Composition semi-implicit methods for chaotic problems simulation

Composition methods are an effective way to obtain ODE solvers of high accuracy order along with extrapolation and Runge-Kutta formulas. This paper considers theoretical layouts and experimental research of semi-implicit composition methods of different order applied to the simulation of chaotic dynamical systems. The comparison of stability regions is given for semi-implicit and implicit midpoint composition methods with different order of accuracy. Two chaotic test problems are considered: Lorenz attractor system and Sprott Case G system. An alternative way to estimate local truncation error for semi-implicit adaptive solvers is described. A performance of various numerical integration methods was compared by plotting computational costs via truncation error. This comparison shows that semi-implicit composition methods can be more computationally effective for chaotic problems solution than traditional solvers.