Computer Simulations and Space Weather

In the 1990s, the use of computer simulations as a virtual environment to model complex physical systems was gaining momentum, driven to a large extent by increase in computational power.  A common simulation technique consists of dividing the simulation domain into a computational grid, initializing the system and then updating the state of the system over time.

At the time, I was looking into computer simulation techniques with an eye towards applications in plasma physics, such as fusion, space physics such as space weather, among others. A common characteristic among these applications is that different parts of the system evolve at different rates in time. An ideal algorithm would intelligently adapt the time step at each computational grid based on local conditions to achieve a desired accuracy. As it turns out, this is a very challenging task for the algorithms. The standard techniques, generally called time-stepped based, faithfully update the system at equal time steps, leading to great inefficiency, and putting realistic modeling of many systems out of reach.

Imagine simulating car traffic on the freeway. It would be quite wasteful to update the simulation as often in the bumper-to-bumper traffic part of the freeway, where there is slow change over time, as in the part where cars are traveling at full speed.  The common approach to address this problem has been to create patches in the simulation with finer spatial grids and then update the patches at smaller time steps. This leads to well-known numerical issues but that is the best that was available. And it is even more problematical to have such algorithms adaptively move the patches in time (e.g., have the fine resolution patch follow the traffic jams in time).

Having reached the conclusion that, the wide spread use of time-stepped approach is fundamentally flawed for simulating temporally in-homogeneous systems, I started to look into alternatives. In the process, I came across a technique, called discrete event simulation, which was being used in applications where the evolution of system occurs in distinct events and the system is assumed to have no change in between the events. Applications include video games, battle field simulation, traffic flow modeling, among others. For example, in the battle field simulation, the event of interest may be whether the tank hits its target and all other aspects such as the detailed motion tracking of the tank could be ignored. 

This technique, which is action-based, is completely different than the time-stepped methodology where the dynamics of the system is continuously tracked over time. The question was whether it was feasible to adapt the discrete event methodology for simulation of problems that were traditionally addressed by time-stepped approaches. 

It took about a year before we had our proof of concept, which was published in 2005 (Karimabadi et al.,  2005). As it often happens in science, new breakthroughs and fundamentally different approaches, encounter a certain degree of resistance from the scientific community.  But eventually with more proof points the resistance dissipates and turns into acceptance. I distinctly remember that in my early talks on the subject, most of the audience had a hard time wrapping their mind around the concept. This was not surprising and spoke to the novelty of the approach. We have since developed the technique further (e.g., Omelchenko and Karimabadi, 2006, 2007, 2012a, 2012b). It has found applications in other domains such as modeling of wildfires, oil reservoirs,  computational fluid dynamics, plasma discharges, among others. The algorithm provides significant improvements in speed, and accuracy compared to standard techniques. It also exhibits better stability properties. Even today, it remains the only algorithm of its type where it self-adaptively changes the time step on each computational grid.

The power of this technique is illustrated through two movies. The movies are from a simulation of solar wind interaction with the Earth’s magnetic field. The movies show the time step distribution for field (Bottom Video) and particle (TopVideo) updates, respectively. The time step is normalized to that from a standard time-stepped algorithm. Note how the algorithm adjusts the time step dynamically and in regions of low activity the time step is two orders of magnitude larger than the corresponding time-stepped algorithm.  The algorithm figures out at each point in the simulation what the proper temporal resolution required should be and updates the simulation accordingly. 

Omelchenko Y.A. and H. Karimabadi, Spontaneous generation of a sheared plasma rotation in a field-reversed q-pinch discharge, Phys. Rev. Lett. 109, 065004, 2012a.

Omelchenko, Y.A. and H. Karimabadi, HYPERS: A Unidimensional Asynchronous Framework for Multiscale Hybrid Simulations, J. Comp. Phys. 231,1766-1780, 2012b.

Omelchenko, Y.A. and H. Karimabadi, A Time-Accurate Explicit Multiscale Technique for Gas Dynamics, J. Comp. Phys. 226 (1): 282-300, 2007.

Omelchenko Y.A. and H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comp. Phys. 216, 179-194, 2006.

Karimabadi H., J. Driscoll, Y.A. Omelchenko, and N. Omidi, A new asynchronous methodology of modeling of physical systems, J. Comp. Phys. 205( 2), 755-775, 2005.