Advancing parabolic operators in thermodynamic MHD models II: Evaluating a Practical Time Step Limit for Unconditionally Stable Methods

TL;DR

This paper tackles the accuracy and stability challenges of unconditionally stable parabolic solvers in thermodynamic MHD models by introducing a practical time-step limit (PTL) that dynamically cycles the parabolic operators. The PTL bound is computed to preserve the sign of adjacent-cell differences and is evaluated at the location of $F_{max}$, with re-evaluation after each cycle to reduce overhead. Through two real-world MAS simulations, the authors show that applying the PTL improves solution quality for both the implicit BE+PCG and the RKG2 STS schemes, and that RKG2 with PTL can be competitive with BE+PCG in both accuracy and performance, especially at scale. The results indicate that PTL can enable more reliable and efficient use of unconditionally stable schemes for parabolic operators and enhance the practicality of STS methods in multi-physics MHD contexts.

Abstract

Unconditionally stable time stepping schemes are useful and often practically necessary for advancing parabolic operators in multi-scale systems. However, serious accuracy problems may emerge when taking time steps that far exceed the explicit stability limits. In our previous work, we compared the accuracy and performance of advancing parabolic operators in a thermodynamic MHD model using an implicit method and an explicit super time-stepping (STS) method. We found that while the STS method outperformed the implicit one with overall good results, it was not able to damp oscillatory behavior in the solution efficiently, hindering its practical use. In this follow-up work, we evaluate an easy-to-implement method for selecting a practical time step limit (PTL) for unconditionally stable schemes. This time step is used to `cycle' the operator-split thermal conduction and viscosity parabolic operators. We test the new time step with both an implicit and STS scheme for accuracy, performance, and scaling. We find that, for our test cases here, the PTL dramatically improves the STS solution, matching or improving the solution of the original implicit scheme, while retaining most of its performance and scaling advantages. The PTL shows promise to allow more accurate use of unconditionally stable schemes for parabolic operators and reliable use of STS methods.

Figures (6)

• Figure 1: Left: Estimated speedup of the RKL2 and RKG2(3/2) schemes compared to explicit Euler (approximating one Euler step to have the same computational cost as one STS iteration). Middle: Amplification factors for the BE, RKL2, and RKG2 schemes when applied to a 1D uniform grid heat equation discretized with a second-order central finite difference at a time-step 500 times larger than the explicit Euler limit. Right: Same as middle, but on a log scale to show the difference in the BE scheme's damping rate compared to the exact solution.
• Figure 2: Comparison run results for Test 1. A small section of the solution output of $v_{\phi}$ is shown in the $\theta-\phi$ plane (top) with the black line indicating the cut that is plotted in 1D (bottom). The PTL is shown to improve the quality of the solution when using either RKG2 and BE+PCG. The scheme used for each result is shown at the bottom of each column, where the text color corresponds to the performance result figures in the next section.
• Figure 3: Comparison run results for Test 2. A small section of the solution output of $j_{\phi}$ is shown in the $r-\theta$ plane (top) with the black line indicating the cut that is plotted in 1D (bottom). The PTL is shown to improve the quality of the solution when using both RKG2 and BE+PCG. Notation is the same as Figure \ref{['fig:test1sol']}.
• Figure 4: Timing results for the solution comparison runs in Sec. \ref{['sec:results_sol']} for Test 1 (left) and Test 2 (right).
• Figure 5: Test 2 scaling results for the thermal conduction (left two plots) and viscosity (right two plots) operators on CPUs (left) and GPUs (right).