2026/03/31 by Ningning Li, Yongqian Zhang, Qin Zhao
Mathematics · Computer Science · #math.AP #cs.NA #math.NA #msc:35Q41 #msc:65M12 #msc:81Q05
arxiv created 2026/08/03 · arxiv updated 2026/08/04
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 L2(ℝ), we prove that the approximate solutions constructed by the corresponding time-splitting scheme are strongly convergent in L2(ℝ×[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 L2 stability estimates for the scheme. Furthermore, we prove that the set of time-splitting solutions is relatively compact in C([0,T];L2(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.