the combination of these methods yields an overall time-stepping scheme from tn to tn that is much more accurate than the individual steps which have errors proportional to and this is indeed truethe numerical error goes in fact like ct for constant which means that the error approaches zero very quickly as we reduce the time step sizecompared to the forward euler method (error )the euler-cromer method (error tor the nd-order runge-kuttaor heun'smethod (error note that the th-order runge-kutta method is fully explicit so there is never any need to solve linear or nonlinear algebraic equationsregardless of what looks like howeverthe stability is conditional and depends on there is large family of implicit runge-kutta methods that are unconditionally stablebut require solution of algebraic equations involving at each time step the odespy package has support for lot of sophisticated explicit runge-kutta methodsbut not yet implicit runge-kutta methods more effectsdampingnonlinearityand external forces our model problem is the simplest possible mathematical model for oscillating systems neverthelessthis model makes strong demands to numerical methodsas we have seenand is very useful as benchmark for evaluating the performance of numerical methods real-life applications involve more physical effectswhich lead to differential equation with more terms and also more complicated terms typicallyone has damping force and spring force uboth these forces may depend nonlinearly on their argumentu or in additionenvironmental forces tmay act on the system for examplethe classical pendulum has nonlinear "springor restoring force usin /and air resistance on the pendulum leads to damping force ju ju examples on environmental forces include shaking of the ground ( due to an earthquakeas well as forces from waves and wind with three types of forces on the systemf and sthe sum of forces is written unote the minus sign in front of and swhich indicates that these functions are defined such that they represent forces acting against the motion for examplesprings attached to the wheels in car are combined with effective damperseach providing damping force bu that acts against the spring velocity the corresponding physical force is then bu which points downwards when the spring is being stretched (and points upwards)while acts upwards when the spring is being compressed (and points downwardsfigure shows an example of mass attached to potentially nonlinear spring and dashpotand subject to an environmental force tneverthelessour general model can equally well be pendulum as in fig with ud mg sin  (where cd : is the cross sectional area of the and up cd % bodyand is the density of airnewton' second law for the system can be written with the mass times acceleration on the left-hand side and the forces on the right-hand sidemu