线方法转置：一种快速隐式波传播求解器
Method of Lines Transpose: A Fast Implicit Wave Propagator
AI总结：
本文提出一种O(N)的隐式波方程求解器，通过改进空间四阶积分方法，实现高精度且适用于不规则网格的求解，支持多种边界条件，并在更高维度中利用交替方向隐式方法实现快速求解。
摘要：
在跟进Causley2013的研究基础上，本文详细描述了波方程O(N)、A-稳定、二阶精度的数值实现，基于半离散边界值问题构建。我们改进了之前的算法，将Causley2013中用于收敛的Lax型修正替换为更精确的空间四阶积分，证明其收敛性。我们还展示了所得到的求解器在无结构网格情况下仍保持快速，可结合域分解，并允许实现狄利克雷、Neumann、周期性和出流边界条件。基于一维结果，利用交替方向隐式（ADI）分裂方法，在更高空间维度中实现快速O(N)求解器。我们的求解器基于线对象，结合积分求解器的灵活性，允许在任意空间域上求解问题，通过将边界嵌入到规则的笛卡尔网格中。我们的求解器设计用于与粒子代码耦合，其中尺度分离是一个问题。因此，我们展示了该求解器能够采用远超Courant-Friedrichs-Lewy（CFL）稳定性极限的时步，适用于显式代码。
英文摘要：
As a follow up to \cite{Causley2013}, we provide a detailed description of the numerical implementation of an O(N), A-stable, second order accurate solution of the wave equation, constructed from semi-discrete boundary value problems. We improve on the previous algorithm by replacing the Lax-type correction used in \cite{Causley2013}, which was necessary for convergence when $Δt < Δx/c$, with a more accurate spatial quadrature, which we prove is convergent. We also demonstrate that the resulting solver remains fast even in the case of unstructured meshes, can incorporate domain decomposition, and allows for the implementation of Dirichlet, Neumann, periodic and outflow boundary conditions. Building upon results for the 1d formulation, we utilize alternate direction implicit (ADI) splitting to achieve a fast O(N) solver in higher spatial dimensions. Our solver is built upon line objects and, combined with the flexibility of the integral solver, allows us to solve problems on arbitrary spatial domains, by embedding the boundary in a regular Cartesian mesh. Our solver is designed to couple with particle codes, where scale separation is an issue. We therefore demonstrate the ability of our solver to take time steps well beyond that of the Courant-Friedrichs-Lewy (CFL) stability limit of explicit codes.
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