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.

Appendix: Sixth-order boundary conditions

Let

$$\begin{aligned} L = h \left( \alpha \left( \partial _{xx}{\varvec{u}}\right) _{0}^{n} + \beta \left( \partial _{xx}{\varvec{u}}\right) _{1}^{n} + \gamma \left( \partial _{xx}{\varvec{u}}\right) _{2}^{n} \right) + \lambda \left( \partial _{x}{\varvec{u}}\right) _{0}^{n} + \mu \left( \partial _{x}{\varvec{u}}\right) _{1}^{n} + \xi \left( \partial _{x}{\varvec{u}}\right) _{2}^{n} + \frac{a {\varvec{u}}_{0}^{n} + b {\varvec{u}}_{1}^{n} + c {\varvec{u}}_{2}^{n} }{h}, \end{aligned}$$

(26)

and substitute (6)–(8) to (26), we have

$$\begin{aligned} \begin{aligned} L&= \frac{a+b+c}{h}{\varvec{u}}_{1}^{n}+\left( \lambda +\mu +\xi +c-a\right) \left( \partial _{x}{\varvec{u}}\right) _{1}^{n} + \left( \alpha +\beta +\gamma +\xi -\lambda +\frac{c+a}{2}\right) h\left( \partial _{xx}{\varvec{u}}\right) _{1}^{n} \\&\quad + \left( \gamma -\alpha +\frac{\xi +\lambda }{2!}+\frac{c-a}{3!}\right) h^{2}\left( \partial _{3x}{\varvec{u}}\right) _{1}^{n} + \left( \frac{\gamma +\alpha }{2!}+\frac{\xi -\lambda }{3!}+\frac{c+a}{4!} \right) h^{3}\left( \partial _{4x}{\varvec{u}}\right) _{1}^{n} \\&\quad + \left( \frac{\gamma -\alpha }{3!}+\frac{\xi +\lambda }{4!}+\frac{c-a}{5!} \right) h^{4}\left( \partial _{5x}{\varvec{u}}\right) _{1}^{n} + \left( \frac{\gamma +\alpha }{4!}+\frac{\xi -\lambda }{5!}+\frac{c+a}{6!} \right) h^{5}\left( \partial _{6x}{\varvec{u}}\right) _{1}^{n} \\&\quad + \left( \frac{\gamma -\alpha }{5!}+\frac{\xi +\lambda }{6!}+\frac{c-a}{7!} \right) h^{6}\left( \partial _{7x}{\varvec{u}}\right) _{1}^{n} + \cdots , \end{aligned} \end{aligned}$$

The last part of the above equation can be replaced by $O\left( h ^{6}\right)$, then let the coefficients be equal to zero, one gets some pairs of sixth-order boundary conditions. Obviously, there are more than four solutions, here we give only four solutions, i.e.,

$$\begin{aligned} \left\{ \begin{array}{l} a = -39, b = 48, c = -9, \alpha = -2, \beta = 8, \gamma = 0, \lambda = -16, \mu = -16, \xi = 2, \\ a = -303, b = 336, c = -33, \alpha = -16, \beta = 56, \gamma = 2, \lambda = -126, \mu = -144, \xi = 0,\\ a = -\frac{645}{2}, b = 360, c = -\frac{75}{2}, \alpha = -17, \beta = 60, \gamma = 2, \lambda = -134, \mu = -152, \xi = 1,\\ a = -\frac{381}{2}, b = 216, c = -\frac{51}{2}, \alpha = -10, \beta = 36, \gamma = 1, \lambda = -79, \mu = -88, \xi = 2. \end{array} \right. \end{aligned}$$

In addition, there are two fifth-order one-side boundary conditions (Li et al. 2015)

$$\begin{aligned} \left\{ \begin{array}{l} 2h\left( \partial _{xx}{\varvec{u}}\right) _{0}^{n}-4h\left( \partial _{xx}{\varvec{u}}\right) _{1}^{n}+14\left( \partial _{x}{\varvec{u}}\right) _{0}^{n}+16\left( \partial _{x}{\varvec{u}}\right) _{1}^{n}+\frac{31{\varvec{u}}_{0}^{n}-32{\varvec{u}}_{1}^{n}+{\varvec{u}}_{2}^{n}}{h}=\mathbf {0},\\ -2h\left( \partial _{xx}{\varvec{u}}\right) _{J}^{n}+4h\left( \partial _{xx}{\varvec{u}}\right) _{J-1}^{n}+14\left( \partial _{x}{\varvec{u}}\right) _{J}^{n}+16\left( \partial _{x}{\varvec{u}}\right) _{J-1}^{n}-\frac{31{\varvec{u}}_{J}^{n}-32{\varvec{u}}_{J-1}^{n}+{\varvec{u}}_{J-2}^{n}}{h}=\mathbf {0}, \end{array} \right. \end{aligned}$$

and four fourth-order one-side boundary conditions (Chu and Fan 1998)

$$\begin{aligned} \left\{ \begin{array}{l} 2h\left( \partial _{xx}{\varvec{u}}\right) _{0}^{n}-2\left( \partial _{x}{\varvec{u}}\right) _{0}^{n}-4\left( \partial _{x}{\varvec{u}}\right) _{1}^{n}-\frac{ 7{\varvec{u}}_{0}^{n}-8{\varvec{u}}_{1}^{n}+7{\varvec{u}}_{2}^{n}}{h}=\mathbf {0},\\ 2h\left( \partial _{xx}{\varvec{u}}\right) _{J-1}^{n}+2\left( \partial _{x}{\varvec{u}}\right) _{J}^{n}+4\left( \partial _{x}{\varvec{u}}\right) _{J-1}^{n}-\frac{ 7{\varvec{u}}_{0}^{n}-8{\varvec{u}}_{1}^{n}+7{\varvec{u}}_{2}^{n}}{h}=\mathbf {0}, \\ h\left( \partial _{xx}{\varvec{u}}\right) _{0}^{n}+5h\left( \partial _{xx}{\varvec{u}}\right) _{1}^{n}-6\left( \partial _{x}{\varvec{u}}\right) _{1}^{n}-\frac{ 9{\varvec{u}}_{0}^{n}-12{\varvec{u}}_{1}^{n}+9{\varvec{u}}_{2}^{n}}{h}=\mathbf {0},\\ h\left( \partial _{xx}{\varvec{u}}\right) _{J}^{n}+5h\left( \partial _{xx}{\varvec{u}}\right) _{J-1}^{n}+6\left( \partial _{x}{\varvec{u}}\right) _{J-1}^{n}-\frac{ 9{\varvec{u}}_{J}^{n}-12{\varvec{u}}_{J-1}^{n}+9{\varvec{u}}_{J-2}^{n}}{h}=\mathbf {0}. \end{array} \right. \end{aligned}$$