How to do efficient NDSolve Error control

*To*: mathgroup at smc.vnet.net*Subject*: [mg91757] How to do efficient NDSolve Error control*From*: pratip <pratip.official at gmail.com>*Date*: Sun, 7 Sep 2008 05:37:03 -0400 (EDT)

Lets take this system of ODE and solve with DSolve exsol = DSolve[{x1'[t] == x2[t], x2'[t] == c^2 x1[t] - (c^2 + p^2) Sin[p t], x1[0] == 0, x2[0] == Pi}, {x1[t], x2[t]}, t] // FullSimplify solution is as follows {{x1[t] -> Sin[p t] + ((-p + \[Pi]) Sinh[c t])/c, x2[t] -> p Cos[p t] + (-p + \[Pi]) Cosh[c t]}} Here is the values of the parameters dp = {c -> 100, p -> 2} Now solve with NDSolve sol = NDSolve[{x1'[t] == x2[t], x2'[t] == c^2 x1[t] - (c^2 + p^2) Sin[p t], x1[0] == 0, x2[0] == Pi} /. dp, {x1, x2}, {t, 0, 2}] Lets Plot the residual error Res = {-x1'[t] + x2[t], -x2'[t] + c^2 x1[t] - (c^2 + p^2) Sin[p t]}; Plot[Evaluate[Res /. sol /. dp], {t, 0, 2}] One can see the exponential increase in error. Please tell me if it is possible to manage this error for any parameter values. I want to know how I can use NDSolve to simulate such ODE with very little residual error just like any other normal system of ODE. Should I choose some special method options? Another very funny thing der = D[exsol, t]; Res /. exsol /. der /. dp // Flatten // FullSimplify gives {0,0}. But lets take the residual with out simplifying. res = Res /. exsol /. der /. dp // Flatten; Plot[Evaluate[res], {t, 0, 1.2}, PlotStyle -> Thick] The Plot shows how badly exponential error shows up in the boundary. Is it a problem with Plot command? why it can not understand without simplifying that "res={0,0}". I hope you guys have an answer.