[petsc-dev] perhaps where the custom adapativity perversity began

Jed Brown jed at jedbrown.org
Sun Apr 30 15:34:25 CDT 2017

Lisandro Dalcin <dalcinl at gmail.com> writes:
> Could this be improved? Well, I could use a higher-order BDF formula
> keeping track of an extra past solution vector and estimate the LTE of
> a BDF-2 method, which would be of equivalent order (up to a constant,
> possibly large?) to the LTE of second-order methods like THETA and
> ALPHA. But I'm not sure we really want this, unless there is a
> reliable way to correlate the constants in front of LTEs for BDF-2 and
> THETA/ALPHA. Otherwise, we will be accepting steps without being sure
> the LTE is below the user-prescribed tolerance.

One reliable, if somewhat expensive, option would be to use
extrapolation similar to TSARKIMEX1BEE.  If this feels like reinventing
a wheel using stone tools, it is; we could just add a TSARKIMEX tableau
for Theta with extrapolation.

Since Theta is supposed to be an implementation that newcomers use as an
example to learn how TS works, we should try to keep it as clean/simple
as possible.  I think it would also be okay to have no error estimator
-- it seems likely that user implementations may not have an error
estimator (though maybe they should think about it up-front).
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