STABILITY ANALYSIS OF DIFFERENCE SCHEMES FOR VARIABLE COEFFICIENT SCHROEDINGER TYPE EQUATIONS.

We consider the stability of difference schemes for the solution of the initial boundary value problem for the equation u//t equals (A(x, t)u//x)//x plus B(x, t)u//x plus C(x, t)u plus f(x, t), where u, A, B, C and f are complex valued functions. Using energy methods, we establish the stability of a general two level scheme which includes Euler's method, Crank-Nicolson's method and the backward Euler method. If the coefficient A(x, t) is purely imaginary, the explicit Euler method is unconditionally unstable. For this case, we propose a new scheme with appropriately chosen artificial dissipation, which we prove to be conditionally stable.