375e-12 | 2.3165e-13 | 1.2712e-14 |
| | Case V: a third new formula of RK4 method | | 5.0912e-08 | 3.2819e-09 | 2.0836e-10 | 1.3123e-11 | 8.2334e-13 | 5.1736e-14 | 2.7756e-15 | | 7.1376e-08 | 4.6056e-09 | 2.9267e-10 | 1.8444e-11 | 1.1573e-12 | 7.2387e-14 | 3.8858e-15 | | 8.0264e-08 | 5.1784e-09 | 3.2915e-10 | 2.0747e-11 | 1.3018e-12 | 8.1379e-14 | 4.5519e-15 | | 8.4581e-08 | 5.4532e-09 | 3.4657e-10 | 2.1845e-11 | 1.3706e-12 | 8.5598e-14 | 5.4401e-15 | | 8.7029e-08 | 5.6062e-09 | 3.5619e-10 | 2.2449e-11 | 1.4083e-12 | 8.8041e-14 | 5.9952e-15 | | 8.8691e-08 | 5.7088e-09 | 3.6258e-10 | 2.2848e-11 | 1.4331e-12 | 9.0150e-14 | 6.3283e-15 | | 9.0020e-08 | 5.7907e-09 | 3.6768e-10 | 2.3166e-11 | 1.4526e-12 | 9.2037e-14 | 6.8834e-15 | | 9.1220e-08 | 5.8652e-09 | 3.7233e-10 | 2.3456e-11 | 1.4706e-12 | 9.3481e-14 | 7.3275e-15 | | 9.2394e-08 | 5.9391e-09 | 3.7696e-10 | 2.3747e-11 | 1.4888e-12 | 9.5257e-14 | 8.1046e-15 | | 9.3601e-08 | 6.0160e-09 | 3.8182e-10 | 2.4052e-11 | 1.5086e-12 | 9.5146e-14 | 5.9952e-15 |
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