论文标题
隐式阐明多段无穷无尽的天才方法
Implicit-explicit multirate infinitesimal GARK methods
论文作者
论文摘要
这项工作着重于开发一类新的高级准确方法，以用于多次微分方程系统的多段时间整合。与该领域的其他最新工作不同，提出的方法支持混合隐式解释（IMEX）治疗缓慢的时间尺度。除了允许这种缓慢的时间尺度灵活性之外，提出的方法还通过定义了一系列修改的“快速”初始值问题，利用了快速时间尺度的所谓“无穷小”公式，可以使用任何可行的算法来解决这些问题。我们将提出的类命名为隐式 - 阐明多段无限的通用结构添加剂runge-kutta（IMEX-MRI-GARK）方法。除了定义这些方法外，我们还可以证明它们可以被视为天然可归方法的特定实例，并在IMEX-MRI-GARK系数上得出一组订单条件，以确保整个多段方法的第三和第四级准确性。此外，我们提供了三种特定的IMEX-MRI-GARK方法，第三阶和第四顺序之一。我们以数值模拟进行了关于两个多条测试问题的数值仿真，证明了该方法的预测融合率，并将其效率与遗留IMEX多培养基方案以及最近的第三和第四阶隐含MRI-GARK方法进行了比较。
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.
