Based on the Lie-algebra, a new time-compact scheme is proposed to solve the one-dimensional Dirac equation. This time-compact scheme is proved to satisfy the conservation of discrete charge and is unconditionally stable. The time-compact scheme is of fourth-order accuracy in time and spectral order accuracy in space. Numerical examples are given to test our results.

The Dirac equation [

In this paper, we consider the one-dimensional Dirac equation [

The Dirac equation (

For Dirac equation (

The arrangement of the rest for this paper is organized as follows. The time-compact scheme with fourth-order accuracy is presented in Section

Recently, there has been growing interest in high-order compact method for solving partial differential equation, especially the time-compact methods [

In this section, we will use the time-compact scheme with fourth-order accuracy to solve the Dirac equation (

In practical computation, the computational domain is

Setting

Obviously, (

For operators

In order to solve the operator

Since

According to the form of (

Next we begin to solve (

In the second step, we solve (

The third step is to solve (

In the fourth step, we solve (

The methods of the fifth, sixth, and seventh steps are obtained as the third, second, and first steps, respectively.

The time-compact scheme for solving the Dirac equation at

The time-compact scheme (

Introduce the definition of the discrete inner product; that is,

In this section, we test the order of accuracy and stability of the time-compact scheme. In order to test the accuracy, we choose the electromagnetic potentials in (

We solve problem (

(Example 1) comparison of errors, orders, and CPU times for different schemes with

| | Order | CPU time (s) | |||
---|---|---|---|---|---|---|

TCS | SSM | TCS | SSM | TCS | SSM | |

0.100000 | | | — | — | 7.310 | 3.300 |

0.050000 | | | 3.96 | 2.01 | 14.565 | 6.427 |

0.025000 | | | 4.00 | 2.00 | 29.199 | 12.680 |

0.012500 | | | 4.02 | 2.02 | 58.864 | 25.789 |

0.006250 | | | 4.10 | 2.00 | 116.640 | 51.960 |

0.003125 | — | | — | 2.00 | — | 95.155 |

From Table

In every time step, the Strang splitting method with second-order accuracy needs three steps, and the time-compact scheme with fourth-order accuracy needs seven steps; that is, the total number of steps of the second-order accuracy scheme is

In order to test convergence of the algorithm with fourth-order

Fourth-order accuracy analysis in time for

The discrete charge calculated by the time-compact scheme is given at different time. As the calculated results have shown, one can see that the time-compact scheme conserves the discrete charge.

From Theorem

When

Oscillation of Dirac equation (

Based on Lie-algebra, the time-compact scheme is presented for solving the one-dimensional linear Dirac equation. Then we test whether the time-compact scheme has fourth-order accuracy in time and is proved to keep the conservation of discrete charge. From the numerical results, the time-compact scheme performs much better than the Strang splitting method in the error analysis, in terms of accuracy and efficiency. In addition, the time-compact scheme is unconditionally stable, and numerical experiment is presented to discuss the changes of the frequency oscillation with different

The authors declare that they have no competing interests.

This work was supported in part by the National Natural Science Foundation of China (no. 11671044) and in part by the Beijing Municipal Education Commission under Grant no. PXM2016_014224_000028.