This work is a first taste of using null-space technique to deal with a basic model of unsteady partial differential equations (PDEs), i.e., the unsteady Stokes equations. Any inf–sup stable semi-discretization of the unsteady Stokes equations yields a system of differential–algebraic equations (DAEs), i.e., the unsteady discrete Stokes equations. The exact solution to the unsteady discrete Stokes equations is explicitly constructed with the help of the null-space of the discrete divergence operator and a matrix exponential. In practical implementation, the explicit use of the null-space of the discrete divergence operator is avoided. Therefore, the main workload of solving the unsteady discrete Stokes equations is the matrix exponential vector product. The matrix exponential vector product is written in terms of an integral relation, which is approximated by a linear combination of values of the integrand evaluated at a select number of complex numbers. Each evaluation of the integrand needs to solve a linear system. In numerical experiments, the null-space method outperforms frequently used time-stepping methods in terms of accuracy and computing time.

这项工作是使用零空间技术处理非稳态偏微分方程（PDE）的基本模型（即非稳态Stokes方程）的初衷。非定常Stokes方程的任何insup稳定半离散都会产生一个微分代数方程组（DAE），即非定常离散Stokes方程。借助离散发散算子的零空间和矩阵指数，可以明确构造非定常离散Stokes方程的精确解。在实际实现中，避免了离散发散算子的零空间的明确使用。因此，求解非稳态离散Stokes方程的主要工作量是矩阵指数矢量积。矩阵指数矢量乘积是按照积分关系写的，这是通过在选定数量的复数上求出的被积数的值的线性组合来近似得出的。对被积物的每次评估都需要求解线性系统。在数值实验中，就准确性和计算时间而言，零空间方法优于常用的时间步长方法。