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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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