An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit 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 $\mathcal{O}(N \log N)$ for the solve in 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：采用加性龙格-库塔方法进行时间步进：线性刚性项通过显式单对角隐式龙格-库塔（ESDIRK）方法积分，非线性项则通过显式龙格-库塔（ERK）方法处理。在每个时间步中，隐式求解由近期发展的分层庞加莱-斯捷克洛夫（HPS）方法执行。该方法是一种椭圆型方程快速直接求解器，通过将空间域分解为分层的子域树，并在各子域上构建谱配点求解器。所提方法自然融合了上述思想：由于ESDIRK中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，因此可高效复用预计算逆矩阵，从而形成一种算法——预计算阶段（构建椭圆问题求解算子）复杂度为$\mathcal{O}(N^{1.5})$（二维情形），而每个时间步的求解复杂度为$\mathcal{O}(N \log N)$。本文证明了该方法在时间一阶精度与空间任意阶精度条件下的稳定性，数值实验进一步验证了其对更广泛时间离散方法类别的稳定性。此外，一维与二维数值算例均展示了该方法在精度与效率方面的优异性能。