Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics
In the present investigation, new explicit approaches by the Milstein method and increment function of the Jacobian derivative of the drift coefficient are designed. Several numerical tests such as Cox–Ingersoll–Ross process, stochastic Brusselator, and Davis-Skodje system are presented to illustrate the accuracy and the efficiency of our schemes. Furthermore, we show that the strong convergence rate of our procedures is approximately one.
Many applied problems in different fields of science, engineering, and technology are required to find mathematical models [1–3]. So, the design of numerical techniques for solving these equations is a challenging task in the numerical analysis [4–6]. Some robust and efficient schemes for solving problems are brought forward [7, 8].
Stochastic ordinary differential equations (SODEs) play a pivotal role in explaining some physical phenomena such as chemical reactions , financial mathematics , mathematical ecology , epidemiology , medicine , and population dynamics . Generally, SODEs cannot be solved analytical, but many numerical solutions can be found, for instance, the split-step theta Milstein method , the least-squares method , the discrete Temimi–Ansari method , the improved Euler-Maruyamamethod , the five-stage Milstein method , the split-step Milstein method , the split-step Adams–Moulton Milstein method , the split-step forward Milstein method , and the Runge-Kutta method [23, 24].
In the last two decades, the explicit type of numerical schemes for stiff SODEs has attracted many researchers’ attention see [25–27], for example. ODE solvers are the key to the success of this kind of numerical scheme. Several studies have recently introduced new explicit Milstein-type schemes using suitable ODE solvers. Yin and Gan  constructed an error-corrected Milstein scheme by arming the Milstein method with an error correction term. Nouri et al.  developed and applied a new explicit split-step Milstein method for stiff SODEs such as the stochastic Davis-Skodje system. These explicit numerical methods have a broad stability region, so they have been proposed to solve stiff SODEs.
There are several attempts to construct numerical methods based on split-step forward Milstein schemes  to improve the numerical results for stiff SODEs.
2. Schemes Formulation
In [23, 24], Milstein’s scheme showed in the following manner:where , , andwhere is a normally distributed random variable . To construct new numerical methods, we divide scheme (2) as follows:where
Remark 1. (see ) In the above equations, , , and denote stochastic Itô and Stratonovich integrals, respectively.
Now using , we present the following fully explicit schemes based on the relations (4), the first improved three-stage Milstein (ITSM-i) schemeThe second improved three-stage Milstein (ITSM-ii) scheme is as follows:The third improved three-stage Milstein (ITSM-iii) scheme is as follows:The fourth improved three-stage Milstein (ITSM-iv) scheme is as follows:The fifth improved three-stage Milstein (ITSM-v) scheme is as follows:The sixth improved three-stage Milstein (ITSM-vi) scheme is as follows:By denoting in approximation methods (6)–(11), is a Jacobian Matrix of drift coefficient .
3. Numerical Experiments
In the present segment, some numerical examples are reported to demonstrate the efficiency and applicability of the proposed schemes. We compare the numerical methods (6)–(11) with the procedures TSM 1a-TSM 1f . For the computations, we have used Matlab R2010a.
Example 1. Consider the following nonlinear stiff SDE ,The exact solution is given byIn Figure 1, we compare the means of absolute errors (MAEs) and roots of mean-square errors (RMSEs) of our methods with slopes of dashed line 1.0. We use discrete Brownian paths over with . Denoting as the numerical approximation to at endpoint in the i-th simulation of all M simulations with step-size , we compare the analytic solution (13) with the numerical approximation using , over different sample paths. Here, the MAEs and RMSEs are denoted as follows [18, 23]:and
Example 2. Consider the Cox–Ingersoll–Ross (CIR) process [31, 32]We know that the CIR process (16) is strictly positive if . For initial value , constants , , and , Table 1 presents the percentage of negative values produced by the Milstein, (6)–(11) and TSM 1a-TSM 1f  schemes, for three different intervals and different values of the step-sizes , along paths. The results in Table 1 clearly show that our schemes other than the ITSM-ii method (7) can indeed guarantee positivity. Table 1 also shows that for step-size , TSM 1a-TSM 1f schemes other than TSM 1b method ensure the positivity of the numerical solution.
Example 3. Let us consider the following stochastic Brusselator :with and . Figure 2 displays the RMSEs with step-sizes , at , initial data , and sample paths. The numerical methods with step-size are used to generate the reference solutions. In Figure 2, besides our results, also the results of the TSM 1a-TSM 1f and Milstein methods are presented. This figure shows that the convergence rate of the methods for the stochastic Brusselator system (17) is approximately 1.0.
Example 4. We investigate a Davis-Skodje system [29, 34, 35].with initial . For analysis of the Davis-Skodje system (18), we choose the parameters as follows:(i)Case I: ,, Case II: ,,The mean of 1000 sample paths of the nonlinear system (18) for two cases with is shown in Figure 3. We can see that our schemes converge quickly towards the asymptotic solution .
New explicit schemes have proved helpful for efficiently solving stochastic problems with both stiff and nonstiff components. The present paper proposes new methods for the numerical approximation of SODEs driven by multidimensional standard Brownian motion. For construction methods (6)–(11), explicit Milstein schemes and increment function of the Jacobian derivative of drift coefficient are used. The efficiency and applicability of the suggested methods are confirmed by some numerical examples such as Cox–Ingersoll–Ross process, stochastic Brusselator, and Davis-Skodje system. In Table 1, we observe the positive solution of the CIR model using numerical methods Milstein (6)–(11) and TSM 1a-TSM 1f . From the results of Table 1, we find that our methods preserve the positivity of the CIR model for all step sizes. For two set parameter values of the Davis-Skodje system, Figure 3 reveals that our schemes converge quickly towards the asymptotic solution. Also, our numerical tests show that the strong convergence rate of the schemes (6)–(11) is approximately 1.0 (see Figures 1 and 2).
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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