This paper studies the quantum lattice Boltzmann scheme for the nonlinear Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in $C([0,T];L^2(R^1))$ to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero. In the proof, at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding quantum lattice Boltzmann scheme, which leads to the compactness of the set of the solutions for the quantum lattice Boltzmann scheme. Finally, the limit of any convergent subsequence of the solutions for the quantum lattice Boltzmann scheme is shown to coincide with the strong solution to a Cauchy problem for the nonlinear Dirac equations.

翻译：本文研究Gross- Neveu 模型的非线性Dirac方程式的量子 lattice Boltzmann 方程式1+1美元。 假设该方程式的初始数据以美元=2美元为单位, 假设该方程式的初始数据以美元=2美元为单位。 然后, 对于任何美元=0美元的量子 lattice Boltzmann 方程式的相应解决方案, 均以美元( [0,T];L2, (R1])美元为单位, 以美元=非线性Dirac 方程式的强性解决方案。 事实证明, 最初引入了 Glimm 型函数, 以建立相应的量子 lattice Boltzmann 方程式两种解决方案之间差异的稳定性估计值, 导致量子 lattice Boltzmann 方程式的一套解决方案的紧凑性。 最后, 量子 lattice Boltzmann 方程式解决方案的任何相趋同的次序列后, 的极限与非线性方程式的非线性方程式问题的强性解决方案相吻合。