Analysis of splitting schemes for 2D and 3D Schrödinger problems

. New splitting ﬁnite diﬀerence schemes for 2D and 3D linear Schrödinger problems are investigated. The stability and convergence analysis is done in the discrete L 2 norm. It is proved that the 2D scheme is unconditionally stable and conservative in the case of zero boundary condition. The splitting scheme is generalized for 3D problems. It is proved that in this case the scheme is only ρ -stable and consequently discrete conservation laws are no longer valid. Results of numerical experiments are presented.


Introduction
Schrödinger problems are solved in variety of areas including nonlinear optics, laser physics and quantum mechanics. Thus fast numerical algorithms with good approximation properties are of practical importance.
One time step of the discrete scheme is implemented in two sub-steps: only tridiagonal systems of linear equations are solved: where the discrete operators are defined by the following formulas Here L y is the averaging operator. Equation (5) defines artificial boundary conditions for U n jk .

Stability, convergence and conservativity
We define the following discrete norms for grid functions W satisfying boundary conditions W | ∂Ω h = 0: It is well known that for both eigenproblems in domain (0, 1) 2 the orthonormal eigenvectors are defined as V p (x) = √ 2 sin pπx.
Next we derive discrete conservation laws for solution of the scheme (4)-(6) in the case of zero boundary condition µ ≡ 0. It is well-known that the following norms can be calculated by using the Fourier coefficients: Since |c n+1 pq | = |c n pq |, then it follows from (8) that splitting scheme satisfies discrete analogues of charge and energy conservation laws: The following convergence theorem is valid. A proof of it is similar to 3D case analysis presented in the further section.
Theorem 2. Suppose, that exact solution u(x, y, t) of problem (1)-(3) is sufficiently smooth. Then solution of linear alternating direction implicit discrete scheme (4)-(6) converges to u(x, y, t) and the following estimate is valid max n=1,...,Nt Example. We use Example 2 given in [1]. 2D Schrödinger problem (1)- (3) is solved in the rectangular domain Ω = (−2.5, 2.5) × (−2.5, 2.5) with initial and boundary conditions obtained from the exact particular solution where k = 2.5. Results of computational experiments are presented in Table 1, they show the second-order convergence in time and the fourth-order convergence in space in both L 2 and L ∞ norms.

Scheme for 3D Schrödinger problem
Square gridΩ h = (x j , y k , z l ) is used for discretization of domainΩ. We propose the following 3D alternating direction implicit (ADI) scheme: with boundary conditions for U n and U n
Since λ j > 0, γ j > 0, it can be verified that |α n pqr | > 1, so scheme is unconditionally unstable. Thus we can consider the ρ-stability of this scheme, only.
The following estimate can be easily derived:

Convergence
The error Z n jkl = u n jkl − U n jkl satisfies the problem where R n jkl is the approximation error. We write Z n and R n as sums: Substituting (19) into (17) after simple computations we get z n+1 pqr = α n pqr z n pqr + τ β n pqr r n pqr , where β n pqr = i (γp+ iτ 2 λp)(γq+ iτ 2 λq)(γr + iτ 2 λr ) . Let us assume that |α n pqr | δ. Since γ j 2/3, we get the estimate |β pqr | 27/8. By using (20), we can express Z n+1 as sum of two discrete functions, then use the triangle inequality and finally by using the discrete Parseval's identity we deduce the stability estimate By using (21) and since R n C a (τ 2 + h 4 ) we get the following estimate If we choose τ = Ch 2+η and use inequality (16), then we obtain the estimate α n pqr 1 + Ch η = 1 +Cτ 1/(2+η) = δ, which is acceptable for the stability with respect to the initial condition. But by extending inequality (22) we get thus controlling the error growth and getting convergence of the discrete solution can be problematic.