We study the discretization of a linear evolution partial differential equation when its Green function is known. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method (DTCM) that we developed in J. Math. Phys. (2010). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.

翻译：当已知绿功能时,我们研究线性进化部分差异方程式的离散性。我们提供空间近似和时间步近似值的误差估计。我们表明,事实上,绿函数近似于绿函数本身。对于适合时间的抛物线方程式,我们解释如何使用我们在J. Math. Phys. (2010年) 中开发的Dyson-Taylor 运算法(DTCM ) 获得绿色函数的良好、明确的近似值。这一短时间近似值,如果加上一个“靴子”参数,在任何规定的容忍范围内就任何固定时间间隔给出了大致的解决方案。