（1+1）维Dirac方程透明边界条件下的保模色散差分格式
A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in (1+1)D
摘要：
本文给出了（1+1）维Dirac方程的一个有限差分格式。它可以处理空间和时间相关的质量和潜在的条款，并利用精确的离散透明边界条件（DTBC）。该方法基于时空交错的蛙跳格式，避免了费米子加倍，并精确地保持了质量为零时连续介质问题（Weyl方程）的色散关系。考虑到边界区域，每个具有恒定的质量和潜在的条款，相关的DTBC推导出第一次应用此有限差分格式，然后使用Z变换的离散时间变量。由此产生的常系数差分方程在空间中可以精确地解决在两个半无限的外部区域。在$l_2$中只允许无穷远处为零的解等价于施加输出边界条件。逆Z变换导致精确的DTBC在离散时间的卷积的形式，抑制在边界处的寄生反射，并加强整个时空方案的稳定性。给出了交错网格上Dirac旋量范数的一个精确保持泛函。高斯波包的模拟，离开计算域没有反射，证明质量的DTBC数值，以及一个忠实的代表性的能量-动量色散关系的网格上的重要性。
英文摘要：
A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stability of the whole space-time scheme. An exactly preserved functional for the norm of the Dirac spinor on the staggered grid is presented. Simulations of Gaussian wave packets, leaving the computational domain without reflection, demonstrate the quality of the DTBCs numerically, as well as the importance of a faithful representation of the energy-momentum dispersion relation on a grid.
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