Numerical method
Where should a scheme spend its dissipation?
Every shock-capturing scheme buys stability with artificial dissipation, and the usual accounting treats that as a global property to trade against accuracy. We think the interesting question is not how much a scheme has, but where it puts it.
FigureThe structure of a reacting wave: a leading discontinuity, an induction region, and the heat release that sustains it. A schematic, not a result.
Why it is hard
In a classical scheme the smearing and the stabilising upwinding are one mechanism, so wherever the scheme is stable it is also dissipative. For a steady shock that is a fair price. For a self-excited flow it is not: the acoustic feedback driving the instability travels through the smooth region, where the dissipation quietly damps the physics. The answer looks plausible and is wrong.
How we attack it
We work with flux constructions where the stabilising term is algebraically separable, so it can be switched rather than scaled — turning an accuracy–stability trade into a placement problem. Placement is decided by a locally evaluated measure of departure from equilibrium, computed inside the flux itself from values already in registers. No stencil-wide reduction, nothing to communicate.
What we have established
- Confining the stabilising term to the two or three faces straddling a discontinuity leaves the rest of the domain effectively free of it.
- Its support then shrinks with mesh spacing instead of staying a fixed fraction of the domain.
- Where the measure is evaluated matters more than which flux it is attached to — which overturned our own starting hypothesis.