A stable and high-order accurate solver for linear and nonlinear parabolic equations is presented. An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an implicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N)$ for each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods.Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

翻译：提出了一种适用于线性和非线性抛物型方程的稳定且高阶精度求解器。采用加性龙格-库塔方法进行时间推进，其中线性刚性项通过隐式单对角隐式龙格-库塔（ESDIRK）方法积分，非线性项则通过显式龙格-库塔（ERK）方法积分。在每个时间步中，隐式求解由近期发展的层次化庞加莱-斯捷克洛夫（HPS）方法完成。这是一种椭圆型方程的快速直接求解器，它将空间域分解为层次化子树结构，并在子域上构建局部谱配置求解器。由于ESDIRK方法的单对角系数与固定时间步长确保了HPS隐式求解中的系数矩阵在所有时间阶段保持不变，上述思想在该方法中自然结合。这意味着预计算的逆矩阵可被高效复用，从而形成一种复杂度（在二维情形下）为$\mathcal{O}(N^{1.5})$的预计算方案（用于构建椭圆问题的求解算子），并实现每个时间步$\mathcal{O}(N)$的计算复杂度。本文证明了该方法对时间一阶精度和空间任意阶精度的稳定性，数值实验进一步表明其对更广泛时间离散方法类别的稳定性。文中还给出了支持该方法在一维和二维情形下精度与效率的数值实验。