Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials

Abstract

.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 \(\varepsilon \in (0, 1]\) representing the amplitude of the potentials. We begin with a semidiscretization of the Dirac equation in time by a time-splitting method, followed by a full-discretization in space by with the Fourier pseudospectral method in space. By employing the unitary flow property of the second-order time-splitting method for the Dirac equation, we prove uniform error bounds for \(\varepsilon\in(0, 1]\) at \(C(t)\tau^2\) and \(C(t)(h^m+\tau^2)\) for the semidiscretization and full-discretization, respectively, for any time \(t\in [0,T_\varepsilon ]\) with \(T_\varepsilon = T/\varepsilon\) and \(T \gt 0\) . In the expressions, \(\tau\) is the time step, \(h\) is the mesh size, \(m\geq 2\) depends on the regularity of the solution, and \(C(t) = C_0 + C_1\varepsilon t\le C_0+C_1T\) grows at most linearly with respect to \(t\) with \(C_0\ge 0\) and \(C_1\gt 0\) two constants independent of \(t\) , \(h\) , \(\tau\) , and \(\varepsilon\) . Then by adopting the regularity compensation oscillation 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 \(O(\varepsilon \tau^2)\) and \(O(h^m +\varepsilon \tau^2)\) for the semidiscretization and full-discretization, respectively, up to the long-time \(T_\varepsilon\) . 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.KeywordsDirac equationlong-time dynamicstime-splitting methodsimproved uniform error boundregularity compensation oscillation (RCO)MSC codes35Q4165M7065N3581Q05

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