DuFort–Frankel averaging is a tactic to stabilize Richardson’s unstable three-level leapfrog timestepping scheme. By including the next time level in the right-hand-side evaluation, it is implicit, but it can be rearranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection–diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Finally, the rabies problem is revisited and a stable method is found for a restricted range of parameter values, although no prescriptive recipe is known which selects this particular choice.

中文翻译：

---

Dufort-FRANKEL 有限差分的慢燃不稳定性

DuFort-Frankel 平均是一种稳定 Richardson 不稳定的三级跳跃式时间步长方案的策略。通过在右侧评估中包含下一个时间级别，它是隐含的，但可以重新排列以给出明确的更新公式，从而显然是两全其美。教科书证明了热方程的无条件稳定性，并且对各种对流-扩散方程的广泛使用产生了许多有用的结果。尽管如此，对于某些问题，该方案可能会以一种有趣且令人惊讶的方式失败，从而在很长一段时间内导致不稳定。对涉及描述狂犬病流行传播的一对进化方程的简单问题的分析可以深入了解这种情况是如何发生的。一个更简单的修正扩散方程也有同样的不稳定性。最后，