Stability analysis for implicit second order finite difference schemes

Recent applications of iterative learning control and repetitive processes lead to implicit second order finite difference schemes which require practical stability testing. A von Neumann type stability analysis is employed to reduce the problem to a second order polynomial. The conditions under which its zeros lie within the unit circle can be recast by application of the bilinear transformation. Then the problem is reduced to a test for a Hurwitz polynomial. Its coefficients depend not only on the spatial frequency but also on parameters of the initial problem like step sizes in time and space. The admissible ranges of these parameters follow finally from simple inequalities. The method is demonstrated by examples.