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

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 $\mathrm{L}^{2}(\mathbb{R}\times[0,T])$ to the global strong solution of the nonlinear Dirac equation for any $T>0$. 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 relatively compact in $\mathrm{C}([0,T];\mathrm{L}^{2}(\mathbb{R}))$ for any $T>0$. Finally, we show that the limit of any convergent subsequence of the time-splitting solutions is the strong solution to the Cauchy problem of the nonlinear Dirac equation.