A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime (Q2814457)

The authors propose a multiscale time integrator Fourier pseudospectral method (MTI-FP) for the resolution of the 1D linear Dirac equation \( i\partial _{t}\phi (t,x)=[-\frac{i}{\varepsilon }\sum_{j=1}^{d}\sigma _{j}\partial _{j}+\frac{1}{\varepsilon ^{2}}\sigma _{3}+V(t,x)I_{2}-\sum_{j=1}^{d}A_{j}(t,x)\sigma _{j}]\phi (t,x)\) in \(\mathbb{ R}^{d}\) with the initial condition \(\phi (0,x)=\phi _{0}(x)\) in \(\mathbb{R}\). The authors claim that their approach also works in the 2D case. Here \(\sigma _{1}=\left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)\), \(\sigma _{2}=\left( \begin{smallmatrix} 0 & -i \\ i & 0 \end{smallmatrix} \right)\) and \(\sigma _{3}=\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)\) are the Pauli matrices, \(V\) is the real-valued electrical potential and \(A\) is the real-valued magnetic potential vector.NEWLINENEWLINEThe authors first observe that this Dirac equation may be rewritten as \(i\partial _{t}\phi (t,x)=\frac{1}{\varepsilon ^{2}}T\phi (t,x)+W(t,x)\phi (t,x)\) with \( T=-i\varepsilon \sigma _{1}\partial _{x}\) and \(W(t,x)=V(t,x)I_{2}-A_{1}(t,x) \sigma _{1}\). They choose a time step \(\Delta t>0\) and define \(t_{n}=n\Delta t\). They observe that the solution may be rewritten on the interval \(\left[ t_{n},t_{n+1}\right] \) as \(\phi (t_{n}+s,x)=e^{-is/\varepsilon ^{2}}(\Psi _{+}^{1,n}(s,x)+\Psi _{-}^{1,n}(s,x))+e^{is/\varepsilon ^{2}}(\Psi _{+}^{2,n}(s,x)+\Psi _{-}^{2,n}(s,x))\) where the functions \(\Psi _{\pm }^{i,n}\) are the solutions to a coupled system of two equations. This leads to the construction of the MTI-FP method, through the use of different approximations for the resolution of the system.NEWLINENEWLINEThe main result of the paper proves two uniform error estimates using two different mathematical approaches. In the last part of their paper, the authors present the results of numerical simulations through the MTI-FP method, essentially analyzing the temporal errors. They also describe the convergence of this Dirac equation to its limiting models when \(\varepsilon\) goes to 0, the authors here analyzing the behavior of error functions.