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On Optimal Convergence Rates for the Nonlinear Schrödinger Equation with a Wave Operator via Localized Orthogonal Decomposition

Hanzhang Hu, Zetao Ma, Lei Zhang

Abstract

In this paper, we develop a Localized Orthogonal Decomposition (LOD) method for the two-dimensional time-dependent nonlinear Schrödinger equation with a wave operator. We prove that our method preserves conservation laws and admits a unique numerical solution; furthermore, we obtain unconditional (i.e., time-step restriction-free) optimal-order superconvergent \(L^p\) error estimates. To complement the theoretical analysis, we present a series of numerical simulations that verify the analytical results and further illustrate structural aspects of the problem.

On Optimal Convergence Rates for the Nonlinear Schrödinger Equation with a Wave Operator via Localized Orthogonal Decomposition

Abstract

In this paper, we develop a Localized Orthogonal Decomposition (LOD) method for the two-dimensional time-dependent nonlinear Schrödinger equation with a wave operator. We prove that our method preserves conservation laws and admits a unique numerical solution; furthermore, we obtain unconditional (i.e., time-step restriction-free) optimal-order superconvergent error estimates. To complement the theoretical analysis, we present a series of numerical simulations that verify the analytical results and further illustrate structural aspects of the problem.
Paper Structure (14 sections, 10 theorems, 107 equations, 13 figures, 2 tables)

This paper contains 14 sections, 10 theorems, 107 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and LOD method
  3. Some Notations and Useful Lemmas
  4. Ideal LOD Space and Approximation Properties
  5. Localization of the Orthogonal Decomposition
  6. The Crank--Nicolson LOD Scheme
  7. Existence, Uniqueness and $L^{\infty}$ Boundedness of LOD Solutions
  8. Numerical Experiments
  9. Proof of Main Theorems
  10. Proof of Theorem \ref{['thm:conservation_property']}
  11. Proof of Theorem \ref{['thm2.2']}
  12. Optimal Lp Error Estimate of the LOD Solution
  13. Proof of Theorem \ref{['main_thm_2.1']}
  14. Conclusion

Key Result

lemma 1

(Gronwall's inequality Zhou) Suppose that the discrete mesh function $\{w_n,n=1,2,\cdots,N\}$, $N\tau=T$, satisfies the inequality where $A$ and $B_k\ (k=1,2,\cdots,N)$ are nonnegative constants. Then, where $\tau$ is sufficiently small, such that $\tau(\underset{k=1,2,\cdots,N}{\max} B_k)\leq \frac{1}{2}$ .

Figures (13)

  • Figure 3.1: Relative $H^1$ error order.
  • Figure 3.2: Relative $L^2$ error order.
  • Figure 3.3: Energy conservation $E^n$ for $m = 2, 3, 4, 5, 6, 7, 8$, with $T=1.0$, $h=1/64$, and $\tau = 1.0 \times 10^{-2}$.
  • Figure 3.4: Error in energy conservation, $E^n - E^{0}$, with $T=1.0$, $h=1/64$, and $\tau = 1.0 \times 10^{-2}$.
  • Figure 3.5: Error in energy conservation, $E^n - E$, with $T=1.0$, $h=1/64$, and $\tau = 1.0 \times 10^{-2}$.
  • ...and 8 more figures

Theorems & Definitions (16)

  • lemma 1
  • lemma 2
  • remark 1
  • theorem 1
  • theorem 2
  • theorem 3
  • proof
  • lemma 3
  • lemma 4: Schaefer's fixed point theorem Evans-PDE
  • proof
  • ...and 6 more