Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmetric systems that arise from non-self-adjoint problems when iterative methods are employed; such methods are the only feasible methods for very large scale computation with PDEs. We then go on to consider non-symmetric all-at-once systems that arise in approximation of time-dependent problems, discuss causality and the parallel-in-time paradigm, suggesting an approach that involves preconditioning initial value problems with time-periodic problems.

数值线性代数——特别是方程的计算解——构成了偏微分方程计算方法的重要组成部分。在这里，我们讨论了当使用迭代方法时，由自伴随问题产生的对称方程组的解与由非自伴随问题产生的非对称方程组的解之间的对比；此类方法是使用 PDE 进行超大规模计算的唯一可行方法。然后，我们继续考虑在时间相关问题的近似中出现的非对称一次性系统，讨论因果关系和时间并行范式，提出一种涉及预处理具有时间周期问题的初始值问题的方法.