Abstract
We explore the performance and advantages/disadvantages of using unconditionally stable explicit super time-stepping (STS) algorithms versus implicit schemes with Krylov solvers for integrating parabolic operators in thermodynamic MHD models of the solar corona. Specifically, we compare the second-order Runge-Kutta Legendre (RKL2) STS method with the implicit backward Euler scheme computed using the preconditioned conjugate gradient (PCG) solver with both a point-Jacobi and a non-overlapping domain decomposition ILU0 preconditioner. The algorithms are used to integrate anisotropic Spitzer thermal conduction and artificial kinematic viscosity at time-steps much larger than classic explicit stability criteria allow. A key component of the comparison is the use of an established MHD model (MAS) to compute a real-world simulation on a large HPC cluster. Special attention is placed on the parallel scaling of the algorithms. It is shown that, for a specific problem and model, the RKL2 method is comparable or surpasses the implicit method with PCG solvers in performance and scaling, but suffers from some accuracy limitations. These limitations, and the applicability of RKL methods are briefly discussed.
Highlights
Complex physical systems often have processes that act on widely different time-scales
In addition to recording the total time for each portion of the run tested, we record the ‘non-compute’ time for each portion which we define as the time the algorithms are involved in inter-processor communication
Through testing using MPI barriers, we have found that it is not the AllReduce MPI routine itself in the dot products that is the main cause of the preconditioned conjugate gradient (PCG)’s lack of scaling, but rather it is the routine’s global synchronization acting as a barrier, causing all imbalances encountered before the routine to ‘pile up’
Summary
Complex physical systems often have processes that act on widely different time-scales. When implementing the models with numerical methods, accurately capturing the dynamics of these wide-ranging time-scales can make the simulations computationally intractable due to the fine time-resolutions required. In order to make the simulations computationally feasible, they need to be integrated at time-steps which exceed the fastest time-scales. This is possible because often, a system can be integrated without accurately capturing the detailed evolution of the faster time-scales and still obtain useful solutions. Integrating at large time-steps can violate the explicit numerical stability requirements, and if too large, can cause the model to loose convergence and become very inaccurate. Accuracy considerations must be taken into account when integrating past the fast time-scales, and numerical methods that can overcome the explicit limits are required. The operators are first isolated from the rest of the terms in their equations of motion through operator-splitting [2] and are written as
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