fig effect of nonlinear (leftand linear (rightspring on sliding friction u_ v_ dt / uvt eulercromer( =fs=sf=fm=mt=tu_ =u_ v_ =v_ dt=dtplot_u(utrunning the sliding_friction function gives us the results in fig with ud tanh , (leftand the linearized version ud ku (righta finite difference methodundampedlinear case we shall now address numerical methods for the second-order ode without rewriting the ode as system of first-order odes the primary motivation for "yet another solution methodis that the discretization principles result in very good schemeand more importantlythe thinking around the discretization can be reused when solving partial differential equations the main idea of this numerical method is to approximate the second-order derivative by finite difference while there are several choices of difference approximations to first-order derivativesthere is one dominating formula for the second-order derivativeu tn unc un un ( the error in this approximation is proportional to letting the ode be valid at some arbitrary time point tn tn tn