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An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient. (English) Zbl 0746.65065
In his earlier paper [Math. Numer. Sinica 11, No. 2, 128-131 (1989; Zbl 0687.65118)] the author established a kind of three-level explicit difference scheme which is unconditionally stable for the Schrödinger equation with a constant coefficient. Here this is generalized to the problem $$iU_ t(x,t)-(a(x,t)U_ x(x,t))_ x=0$$, $$U(x,0)=U_ 0(x)$$, $$U(0,t)=g_ 0(t)$$, $$U(1,t)=g_ 1(t)$$, $$a(x,t)>0$$. The discrete energy method is used to justify. A numerical example is presented.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35J10 Schrödinger operator, Schrödinger equation
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