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2022, Volume 15, Issue 4: 773-795. Doi: 10.3934/dcdss.2021044
This issue Previous Article ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems Next Article Bayesian topological signal processing

Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations

  • 1.

    School of Mathematics, Shandong University, Jinan, Shandong 250100, China

  • 2.

    College of Science, National University of Defense Technology, Changsha, Hunan 410073, China

  • * Corresponding author: wdzhao@sdu.edu.cn

    * Corresponding author: wdzhao@sdu.edu.cn
Received:  November 30, 2020
Revised:  December 31, 2020
Early access:  April 2021
Published:  April 2022

This research is partially supported by the NSF of China (Nos. 12071261, 12001539, 11831010, 11871068), the Science Challenge Project (No. TZ2018001), the national key basic research program (No. 2018YFA0703903, No. 2018YFB0704304), the NSF of Hunan Province (No. 2020JJ5647), and China Postdoctoral Science Foundation (No. 2019TQ0073)

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    Abstract

  • In this work, by combining with stochastic approximation methods, we proposed a new explicit multistep scheme for solving the forward backward stochastic differential equations. Compared with the one constructed by using derivative approximation method, the new one covers the approximation of the stochastic part and is more accurate and easier to realize. Several numerical tests are presented to show the stability and effectiveness of the proposed scheme.

    Mathematics Subject Classification: Primary: 65C30; Secondary: 60H10, 60H35.

    Citation:
    shu

    \begin{equation} \\ \end{equation}
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  • Table 1.  The errors of Scheme 4.1 with $ b = 1 $, $ \sigma = 1 $ for Ex. 5.1

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 5.565e-02 1.152e-01 3.385e-02 6.502e-02 4.796e-02 1.732e-01 2.146e-02 1.537e-01
    $ \frac{1}{20} $ 4.241e-02 9.257e-02 2.602e-02 4.585e-02 3.774e-02 1.336e-01 1.520e-02 1.175e-01
    $ \frac{1}{25} $ 3.422e-02 7.616e-02 2.111e-02 3.400e-02 3.091e-02 1.092e-01 1.170e-02 9.380e-02
    $ \frac{1}{30} $ 2.857e-02 6.377e-02 1.770e-02 2.692e-02 2.596e-02 9.268e-02 9.542e-03 7.724e-02
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.122e-02 9.952e-03 4.370e-03 1.590e-02 1.355e-02 3.715e-02 4.945e-03 3.855e-02
    $ \frac{1}{20} $ 6.722e-03 5.644e-03 2.781e-03 8.588e-03 8.000e-03 2.395e-02 2.719e-03 2.267e-02
    $ \frac{1}{25} $ 4.505e-03 3.557e-03 1.958e-03 5.055e-03 5.221e-03 1.648e-02 1.683e-03 1.455e-02
    $ \frac{1}{30} $ 3.248e-03 2.443e-03 1.471e-03 3.671e-03 3.662e-03 1.197e-02 1.120e-03 9.942e-03
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.518e-03 5.198e-03 7.993e-04 5.353e-03 4.316e-03 1.266e-02 1.525e-03 1.234e-02
    $ \frac{1}{20} $ 7.410e-04 2.531e-03 4.301e-04 2.404e-03 2.093e-03 6.538e-03 6.871e-04 5.892e-03
    $ \frac{1}{25} $ 4.196e-04 1.308e-03 2.707e-04 1.164e-03 1.162e-03 3.760e-03 3.483e-04 3.119e-03
    $ \frac{1}{30} $ 2.611e-04 7.228e-04 1.838e-04 6.463e-04 7.088e-04 2.337e-03 1.944e-04 1.798e-03
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.332e-04 2.656e-03 1.880e-04 2.142e-03 1.442e-03 4.931e-03 5.740e-04 4.726e-03
    $ \frac{1}{20} $ 5.901e-05 1.020e-03 8.249e-05 8.381e-04 5.794e-04 1.943e-03 2.188e-04 1.914e-03
    $ \frac{1}{25} $ 3.039e-05 4.278e-04 4.547e-05 3.501e-04 2.743e-04 9.415e-04 9.360e-05 8.531e-04
    $ \frac{1}{30} $ 1.741e-05 2.000e-04 2.679e-05 1.623e-04 1.443e-04 4.965e-04 4.636e-05 4.259e-04
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 5.377e-05 9.346e-04 5.914e-05 8.994e-04 5.077e-04 2.010e-03 2.425e-04 2.011e-03
    $ \frac{1}{20} $ 2.249e-05 3.264e-04 1.837e-05 3.228e-04 1.724e-04 6.706e-04 8.127e-05 7.200e-04
    $ \frac{1}{25} $ 1.061e-05 1.220e-04 9.341e-06 1.209e-04 7.104e-05 2.676e-04 2.789e-05 2.708e-04
    $ \frac{1}{30} $ 5.107e-06 5.131e-05 5.008e-06 4.895e-05 3.315e-05 1.223e-04 1.163e-05 1.153e-04
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 2.551e-05 3.935e-04 1.878e-05 4.139e-04 1.964e-04 8.533e-04 1.217e-04 9.782e-04
    $ \frac{1}{20} $ 6.342e-06 1.318e-04 4.988e-06 1.370e-04 5.911e-05 2.457e-04 3.251e-05 2.984e-04
    $ \frac{1}{25} $ 2.819e-06 4.485e-05 2.455e-06 4.635e-05 2.144e-05 8.782e-05 9.493e-06 9.755e-05
    $ \frac{1}{30} $ 1.264e-06 1.644e-05 1.150e-06 1.682e-05 8.642e-06 3.459e-05 3.280e-06 3.570e-05
     | Show Table
    DownLoad: CSV

    Table 2.  The errors of Scheme 4.1 with $ b = 5 $, $ \sigma = 1 $ for Ex. 5.1

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 2.182e-01 4.303e-01 3.216e-02 6.829e-02 1.971e-01 3.969e-01 2.207e-02 1.539e-01
    $ \frac{1}{20} $ 1.651e-01 3.578e-01 2.661e-02 4.673e-02 1.557e-01 3.074e-01 1.508e-02 1.177e-01
    $ \frac{1}{25} $ 1.302e-01 3.013e-01 2.216e-02 3.389e-02 1.269e-01 2.660e-01 1.147e-02 9.416e-02
    $ \frac{1}{30} $ 1.058e-01 2.592e-01 1.909e-02 2.865e-02 1.061e-01 2.331e-01 8.924e-03 7.698e-02
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.586e-01 3.414e-01 4.025e-03 1.811e-02 1.457e-01 2.964e-01 5.299e-03 3.867e-02
    $ \frac{1}{20} $ 8.731e-02 1.997e-01 3.250e-03 8.854e-03 8.955e-02 1.817e-01 2.694e-03 2.259e-02
    $ \frac{1}{25} $ 5.620e-02 1.304e-01 2.467e-03 5.938e-03 5.992e-02 1.210e-01 1.573e-03 1.433e-02
    $ \frac{1}{30} $ 4.126e-02 9.162e-02 1.828e-03 4.243e-03 4.276e-02 8.680e-02 1.039e-03 9.889e-03
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.148e-01 3.305e-01 7.149e-04 5.242e-03 8.573e-02 2.405e-01 1.398e-03 1.171e-02
    $ \frac{1}{20} $ 4.955e-02 1.552e-01 5.287e-04 2.797e-03 4.227e-02 1.147e-01 6.942e-04 5.813e-03
    $ \frac{1}{25} $ 2.661e-02 8.080e-02 4.105e-04 1.459e-03 2.400e-02 6.287e-02 3.479e-04 3.112e-03
    $ \frac{1}{30} $ 1.577e-02 4.657e-02 3.057e-04 8.920e-04 1.486e-02 3.814e-02 1.722e-04 1.725e-03
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.003e-01 2.872e-01 2.383e-04 1.778e-03 6.024e-02 1.763e-01 4.832e-04 4.238e-03
    $ \frac{1}{20} $ 3.491e-02 1.039e-01 1.013e-04 9.635e-04 2.676e-02 7.735e-02 2.074e-04 1.777e-03
    $ \frac{1}{25} $ 1.614e-02 4.563e-02 9.230e-05 3.843e-04 1.304e-02 3.675e-02 8.852e-05 7.994e-04
    $ \frac{1}{30} $ 8.115e-03 2.140e-02 5.403e-05 2.389e-04 6.938e-03 1.913e-02 5.136e-05 4.561e-04
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 9.551e-02 3.260e-01 8.368e-05 7.301e-04 5.461e-02 1.597e-01 1.600e-04 1.648e-03
    $ \frac{1}{20} $ 2.574e-02 9.071e-02 1.891e-05 2.792e-04 1.689e-02 5.753e-02 7.795e-05 6.725e-04
    $ \frac{1}{25} $ 9.002e-03 2.865e-02 2.061e-05 1.547e-04 6.151e-03 2.346e-02 1.876e-05 2.037e-04
    $ \frac{1}{30} $ 3.744e-03 1.255e-02 1.359e-05 6.370e-05 2.847e-03 1.076e-02 1.232e-05 1.136e-04
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 9.859e-02 3.329e-01 3.338e-05 3.397e-04 4.665e-02 1.441e-01 7.870e-05 7.999e-04
    $ \frac{1}{20} $ 2.129e-02 7.413e-02 1.435e-05 9.758e-05 1.260e-02 3.741e-02 3.018e-05 2.791e-04
    $ \frac{1}{25} $ 6.412e-03 2.066e-02 6.947e-06 6.856e-05 4.122e-03 1.316e-02 9.675e-06 7.952e-05
    $ \frac{1}{30} $ 2.180e-03 7.511e-03 3.173e-06 2.020e-05 1.615e-03 5.427e-03 3.225e-06 2.782e-05
     | Show Table
    DownLoad: CSV

    Table 3.  The errors of Scheme 4.1 with $ b = 10 $, $ \sigma = 1 $ for Ex. 5.1

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 5.181e-02 1.250e-01 2.796e-02 6.815e-02 4.717e-02 1.944e-01 2.588e-02 1.576e-01
    $ \frac{1}{20} $ 3.843e-02 9.541e-02 2.077e-02 5.144e-02 3.586e-02 1.524e-01 1.960e-02 1.232e-01
    $ \frac{1}{25} $ 3.050e-02 7.720e-02 1.652e-02 4.127e-02 2.897e-02 1.251e-01 1.578e-02 1.006e-01
    $ \frac{1}{30} $ 2.523e-02 6.484e-02 1.372e-02 3.472e-02 2.425e-02 1.059e-01 1.320e-02 8.584e-02
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 4.917e-01 1.039e+00 5.761e-03 1.401e-02 3.993e-01 8.410e-01 4.412e-03 3.729e-02
    $ \frac{1}{20} $ 3.448e-01 7.170e-01 2.803e-03 8.561e-03 2.854e-01 6.164e-01 2.727e-03 2.274e-02
    $ \frac{1}{25} $ 2.357e-01 4.800e-01 1.939e-03 4.953e-03 2.071e-01 4.352e-01 1.683e-03 1.438e-02
    $ \frac{1}{30} $ 1.667e-01 3.541e-01 1.389e-03 3.686e-03 1.532e-01 3.099e-01 1.164e-03 1.004e-02
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 6.489e-01 1.674e+00 1.629e-03 5.857e-03 4.829e-01 1.088e+00 1.465e-03 1.198e-02
    $ \frac{1}{20} $ 3.241e-01 9.866e-01 5.183e-04 2.023e-03 2.603e-01 7.271e-01 5.862e-04 5.554e-03
    $ \frac{1}{25} $ 1.899e-01 5.217e-01 2.742e-04 1.135e-03 1.392e-01 4.518e-01 3.464e-04 3.092e-03
    $ \frac{1}{30} $ 1.157e-01 3.160e-01 1.965e-04 6.306e-04 9.566e-02 2.844e-01 1.763e-04 1.700e-03
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 9.734e-01 2.587e+00 3.257e-04 2.688e-03 5.946e-01 1.525e+00 5.273e-04 4.400e-03
    $ \frac{1}{20} $ 4.297e-01 1.193e+00 1.797e-04 4.954e-04 3.055e-01 7.672e-01 1.274e-04 1.489e-03
    $ \frac{1}{25} $ 2.010e-01 5.811e-01 4.152e-05 3.768e-04 1.549e-01 3.857e-01 1.001e-04 8.915e-04
    $ \frac{1}{30} $ 1.125e-01 2.918e-01 2.880e-05 1.476e-04 8.144e-02 2.288e-01 4.368e-05 4.084e-04
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 1.432e+00 4.053e+00 8.015e-05 8.045e-04 7.547e-01 2.122e+00 2.440e-04 2.041e-03
    $ \frac{1}{20} $ 5.296e-01 1.611e+00 9.127e-05 2.672e-04 3.072e-01 9.677e-01 9.220e-05 7.931e-04
    $ \frac{1}{25} $ 2.040e-01 7.428e-01 1.189e-05 9.266e-05 1.382e-01 4.365e-01 2.198e-05 2.226e-04
    $ \frac{1}{30} $ 9.414e-02 3.360e-01 5.087e-06 4.835e-05 6.835e-02 1.994e-01 1.145e-05 1.142e-04
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{15} $ 2.117e+00 6.804e+00 2.143e-05 4.101e-04 1.073e+00 2.566e+00 8.551e-05 8.529e-04
    $ \frac{1}{20} $ 6.795e-01 2.344e+00 5.912e-05 2.660e-04 3.827e-01 1.115e+00 4.831e-05 3.871e-04
    $ \frac{1}{25} $ 2.739e-01 8.312e-01 4.996e-06 4.322e-05 1.464e-01 4.481e-01 8.071e-06 5.051e-05
    $ \frac{1}{30} $ 9.606e-02 3.503e-01 8.488e-07 1.913e-05 6.020e-02 2.035e-01 4.445e-06 4.193e-05
     | Show Table
    DownLoad: CSV

    Table 4.  The errors of Scheme 4.1 with $ b = 1 $, $ \sigma = 1 $ for Ex. 5.2

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 4.105e-03 5.714e-03 8.365e-04 4.125e-03 3.933e-03 6.808e-03 8.581e-04 6.639e-03
    $ \frac{1}{16} $ 3.060e-03 4.362e-03 6.273e-04 3.089e-03 2.975e-03 5.346e-03 6.503e-04 5.083e-03
    $ \frac{1}{20} $ 2.436e-03 3.518e-03 4.977e-04 2.471e-03 2.399e-03 4.412e-03 5.380e-04 4.120e-03
    $ \frac{1}{24} $ 2.024e-03 2.942e-03 4.181e-04 2.056e-03 2.007e-03 3.747e-03 4.556e-04 3.462e-03
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 3.154e-04 2.073e-04 5.813e-05 2.748e-04 3.920e-04 7.100e-04 7.049e-05 5.513e-04
    $ \frac{1}{16} $ 1.830e-04 1.192e-04 3.494e-05 1.563e-04 2.300e-04 4.181e-04 3.991e-05 3.240e-04
    $ \frac{1}{20} $ 1.197e-04 7.880e-05 2.311e-05 1.007e-04 1.507e-04 2.742e-04 2.538e-05 2.129e-04
    $ \frac{1}{24} $ 8.374e-05 5.548e-05 1.628e-05 7.054e-05 1.063e-04 1.926e-04 1.848e-05 1.508e-04
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 3.329e-05 5.716e-05 6.428e-06 3.795e-05 5.813e-05 1.083e-04 1.030e-05 8.267e-05
    $ \frac{1}{16} $ 1.476e-05 2.619e-05 3.175e-06 1.728e-05 2.841e-05 4.990e-05 4.649e-06 3.881e-05
    $ \frac{1}{20} $ 7.830e-06 1.360e-05 1.766e-06 9.119e-06 1.577e-05 2.674e-05 2.424e-06 2.101e-05
    $ \frac{1}{24} $ 4.653e-06 7.780e-06 1.081e-06 5.333e-06 9.548e-06 1.593e-05 1.403e-06 1.256e-05
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 2.616e-06 1.647e-05 1.002e-06 8.273e-06 1.283e-05 2.250e-05 2.193e-06 1.833e-05
    $ \frac{1}{16} $ 7.797e-07 5.906e-06 4.216e-07 3.099e-06 4.719e-06 8.419e-06 8.240e-07 7.104e-06
    $ \frac{1}{20} $ 3.360e-07 2.582e-06 2.034e-07 1.381e-06 2.095e-06 3.952e-06 3.552e-07 3.226e-06
    $ \frac{1}{24} $ 1.662e-07 1.289e-06 1.082e-07 6.960e-07 1.059e-06 2.034e-06 1.752e-07 1.657e-06
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 5.529e-07 2.942e-06 2.335e-07 2.457e-06 2.788e-06 6.972e-06 6.110e-07 5.315e-06
    $ \frac{1}{16} $ 1.562e-07 8.938e-07 7.126e-08 7.612e-07 8.694e-07 2.140e-06 1.973e-07 1.763e-06
    $ \frac{1}{20} $ 5.644e-08 4.817e-07 3.114e-08 4.712e-07 3.377e-07 7.491e-07 7.317e-08 6.889e-07
    $ \frac{1}{24} $ 2.403e-08 2.232e-07 1.485e-08 2.823e-07 1.515e-07 3.390e-07 3.036e-08 3.055e-07
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 1.913e-07 1.582e-06 1.170e-07 1.451e-06 8.519e-07 3.274e-06 2.211e-07 2.042e-06
    $ \frac{1}{16} $ 9.004e-08 1.721e-06 1.030e-07 2.062e-06 2.131e-07 4.811e-07 5.975e-08 5.508e-07
    $ \frac{1}{20} $ 1.019e-07 1.799e-06 1.008e-07 2.561e-06 6.204e-08 2.111e-07 1.929e-08 1.814e-07
    $ \frac{1}{24} $ 1.070e-07 2.281e-06 1.181e-07 3.290e-06 2.545e-08 2.302e-07 6.984e-09 1.047e-07
     | Show Table
    DownLoad: CSV

    Table 5.  The errors of Scheme 4.1 with $ b = 5 $, $ \sigma = 1 $ for Ex. 5.2

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 1.895e-02 1.840e-02 8.456e-04 4.131e-03 1.869e-02 1.852e-02 8.619e-04 6.649e-03
    $ \frac{1}{16} $ 1.463e-02 1.393e-02 6.377e-04 3.078e-03 1.401e-02 1.423e-02 6.504e-04 5.080e-03
    $ \frac{1}{20} $ 1.186e-02 1.128e-02 5.045e-04 2.460e-03 1.133e-02 1.162e-02 5.377e-04 4.117e-03
    $ \frac{1}{24} $ 9.949e-03 9.466e-03 4.171e-04 2.048e-03 9.567e-03 9.785e-03 4.556e-04 3.461e-03
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 6.605e-03 1.234e-02 5.867e-05 2.740e-04 6.149e-03 1.067e-02 7.012e-05 5.487e-04
    $ \frac{1}{16} $ 3.678e-03 6.963e-03 3.660e-05 1.554e-04 3.602e-03 6.584e-03 4.001e-05 3.244e-04
    $ \frac{1}{20} $ 2.372e-03 4.374e-03 2.420e-05 9.806e-05 2.370e-03 4.370e-03 2.525e-05 2.125e-04
    $ \frac{1}{24} $ 1.692e-03 2.994e-03 1.688e-05 6.797e-05 1.655e-03 3.089e-03 1.840e-05 1.502e-04
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 4.074e-03 7.196e-03 6.300e-06 3.728e-05 3.605e-03 5.281e-03 1.017e-05 8.182e-05
    $ \frac{1}{16} $ 1.832e-03 2.785e-03 3.285e-06 1.720e-05 1.616e-03 2.493e-03 4.597e-06 3.834e-05
    $ \frac{1}{20} $ 9.493e-04 1.456e-03 1.937e-06 8.953e-06 8.535e-04 1.320e-03 2.411e-06 2.089e-05
    $ \frac{1}{24} $ 5.516e-04 8.496e-04 1.200e-06 4.954e-06 5.042e-04 7.793e-04 1.408e-06 1.257e-05
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 2.175e-03 5.736e-03 1.070e-06 8.069e-06 1.745e-03 3.847e-03 2.138e-06 1.805e-05
    $ \frac{1}{16} $ 7.148e-04 1.670e-03 4.136e-07 2.920e-06 5.906e-04 1.404e-03 7.690e-07 6.788e-06
    $ \frac{1}{20} $ 3.020e-04 7.275e-04 2.112e-07 1.415e-06 2.757e-04 6.430e-04 3.398e-07 3.093e-06
    $ \frac{1}{24} $ 1.532e-04 3.651e-04 1.299e-07 6.663e-07 1.425e-04 3.321e-04 1.726e-07 1.613e-06
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 1.738e-03 4.156e-03 2.475e-07 2.509e-06 1.123e-03 2.525e-03 5.901e-07 5.241e-06
    $ \frac{1}{16} $ 4.262e-04 1.137e-03 7.288e-08 7.014e-07 3.470e-04 6.684e-04 1.786e-07 1.642e-06
    $ \frac{1}{20} $ 1.502e-04 3.231e-04 3.029e-08 5.094e-07 1.304e-04 2.659e-04 6.709e-08 6.430e-07
    $ \frac{1}{24} $ 6.136e-05 1.315e-04 1.596e-08 3.320e-07 5.705e-05 1.186e-04 2.704e-08 2.734e-07
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 1.301e-03 8.282e-03 1.262e-07 1.682e-06 6.971e-04 1.946e-03 1.860e-07 1.964e-06
    $ \frac{1}{16} $ 5.095e-04 4.004e-03 1.125e-07 2.145e-06 1.750e-04 4.942e-04 4.792e-08 4.750e-07
    $ \frac{1}{20} $ 2.988e-04 8.144e-04 1.090e-07 2.759e-06 5.535e-05 1.594e-04 1.828e-08 1.708e-07
    $ \frac{1}{24} $ 1.828e-04 2.020e-03 1.426e-07 4.105e-06 2.038e-05 1.826e-04 6.291e-09 1.087e-07
     | Show Table
    DownLoad: CSV

    Table 6.  The errors of Scheme 4.1 with $ b = 10 $, $ \sigma = 1 $ for Ex. 5.2

    k = 1 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 3.768e-02 3.905e-02 8.467e-04 4.106e-03 3.672e-02 3.742e-02 8.551e-04 6.632e-03
    $ \frac{1}{16} $ 2.879e-02 2.949e-02 6.256e-04 3.090e-03 2.831e-02 2.836e-02 6.505e-04 5.086e-03
    $ \frac{1}{20} $ 2.311e-02 2.459e-02 5.001e-04 2.473e-03 2.287e-02 2.393e-02 5.382e-04 4.119e-03
    $ \frac{1}{24} $ 1.920e-02 2.067e-02 4.173e-04 2.059e-03 1.911e-02 2.030e-02 4.571e-04 3.466e-03
    k = 2 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 2.710e-02 4.473e-02 6.102e-05 2.732e-04 2.402e-02 3.978e-02 7.074e-05 5.513e-04
    $ \frac{1}{16} $ 1.502e-02 2.748e-02 3.425e-05 1.532e-04 1.444e-02 2.551e-02 3.974e-05 3.231e-04
    $ \frac{1}{20} $ 1.003e-02 1.821e-02 2.308e-05 1.008e-04 9.176e-03 1.640e-02 2.540e-05 2.130e-04
    $ \frac{1}{24} $ 6.951e-03 1.327e-02 1.635e-05 7.013e-05 6.451e-03 1.205e-02 1.841e-05 1.503e-04
    k = 3 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 3.127e-02 6.145e-02 6.569e-06 3.761e-05 2.586e-02 4.532e-02 1.013e-05 8.171e-05
    $ \frac{1}{16} $ 1.452e-02 3.056e-02 3.294e-06 1.539e-05 1.207e-02 2.181e-02 4.627e-06 3.853e-05
    $ \frac{1}{20} $ 7.818e-03 1.373e-02 1.671e-06 9.015e-06 6.879e-03 1.200e-02 2.398e-06 2.080e-05
    $ \frac{1}{24} $ 4.342e-03 7.597e-03 1.081e-06 5.331e-06 4.134e-03 6.770e-03 1.403e-06 1.256e-05
    k = 4 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 4.507e-02 1.018e-01 1.073e-06 8.016e-06 2.971e-02 5.359e-02 2.129e-06 1.797e-05
    $ \frac{1}{16} $ 1.422e-02 4.148e-02 5.479e-07 2.920e-06 1.011e-02 2.178e-02 7.949e-07 6.842e-06
    $ \frac{1}{20} $ 4.966e-03 1.289e-02 2.020e-07 1.170e-06 4.130e-03 1.039e-02 3.214e-07 2.975e-06
    $ \frac{1}{24} $ 2.407e-03 6.291e-03 1.076e-07 6.872e-07 2.034e-03 4.908e-03 1.724e-07 1.637e-06
    k = 5 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 9.803e-02 1.972e-01 2.484e-07 2.519e-06 3.445e-02 7.197e-02 5.880e-07 5.241e-06
    $ \frac{1}{16} $ 6.435e-02 1.470e-01 6.926e-08 8.647e-07 9.862e-03 3.819e-02 1.770e-07 1.600e-06
    $ \frac{1}{20} $ 2.926e-02 1.051e-01 4.978e-08 4.641e-07 3.456e-03 7.644e-03 7.673e-08 7.162e-07
    $ \frac{1}{24} $ 1.005e-02 3.746e-02 1.496e-08 3.401e-07 1.410e-03 3.138e-03 2.581e-08 2.606e-07
    k = 6 Case 1: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = 0 $ Case 2: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = 0 $ Case 3: $ \bar{\bar{b}} = 0, \bar{\bar{\sigma}} = \sigma $ Case 4: $ \bar{\bar{b}} = b, \bar{\bar{\sigma}} = \sigma $
    $ \Delta t $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $ $ |Y_0-Y^0| $ $ |Z_0-Z^0| $
    $ \frac{1}{12} $ 1.846e-01 4.120e-01 1.227e-07 1.679e-06 4.826e-02 1.027e-01 1.849e-07 1.971e-06
    $ \frac{1}{16} $ 1.899e-01 4.765e-01 1.111e-07 2.096e-06 2.091e-02 6.755e-02 5.838e-08 5.410e-07
    $ \frac{1}{20} $ 1.910e-01 7.164e-01 9.555e-08 2.265e-06 4.921e-03 1.971e-02 2.450e-08 2.008e-07
    $ \frac{1}{24} $ 1.524e-01 4.078e-01 1.704e-07 4.768e-06 1.082e-03 1.026e-02 4.522e-09 1.106e-07
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