Stability Analysis of Difference Schemes for Variable Coefficient Schrödinger Type Equations

Abstract

We consider the stability of difference schemes for the solution of the initial boundary value problem for the equation \[ u_1 = \left( {A(x,t)u_x } \right)_x + B(x,t)u + C(x,t)u + f(x,t)\], 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.