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

1859年的卡林顿事件是一次巨大的太阳耀斑，击中了地球，留下了大量被困在地球磁场中的带电粒子。这些带电粒子产生大电流，通过电感耦合耦合到电报导线上，就像变压器中发生的那样。通过电报线的电流如此之大，以至于电报机都爆炸了。如果今天发生这种情况，除非我们知道关闭和断开电网，否则这样的事件将摧毁现代电力基础设施。然而，太阳耀斑总是在发生，我们不能每次发生太阳耀斑就简单地关闭电网。从探测到的时间开始，太阳光芒需要3天的时间才能到达地球。预测太阳耀斑是否会撞击地球目前是用低分辨率模型完成的，这些模型在预测这些事件方面并不是特别好。提高计算的精确度以在计算中包含更多的物理信息会导致更大的计算，而这些计算根本不可能在三天内完成，即使是在使用最先进方法的现代超级计算机上也是如此。这个项目的中心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。这些通常导致隐式耦合方程的大系统。这类系统的可伸缩解算器的主要瓶颈是解决这些系统所需的通信和大量交互。在这个项目中，我们正在追求一种新的范例来求解非线性偏微分方程组的耦合系统，重要的是要注意到算子的片段通常是线性的。我们开发了一种策略，将这些算子扩展为全局卷积，当与显式时间推进方法相结合时，这些卷积具有无条件稳定性，但可以用三项递推关系来廉价地计算，并且避免了迭代。这种方法已经为退化对流扩散或哈密尔顿-雅可比方程等问题带来了无条件稳定的解。在这里，我们正在努力将这些方法扩展到PDE系统。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.