2021/12/07 by Weizhu Bao, Bao, Weizhu, Yue Feng +3
Computer Science · Engineering · Mathematics · Physics and Astronomy · #35Q41 #65M70 #65N35 #81Q05 #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Gyrotron and Vacuum Electronics Research #Numerical Analysis (math.NA) #Numerical methods for differential equations #cs.NA #math.NA #msc:35Q41 #msc:65M70 #msc:65N35 #msc:81Q05
paper · pdf · doi:10.48550/arxiv.2112.03616
22 pages, 6 figures
arxiv created 2021/12/07 · openalex publication_date 2021/12/07 · arxiv updated 2021/12/08 · openalex created_date 2021/12/31 · openalex updated_date 2026/07/28
We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small electromagnetic potentials characterized by a dimensionless parameter ε∈ (0, 1] representing the amplitude of the potentials. We begin with a semi-discritization of the Dirac equation in time by a time-splitting method, and then followed by a full-discretization in space by the Fourier pseudospectral method. Employing the unitary flow property of the second-order time-splitting method for the Dirac equation, we prove uniform error bounds at C(t)τ2 and C(t)(hm+τ2) for the semi-discretization and full-discretization, respectively, for any time t∈[0,Tε] with Tε = T/ε for T > 0, which are uniformly for ε ∈ (0, 1], where τ is the time step, h is the mesh size, m≥ 2 depends on the regularity of the solution, and C(t) = C0 + C1ε t≤ C0+C1T grows at most linearly with respect to t with C0≥0 and C1>0 two constants independent of t, h, τ and ε. Then by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation and energy method, we establish improved uniform error bounds at O(ετ2) and O(hm +ετ2) for the semi-discretization and full-discretization, respectively, up to the long-time Tε. Numerical results are reported to confirm our error bounds and to demonstrate that they are sharp. Comparisons on the accuracy of different time discretizations for the Dirac equation are also provided.