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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 $i{U}_{t}\left(x,t\right)-{\left(a\left(x,t\right){U}_{x}\left(x,t\right)\right)}_{x}=0$, $U\left(x,0\right)={U}_{0}\left(x\right)$, $U\left(0,t\right)={g}_{0}\left(t\right)$, $U\left(1,t\right)={g}_{1}\left(t\right)$, $a\left(x,t\right)>0$. The discrete energy method is used to justify. A numerical example is presented.
##### MSC:
 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M06 Finite difference methods (IVP of PDE) 35J10 Schrödinger operator