  We present a collection of algorithms which utilize dimensional reduction to
perform mesh refinement and study possibly singular solutions of time-dependent
partial differential equations. The algorithms are inspired by constructions
used in statistical mechanics to evaluate the properties of a system near a
critical point. The first algorithm allows the accurate determination of the
time of occurrence of a possible singularity. The second algorithm is an
adaptive mesh refinement scheme which can be used to approach efficiently the
possible singularity. Finally, the third algorithm uses the second algorithm
until the available resolution is exhausted (as we approach the possible
singularity) and then switches to a dimensionally reduced model which, when
accurate, can follow faithfully the solution beyond the time of occurrence of
the purported singularity. An accurate dimensionally reduced model should
dissipate energy at the right rate. We construct two variants of each
algorithm. The first variant assumes that we have actual knowledge of the
reduced model. The second variant assumes that we know the form of the reduced
model, i.e. the terms appearing in the reduced model, but not necessarily their
coefficients. In this case, we also provide a way of determining the
coefficients. We present numerical results for the Burgers equation with zero
and nonzero viscosity to illustrate the use of the algorithms.
