Spatial resolution of different discretizations over long-time for the Dirac equation with small potentials

We compare the long-time error bounds and spatial resolution of finite difference methods with different discretizations for Dirac equation small electromagnetic potentials characterized by ??(0,1] a dimensionless parameter. begin simple widely used time domain (FDTD) methods, establish rigorous them, which are valid up to at O(1/?). In estimates, we pay particular attention how errors depend explicitly on mesh size h step ? as well parameter ?. Based results, in order obtain “correct” numerical solutions O(1/?), ?-scalability (or meshing strategy requirement) FDTD should be taken h=O(?1/2) ?=O(?1/2). To improve capacity, apply Fourier spectral method discretize space. Error resulting pseudospectral (FDFP) show that they exhibit uniform regime, optimal space suggested Shannon’s sampling theorem. Extensive results reported confirm demonstrate sharp.