Multirate Time Integration Algorithms for Adaptive Simulations of PDEs

禤浩焯三都弗吉尼亚理工学院和州立大学用于PDE自适应模拟的多时间积分算法时间相关偏微分方程(PDE)的大尺度模拟通常涉及覆盖不同子域的多分辨率网格。当采用显式时间积分时，稳定性要求限制了全局模拟时间步长。时间步长界限由最精细的网格面片或最高的波速驱动，通常比计算域中的其他变量所需的小得多。效率和整体模拟能力的提高要求开发新的、自适应的、多速率的时间积分方法。由于时间步进算法需要满足的守恒性和稳定性约束，多速率积分的发展具有挑战性。本课题的总体目标是开发高效的时间步长方法，用于大规模时变偏微分方程组的并行模拟。多速率算法将被构造为：(1)可以在不同的子域中使用不同的时间步长以实现效率；(2)可以以高的时间精度构造方法；(3)线性和非线性稳定性仅施加步长的局部限制(例如局部Courant数)；(4)方法是保守的；以及(5)在多物理模拟中，不同的方法可以应用于不同的过程。研究方法是对Runge-Kutta方法和线性多步方法采用多速率积分框架。多速率积分技术将继承相应单速率积分器的强稳定性，并将构造适合于多物理多尺度模拟的隐式-显式多速率方法。这些方法将在大气污染预测中出现的真实、多尺度、多物理模拟中得到说明。

ABSTRACT0515170 Adrian SanduVirginia Polytechnic Institute and State UniversityMULTIRATE TIME INTEGRATION ALGORITHMS FOR ADAPTIVE SIMULATIONS OF PDESLarge scale simulations of time-dependent partial differential equations (PDEs) often involve grids of multiple resolutions covering different subdomains. When explicit temporal integration is employed, stability requirements restrict the global simulation time step. The time step bound is driven by the finest mesh patch or by the highest wave velocity, and is typically (much) smaller than necessary for other variables in the computational domain. Improvements in the efficiency and overall simulation capabilities require the development of new, adaptive, multirate time integration methods. The development of multirateintegration is challenging due to the conservation and stability constraints which time stepping schemes need to satisfy.The overall goal of the proposed project is to develop efficient time stepping methods for parallel simulation of large-scale time-dependent PDEs. Multirate algorithms will be constructed such that: (1) differenttime steps can be used in different subdomains to achieve efficiency; (2) the methods can be constructed with high order of temporal accuracy; (3) linear and nonlinear stability impose only local restrictions of the stepsize (e.g., local Courant numbers); (4) the methods are conservative; and (5) different methods can be applied to different processes in multi-physics simulations. The research approach is to employ theframework of multirate integration for both Runge-Kutta and linear multistep methods. The multirate integration techniques will inherit the strong stability properties of the corresponding single rate integrators.Moreover, implicit-explicit multirate methods will be constructed, which are appropriate for multiphysics multiscale simulations. The methods will be illustrated in real-life, multi-scale, multi-physics simulationsarising in the prediction of atmospheric pollution.