连续对称问题的射影和粗射影积分
Projective and Coarse Projective Integration for Problems with Continuous Symmetries
摘要：
具有连续对称性的方程（例如，具有与行进解相关联的平移不变性和与自相似解相关联的尺度不变性的系统）在"共同演化“框架（即，与演化解共同行进、共同坍缩或共同爆炸的框架）中的时间积分导致改进的精度，因为在新的空间框架中的时间导数较小。较慢的时间行为允许使用{ \it投影}和{ \it粗投影}集成与较长的投影步骤中的时间演化的计算的偏微分方程和多尺度系统，分别。这些方法也被证明是有效的系统，只有近似或渐近具有连续对称性。在协进化框架下的射影积分思想，说明了一维，几何不变的Nagumo偏微分方程（PDE）。相应的动力学蒙特卡罗模型，从南云动力学的动机，是用来说明粗粒度的方法。一个简单的，一维的扩散问题是用来说明尺度不变的情况。投影积分的效率，在共同发展的框架下的宏观扩散偏微分方程和随机行走粒子为基础的模型再次证明。
英文摘要：
Temporal integration of equations possessing continuous symmetries (e.g. systems with translational invariance associated with traveling solutions and scale invariance associated with self-similar solutions) in a ``co-evolving'' frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with the evolving solution) leads to improved accuracy because of the smaller time derivative in the new spatial frame. The slower time behavior permits the use of {\it projective} and {\it coarse projective} integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems, respectively. These methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries. The ideas of projective integration in a co-evolving frame are illustrated on the one-dimensional, translationally invariant Nagumo partial differential equation (PDE). A corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is used to illustrate the coarse-grained method. A simple, one-dimensional diffusion problem is used to illustrate the scale invariant case. The efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model is again demonstrated.
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