Comparison between analytical and numerically obtained solutions for a damped harmonic oscillator. The left column shows the position of the particle, while the right column shows its velocity, both of which are as a function of time. The rows, from top to bottom, demonstrate the cases of an undamped (﹩{\omega }_{0}{t}_{s}=\infty ﹩), underdamped (﹩{\omega }_{0}{t}_{s}=5﹩), critically damped (﹩{\omega }_{0}{t}_{s}=0.5﹩), and over-damped (﹩{\omega }_{0}{t}_{s}=0.05﹩) system, respectively. The solid lines are the analytical solutions, while the diamond and the square symbols denote the numerical solutions using the Godunov and the Strang splitting methods, respectively. Note that an unusually large time step of ﹩{\rm{\Delta }}t={\omega }_{0}^{-1}﹩ is used to highlight the numerical errors.