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.

翻译：无条件稳定的时间推进方案在多尺度系统中推进抛物型算子时非常有用，且通常具有实际必要性。然而，当时间步长远超过显式稳定性极限时，可能会出现严重的精度问题。在先前的工作中，我们比较了使用隐式方法和显式超时间步进（STS）方法推进热力学MHD模型中抛物型算子的精度和性能。我们发现，虽然STS方法的整体效果优于隐式方法，但它无法有效抑制解中的振荡行为，从而限制了其实际应用。在本后续工作中，我们评估了一种易于实现的方法，用于为无条件稳定方案选择实用时间步长限制（PTL）。该时间步长用于"循环"操作分裂的热传导和粘性抛物型算子。我们使用隐式和STS方案测试了新的时间步长，评估其精度、性能和扩展性。结果表明，在我们的测试案例中，PTL显著改善了STS解，匹配甚至优于原始隐式方案的解，同时保留了其大部分性能和扩展优势。PTL有望使抛物型算子的无条件稳定方案得到更精确的使用，并实现STS方法的可靠应用。