Rothe-Maruyama Difference Scheme for the Stochastic Schr\"{o}dinger Equation

Abstract

In this study, the initial value stochastic Schrödinger type problem in an abstract Hilbert space with the self-adjoint operator is investigated.Rothe-Maruyama method for the numerical solution of this problem is presented. Theorem on the convergence of this difference scheme is established. A numerical example is given.

Time Multipoint Nonlocal Problem for a Stochastic Schrödinger Equation

“…The Euler type finite difference scheme, adapted for stochastic differential equations is known under the name Euler-Maruyama scheme, [20]. The Euler-Maruyama scheme considered here is also implicit and is therefore named as the Rothe-Maruyama scheme as in [29], where u(0) was given which is different from here where u(0) depends on a finite number of future values. Since A is a self adjoint operator, all eigenvalues of A are real numbers.…”

“…Proof. To prove (4.11) and (4.13), we modify the proof in [29], where the case of 0 ≤ α ≤ 1 and 1 ≤ β ≤ 2 was shown.…”