Convergence analysis of a symmetric exponential integrator Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

“…Lemma [38] For $$$ v>1/2 $$$, $$$ u\in {H}^v\left(\Omega \right) $$$ and $$$ U,W\in {\left[{H}^v\left(\Omega \right)\right]}^2 $$$ , we have $$$$ {\left\Vert u{\sigma}_3W\right\Vert}_v\le {C}_v{\left\Vert u\right\Vert}_v{\left\Vert W\right\Vert}_v,\kern1em {\left\Vert {U}^{\ast }{\sigma}_3W\right\Vert}_v\le {C}_v{\left\Vert U\right\Vert}_v{\left\Vert W\right\Vert}_v.\kern0.5em $$$$ Here $$$ {C}_v>0 $$$ is independent of $$$ u $$$, $$$ U $$$ and $$$ W $$$.…”

“…Lemma [38] For $$$ v>1/2 $$$, $$$ {u}_1,{u}_2\in {H}^v\left(\Omega \right) $$$, $$$ {\Psi}_1,{\Psi}_2\in {\left[{H}^v\left(\Omega \right)\right]}^2 $$$ and $$$ F\left(\Psi \right) $$$, $$$ G\left(u,\Psi \right) $$$ of ( 8 ), we have …”

“…For the numerical part, many efficient and accurate numerical methods have been proposed and analyzed, such as the finite difference time domain (FDTD) methods and spectral methods for the Klein-Gordon (KG) equation [9,34,42], the Dirac equation [6,7,28,32,33,35,37,48], the Klein-Gordon-Schrödinger equations [10,17,18] and the Maxwell-Dirac systems [33], and so forth. Recently, some numerical methods for the KGD Equation (1) have been proposed in the literature [15,36,38,49,50]. For the case 𝜀 = 1, some authors studied FDTD schemes and proved that they are second order in both time and space [15,49].…”

“…For the case $$$ \varepsilon =1 $$$, some authors studied FDTD schemes and proved that they are second order in both time and space [15, 49]. In recent papers [38, 50], the authors proposed the exponential wave integrator Fourier pseudo‐spectral (EWIFP) methods for the KGD Equation (1) in the standard regime and nonrelativistic limit regime, respectively. In addition, time‐split Fourier pseudo‐spectral methods (TSFP) are also considered [50].…”

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

“…In practical application, efficient numerical methods are very necessary. Existing numerical methods include finite difference (FD) and compact FD (CFD) methods [8,12,31,41,44], exponential wave integrator (EWI) methods [27,44], time-splitting (TS) methods [42,45]. Generally speaking, compared with the FD and CFD methods, these EWI methods and TS methods perform better in numerical accuracy and regularity requirement.…”