High-order compact methods for the nonlinear Dirac equation

Abstract

In this work, a fourth-order in space and second-order in time compact scheme, a sixth-order in space and second-order in time compact scheme and two linearized compact schemes are proposed for the (1+1)-dimensional nonlinear Dirac equation. The iterative algorithm is used to compute the nonlinear algebraic system and the Thomas algorithm in the matrix form is adopted to enhance the computational efficiency. It is proved that all of the schemes are unconditionally stable in the linear sense. Numerical experiments are given to test the accuracy order of the presented schemes, record the error history for all of the schemes with respect to t, discuss the conservation laws of discrete charge and energy from the numerical point of view, study the stability of the solitary waves by adding a small random perturbation to the initial data, and simulate the collision of two and three solitary waves.

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

“…$$ It is easy to check that $$$ {\mathcal{A}}_h\left({\partial}_x\Phi \left({x}_j\right)\right)=\left(\Phi \left({x}_{j+1}\right)-\Phi \left({x}_{j-1}\right)\right)/2h+O\left({h}^4\right) $$$. A fourth‐order compact approximation is implemented by replacing $$$ {\partial}_x\Phi \left({t}_n,{x}_j\right) $$$ by $$$ {\mathcal{A}}_h^{-1}\left({\delta}_x{\Phi}_j^n\right) $$$ [37, 39].…”

“…) [37,39]. Combining the 4cFD discretization in space with the implicit and semi-implicit temporal discretizations, we have the following two 4cFD schemes for j = 0, 1, … , N − 1:…”

“…The fourth-order compact finite difference (4cFD) method is a simple scheme to attain higher spatial order with the same number of grids for the central difference method [30][31][32]. Recently, the 4cFD method has been used to solve the (nonlinear) Schrödinger equation [33,34], Klein-Gorden equation [35,36], Dirac equation [37], Burgers' equation [30] and so on. For more details, we refer to [31,[38][39][40][41][42] and references therein.…”

Error bounds of fourth-order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

“…and the average vector [26,28]. Combining the fourth-order compact finite difference discretization in space with the implicit/semiimplicit temporal discretization, we have the following two 4cFD schemes for j = 0, 1, • • • , N − 1:…”

“…The fourth-order compact finite difference (4cFD) method is a simple scheme to attain higher spatial order with the same number of grids for the central difference method [29,33,43]. Recently, the 4cFD method has been used to solve the (nonlinear) Schrödinger equation [24,41], Klein-Gorden equation [17,30], Dirac equation [26], Burgers' equation [29] and so on. For more details, we refer to [27,28,33,39,40] and references therein.…”