Compact Extrapolation Schemes for a Linear Schrödinger Equation

This paper proposes a kind of compact extrapolation schemes for a linear Schrödinger equation. The schemes are convergent with fourth-order accuracy both in space and time. Especially, a specific scheme of sixth-order accuracy in space is given. The stability and discrete invariants of the schemes are analyzed. The schemes satisfy discrete conservation laws of original Schrödinger equation. The numerical example indicates the efficiency of the new schemes.


Introduction
Partial differential equations (PDEs) describe many physical phenomena. They are an important research topic in many scientific fields, such as hydrodynamics, plasma physics, nonlinear optics, molecular dynamics, celestial mechanics. The numerical investigations of PDEs can be found in [1]- [3] and references therein. To meet the demands of massive computation with high accuracy, many compact schemes have been presented recently in fluid dynamics, optics and plasma [4]- [6]. The schemes are high-order accurate with small stencil and little cost.
Schrödinger equations are important mathematical physical models [7]. They satisfy some conservation laws related to some physical quantities. Numerical preservation of these conservation laws is as important as high accuracy of numerical solutions [8]- [10]. So in this paper, we apply compact schemes to Schrödinger equations and analyze the discrete invariants of the schemes.
Consider the initial-boundary problems of the linear Schrödinger equation and m t u means the morder partial derivatives of u with respect to x and t , respectively. Proposition 1. Under the periodic boundary condition, the solution of (1) satisfies the following conservation laws: (1) Norm conservation (2) Energy conservation

Compact Extrapolation Schemes
Introduce the following uniform mesh grids

Spatial Discretization
By introducing the following linear operators 193 393120 x k u h . Here, we consider periodic boundary condition. Applying the approximation (4) to Schrödinger Equation (1), we obtain the following semi-discretization system ( )

Temporal Discretization
We use the central difference operator The resulting dominant truncation error is  (7). We adopt the following formula are the solutions of (7) with temporal step-sizes , respectively. To approximate (1) with fourth order in both time and space, the parameters should satisfy the constraints Clearly, if the scheme (7) has the discrete invariants, the extrapolations (8) are numerical stable. In our numerical example, we use two kinds of parameters:

Stability Analysis
Now we consider the stability of (8), which comes from that of (7). According to the Fourier analysis, assume the formal wave solution of (7) is with wave number β and stability factor v . First we can derive Next, with (7) and (9), we obtain Therefore, the scheme (7) is unconditionally stable. Moreover, by its symmetry, it is non-dissipative.  (7), which implies the discrete norm conservation law of (1).

Invariants Analysis
Proof. Let 0 16 36 . Denote two symmetric and cyclic matrices by Then the matrix form of (4) is 4 1 .
By the symmetry of , c the second summation term of (12) is real, while the first term is purely imaginary, which implies Then under the periodic boundary condition,  (7), which implies the discrete energy conservation law of (1). (7)

Proof. Multiplying
The first two summation terms in above equality are purely imaginary, while the last three summation terms are real. Moreover, , .
n n n n n n n n k k Therefore, taking the imaginary parts of (14) we can get

Numerical Result
Denote the schemes (7) with sixth-order and fourth-order in space by CT6 and CT4, respectively. Denote the extrapolation schemes (8) Table 1 lists the numerical results of scheme RE1 combined with CT6 at 3 t = . It conforms that the scheme is convergent with fourth-order in time. Table 2 lists the numerical results of scheme CT6 at 1 t = . We can see that scheme CT6 is convergent with sixth-order in space with respect to the 2 l norm and l ∞ norm. Table 3 lists the numerical results of scheme CT4 at 1 t = . We can see that scheme CT4 is convergent with fourth-order in space. Table 4 lists the numerical results of scheme RE2 combined with CT4 at 1.5 t = . It conforms that the scheme is convergent with fourth-order in time. Figure 1 and Figure 2 plot the residuals of discrete invariants of scheme CT6 and scheme CT4, respectively. In the two figures, we depict the residuals of norm and energy of numerical solutions with π 10 h = and 0.01 τ = , respectively. From the figures, we can see that the two schemes preserve two discrete conservation laws. Figure 3 and Figure 4 plot the residuals of discrete invariants of scheme RE1 combined with scheme CT6 and scheme RE2 combined with scheme CT4, respectively. In the two figures, we depict the residuals of norm and energy of numerical solutions with 10 h = π and 0.003 τ = , respectively. The figures tell us that the two methods preserve two discrete conservation laws too. The compact extrapolation schemes established in this paper have some advantages such as compactness,      high accuracy, less memory and less computational cost. The schemes are also stable, non-dissipative and conservative with respect to the charge and energy conservation laws. We can generalize the methods to other kind of PDEs.