Fifth-stage stochastic runge-kutta method for stochastic differential equations - Noor Amalina Nisa, Ariffin (2018)
Most of the physical systems around us are subjected to uncontrollable factors. Hence, models for these systems are required via stochastic differential equations (SDEs). However, it is often difficult to find analytical solutions of SDEs. In such a case, a numerical method provides an alternative way to solve problems with such systems. The development of numerical methods for SDEs is far from complete. Conversely, numerical methods for their deterministic counterparts are well-developed. The relative paucity of numerical methods in SDEs is due to the complexity of approximating high-order multiple stochastic integrals. A stochastic integral provides information of the Wiener process, which then contributes to the order of the methods. Motivated by the development of high-order Runge-Kutta methods for solving ordinary differential equations (ODEs), this research was aimed to develop a new fifth-stage stochastic Runge-Kutta (SRK5) method for SDEs with a strong order of 2.0. The derivation of this derivative-free method was based on the stochastic Taylor series expansion. The Taylor series expansion for both Taylor series and numerical solutions up to 2.0 order of convergence have been expanded. The analysis of the order conditions for the SRK5 was performed by evaluating the local truncation error in terms of the mean square in MAPLE. The difference between Taylor series solution and numerical solution was evaluated. In order to analyze the order conditions, the local truncation error between both solutions was minimized. All equations arise in order conditions analysis have been solved simultaneously by using MATLAB, and three newly developed SRK5 schemes were presented. A mean-square stability analysis was then performed on the SRK5 scheme in order to ensure the efficiency of the newly-developed numerical scheme. The stability function for each scheme was derived and the change of variables have been applied for stability region plotting purposed. Stability region have been plotted on the uv-plane to visualize the stability property of each scheme. In addition, the simple numerical experiments have been performed to check on the stability property. In order to validate the efficiency of the newly develop numerical schemes, all schemes have been used to solve both linear and non-linear stochastic models respectively in C++. The performances of SRK5 schemes in solving linear SDEs have been measured by comparing the root mean-square error and the global error obtained by solving linear SDE via SRK5 schemes, SRK2.0, SRK4, Milstein and Euler-Maruyama methods. Besides, three different models of fermentation process were solved by using SRK5 schemes, SRK2.0 and SRK4. The errors obtained have been compared. The SRK5 is proved to be a more efficient tool for the numerical approximation of solutions to SDEs.