Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics. (English) Zbl 07832807
Summary: We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are \(O(\varepsilon \tau^2)\) and \(O(h_1^m +h_2^m +\varepsilon \tau^2)\) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, \(\tau\) is the time step, \(h_i(i=1,2)\) stands for spatial sizes in \(x_i\)-directions, \(m\) is dependent on the regularity of solutions, and \(\varepsilon \in(0,1]\). In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied \(\varepsilon\), numerical investigations are presented.
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
strang splitting method; nonlinear space fractional Dirac equation; regularity compensation oscillation (RCO); long-time dynamics; Fourier pseudospectral methodReferences:
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