Implicit Multi-Scale Plasma Simulations Using Low Cost Matrix-Free Methods for Partial Differential Equations

1859年的卡灵顿事件是一次巨大的太阳耀斑，它撞击了地球，并在地球磁场中留下了大量带电粒子。这些带电粒子产生了巨大的电流，通过电感耦合耦合到电报线上，就像在变压器中发生的那样。通过电报线的电流太大，以至于导致电报机爆炸。如果这种情况发生在今天，除非我们知道如何关闭和断开电网，否则这样的事件将摧毁现代电力基础设施。然而，太阳现象一直在发生，我们不能每次都简单地关闭电网。从被探测到的时间算起，一个太阳光束需要3天才能到达地球。预测太阳是否会撞击地球目前是用低分辨率的模型来完成的，这些模型在预测这些事件方面并不擅长。在计算中加入更多的物理因素，从而提高计算精度，这将导致更大的计算，即使是在拥有最先进方法的现代超级计算机上，也不可能在三天内完成。这个项目的重点是创建一类新的方法，使这些计算成为可能。开始解决具有大量时间尺度的模拟问题的一种方法是使用隐式求解器。这些通常会导致隐式耦合方程的大型系统。这类系统的可扩展求解器的主要瓶颈在于解决这些系统所需的通信和大量交互。在这个项目中，我们正在寻求解决非线性偏微分方程耦合系统的新范例，重要的是要注意算子的部分通常是线性的。我们已经开发了一种策略，将这些算子扩展为全局卷积，当与显式时间步进方法结合时，它具有无条件的稳定性，但可以用三项递归关系便宜地评估并避免迭代。这种方法导致了诸如退化平流扩散或汉密尔顿雅可比方程等问题的无条件稳定解。在这里，我们正在努力将这些方法扩展到pde系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Carrington event of 1859 was a massive solar flair that hit the earth and left large amounts of charged particles trapped in the earth's magnetic field. These charged particles created large currents that coupled to the telegraph wires via an inductive coupling, as what happens in transformers. The currents that were pushed through the telegraph wires were so large they caused the telegraph machines to explode. If this were to happen today, unless we know to turn off and disconnect the power grid, such an event would destroy modern power infrastructure. Yet solar flairs happen all the time, we can't simply turn off the power grid every time one happens. From the time of detection, it takes 3 days for a solar flair to travel the distance to the earth. Predicting if a solar flair will hit the earth is currently done with low resolution models, which are not particular good at predicting these events. Increasing the accuracy to include more physics in the calculation leads to bigger calculations that simply can't be done in three days, even on modern super computers with state of the art methods. This project centers on creating a new class of methods that could make these calculations possible. One way to begin to address the issues of simulating problems with a significant number of time scales is to use implicit solvers. These typically lead to large systems of implicitly coupled equations. The main bottleneck in scalable solvers for such systems is in the communication and large number of interactions that are needed to solve these systems. In this project we are pursuing a new paradigm that solves coupled systems of non-linear PDEs, it is important to note that the pieces of the operators are typically linear. We have developed a strategy of expanding these pieces of the operators as global convolutions that give unconditional stability when combined with explicit time stepping methods, but can be cheaply evaluated with three term recurrence relations and avoids iteration. This approach has led to unconditionally stable solvers for problems such as degenerate advection diffusion or the Hamilton Jacobi equations. Here we are working to extend these methods to systems of PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.