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$\mathrm{L}^{2}$--convergence of the time-splitting scheme for nonlinear Dirac equation in 1+1 dimensions

Ningning Li, Yongqian Zhang, Qin Zhao

Abstract

We study the time-splitting scheme for approximating solutions to the Cauchy problem of the nonlinear Dirac equation in 1+1 dimensions. Under the assumption that the initial data for the scheme are convergent in $\mathrm{L}^{2}(\mathbb{R})$, we prove that the approximate solutions constructed by the corresponding time-splitting scheme are strongly convergent in $\mathrm{C}([0,\infty);\mathrm{L}^{2}(\mathbb{R}))$ to the global strong solution of the nonlinear Dirac equation. To achieve this, we first establish the pointwise estimates for time-splitting solutions. Based on these estimates, a modified Glimm-type functional is carefully designed to show that it is uniformly bounded in time, which yields $\mathrm{L}^2$ stability estimates for the scheme. Furthermore, we prove that the set of time-splitting solutions is precompact in $\mathrm{C}([0,T];\mathrm{L}^{2}(\mathbb{R}))$ for any $T>0$. Finally, we show that the limit of any subsequence of the time-splitting solutions is the unique strong solution to the Cauchy problem of the nonlinear Dirac equation.

$\mathrm{L}^{2}$--convergence of the time-splitting scheme for nonlinear Dirac equation in 1+1 dimensions

Abstract

We study the time-splitting scheme for approximating solutions to the Cauchy problem of the nonlinear Dirac equation in 1+1 dimensions. Under the assumption that the initial data for the scheme are convergent in , we prove that the approximate solutions constructed by the corresponding time-splitting scheme are strongly convergent in to the global strong solution of the nonlinear Dirac equation. To achieve this, we first establish the pointwise estimates for time-splitting solutions. Based on these estimates, a modified Glimm-type functional is carefully designed to show that it is uniformly bounded in time, which yields stability estimates for the scheme. Furthermore, we prove that the set of time-splitting solutions is precompact in for any . Finally, we show that the limit of any subsequence of the time-splitting solutions is the unique strong solution to the Cauchy problem of the nonlinear Dirac equation.
Paper Structure (12 sections, 17 theorems, 209 equations, 3 figures)

This paper contains 12 sections, 17 theorems, 209 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Time splitting schemes and main result
  3. Main difficulties and key ideas of the proof
  4. Organization of the paper
  5. A priori estimates
  6. Estimates on the time-splitting solutions along characteristic directions
  7. Pointwise estimates on the time-splitting solutions
  8. $\mathrm{L}^2$ stability estimates on the time-splitting solutions in a discrete characteristic triangle
  9. Compactness of the set of time-splitting solutions in $\mathrm{L}^{2}_{\mathrm{loc}}$
  10. Global $\mathrm{L}^2$ stability of time-splitting solutions
  11. Compactness of the family of time-splitting solutions
  12. Uniqueness of the limit of time-splitting solutions for the NLDE

Key Result

Theorem 1.1

Suppose that $(u_{0},v_{0})\in \mathrm{L}^{2}(\mathbb{R})$, and suppose that Let $(u,v)$ be the unique strong solution to the NLDE Cauchy problem eq:NLDE--eq:NLDE-initail-data, and let $(u^{(\tau)},v^{(\tau)})$ be the time-splitting solution to eq:NLDE--eq:NLDE-initail-data with initial data $(u^{(\tau)}(x,0),v^{(\tau)}(x,0))$, defined by the scheme time-split for $\tau>0$.

Figures (3)

  • Figure 2.1: The discrete characteristic triangle $\Delta(j_{1},n_{1};n_{0})$
  • Figure 3.1: The characteristic triangle $\Lambda(x_1,t_{1};t_{0})$
  • Figure 3.2: Division of the strip $\mathbb{R}\times [0,T]$

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 1.1: Strong solutions
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 31 more