Exponential integrators for problems in magnetohydrodynamics

The present project considers the application of exponential integrators to problems in magnetohydrodynamics (MHD). The equations of MHD model the interaction of a conducting fluid/gas with electric and magnetic fields. Consequently, these equations are widely used in applications ranging from astrophysics to industrial cooling. Three dimensional simulations of the MHD equations can, depending on the problem, take days or even weeks on modern supercomputers. Consequently, developing efficient numerical algorithms for such problems is of paramount importance. In recent years, exponential integrators have been identified as a promising alternative to more traditional numerical methods. Exponential integrators are appealing because they solve the linear part of the problem exactly and are thus able to obtain very accurate results. However, since they require the evaluation of certain matrix functions these methods can be quite expensive. In this project a number of improvements will be made that further increase the utility of these types of methods for conducting numerical simulations. In particular, the developed algorithms will help to significantly reduce the run time of MHD simulations. In addition, we will study the convergence of these methods from a mathematical point of view.

Computer simulations are critical tools in many scientific and engineering fields. They allow researchers and engineers to model complex systems and predict their behavior. This is essential for understanding phenomena that are too difficult, expensive, or dangerous to study through physical experiments. A promising method for performing such simulations are exponential integrators. These methods are particularly efficient for stiff systems, e.g. problems where disparate physical time scales must be resolved. Central to exponential integrators is the efficient computation of matrix functions. Compared to more traditional Krylov subspace methods, Leja interpolation is an alternative that has many advantages, in particular, on modern high-performance computing systems. Despite their potential, exponential integrators based on Leja interpolation face several challenges that have limited their widespread adoption. This project has successfully addressed these issues: - In many applications, working in a matrix-free context is essential for increased efficiency and reduced memory consumption. However, this requires an estimate of the spectrum. We have shown that a relatively small number of power iterations can provide the required estimate at little additional computational cost. - Automatic time stepping is crucial for user-friendly software. The tacit assumption in step size controllers is that taking large time step sizes is favorable. However, for exponential integrators, which can take significantly larger time step sizes than implicit or explicit methods, this is often not true. We have developed a method that adjusts the time step size dynamically during the simulation to reduce computational cost. - Problem specific information can often be used to speed up the solution of differential equations (e.g. preconditioners are widely used for implicit methods). There was only limited research of doing this for exponential integrators. We have developed an approach that can be used to increase the computational performance of exponential integrators when an efficient solution of certain matrix functions (e.g. the matrix exponential) is available for a related, but simplified, problem. We have also analyzed the stability of exponential integrators and found very surprising results for advection dominated problems. This will most likely have a significant influence on which methods are used for such problems and how more efficient methods are constructed in the future. To make all of these advancements accessible, we have developed a software package, LeXInt (https://github.com/Pranab-JD/LeXInt), which provides a Python interface for ease of use. The package leverages modern high-performance computing technologies, such as graphics processing units (GPUs), to maximize efficiency. Our numerical methods have been successfully applied to a variety of complex problems. For example, magnetohydrodynamic simulations in plasma physics, cosmic ray transport in astrophysics, and solving advection-diffusion-reaction equations, which e.g. can be used to study pollution or pattern formation in biological systems.