How to | Check the Results of NDSolve
For most differential equations, the results given by NDSolve are quite accurate. However, because its results are based on numerical sampling and error estimates, there can occasionally be significant errors. When you need to be sure of the quality of a solution, it is a good idea to do some basic checking of the solution.
Different solutions may save solution data at different points, leading to differences at these points. To keep these differences no larger than the numerical error in the solution, use InterpolationOrder->All.
Since errors are often quite small, it is useful to view them on a logarithmic scale. RealExponent[x] is effectively equal to Log10[Abs[x]], but without a singularity at zero, so it is a good choice for viewing differences that might be zero at some points:
The numerical methods used in NDSolve are designed to keep the residual small at any point. You can plot the logs of the residuals:
As you can see, the numerical errors are significantly smaller when using the higher WorkingPrecision. The trade off is, of course, increased calculation time.