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$1+1$维Gross-Neveu模型中非线性狄拉克方程的量子格点玻尔兹曼格式。假设该格式的初始数据在$L^2$范数下收敛。则对于任意$T\ge 0$，当网格尺寸趋近于零时，量子格点玻尔兹曼格式对应的解在$C([0,T];L^2(R^1))$中收敛于非线性狄拉克方程的强解。在证明过程中，首先引入Glimm型泛函建立量子格点玻尔兹曼格式两个解之差的稳定性估计，进而得到该格式解集的紧性。最终证明该格式解的任意收敛子序列的极限与非线性狄拉克方程柯西问题的强解一致。