Solving the difference initial-boundary value problems by the operator exponential method

TL;DR

This work addresses solving linear difference initial-boundary value problems by modifying the operator exponential (OE) method to represent the operator with boundary conditions as a perturbation of the periodic operator. The key idea uses the Lee–Trotter–Kato formula to compute $\exp(t\hat{A}_K)$ via $\exp(t\hat{A}_L)$ and a small boundary operator $\hat{G}_{KL}$, yielding first- and second-order approximations $\hat{S}_1(t)$ and $\hat{S}_2(t)$ with error controlled by commutators. Stability is established under a negative definite-type condition $\mathrm{Re}(\hat{A}_K g,g)<0$, with absolute stability guaranteed for Hermitian or skew-symmetric cases; Neumann-type boundaries can challenge stability in some configurations. Spectral analysis in 1D shows OE eigenvalues approximate the true spectrum well for most harmonics, and the method remains competitive with classical schemes like Euler and Krank–Nickolson, while remaining explicit and scalable to higher dimensions on rectangular grids. Overall, the modified OE method provides an efficient, explicit alternative for solving difference I-B-V problems, with demonstrated applicability to diffusion, Schrödinger-type dynamics, and Ginzburg–Landau-type models.

Abstract

We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions as the perturbation of the same operator for periodic ones. We analyze the error, stability and efficiency of the scheme for a model example of the one-dimensional operator of second difference.

Figures (2)

• Figure 1: Error in even eigenvalues of step operator for the OE method (1) and the Krank--Nickolson method (2) for one-dimensional Laplace operator with Dirichlet boundary conditions $t=\frac{1}{2}$, $N=1024$.
• Figure 2: Relative error of solution of the Schrödinger equation for the OE method (1) and the Krank--Nickolson method (2) with random initial vector, $t=\frac{1}{2}$, $N=128$.

Theorems & Definitions (1)

• Remark