solving ordinary differential equations and inserting the finite difference approximations to and results in unc un un unc un cb un ( where is short notation for tn equation ( is linear in the unknown unc so we can easily solve for this quantity  unc mun un un / ( as in the case without dampingwe need to derive special formula for the initial condition implies also now that and with ( for we get ( / in the more general case with nonlinear damping term /mu ud /we get unc un un cf unc un un which is nonlinear algebraic equation for unc that must be solved by numerical methods much more convenient scheme arises from using backward difference for un un tn because the damping term will then be knowninvolving only un and un and we can easily solve for unc the downside of the backward difference compared to the centered difference ( is that it reduces the order of the accuracy in the overall scheme from to in factthe euler-cromer scheme evaluates nonlinear damping term as when computing nc and this is equivalent to using the backward difference above consequentlythe convenience of the euler-cromer scheme for nonlinear damping comes at cost of lowering the overall accuracy of the scheme from second to first order in using the same trick in the finite difference scheme for the second-order differential equationi using the backward difference in /makes this scheme equally convenient and accurate as the euler-cromer scheme in the general nonlinear case mu ud