Efficiency of Runge-Kutta Methods in Solving Simple Harmonic Oscillators

Abstract

Symmetric methods such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR) and 2-stage Gauss method are beneficial in solving Hamiltonian problems since they are also symplectic. Symplectic methods have advantages over non-symplectic methods in the long term integration of Hamiltonian problems. The study is to show the efficiency of IMR, ITR and the 2-stage Gauss method in solving simple harmonic oscillators (SHO). This study is done theoretically and numerically on the simple harmonic oscillator problem. The theoretical analysis and numerical results on SHO problem showed that the magnitude of the global error for a symmetric or symplectic method with stepsize h is linearly dependent on time t. This gives the linear error growth when a symmetric or symplectic method is applied to the simple harmonic oscillator problem. Passive and active extrapolations have been implemented to improve the accuracy of the numerical solutions. Passive extrapolation is observed to show quadratic error growth after a very short period of time. On the other hand, active extrapolation is observed to show linear error growth for a much longer period of time.