When central finite differencing gives complex values for a real solution!
Numerical computation, visualization and computer algebra have enlivened and empowered classical applied mathematics. By combining the classical paradigm with information technology, the zones of applicability of classical applied mathematics are being extended. Reversing the usual test-bed procedure, tried and trusted numerical algorithms may be used to validate implementations of complicated analytical solutions that involve many steps, each with sources of error. Conversely, analytical solutions sometimes indicate the limited range of applicability of overly trusted numerical algorithms. An example is given in fourth-order nonlinear surface diffusion for which the standard central finite-difference approximation fails to have a real-valued solution even though the un-approximated continuous problem has been solved analytically. This pathology may be circumvented by a modified scheme after reformulating the original problem.
Broadbridge, P. & Goard, J. (2012). When central finite differencing gives complex values for a real solution!. Complex Variables and Elliptic Equations, 57 (2-4), 455-467.