The Use of Self Adaptive Techniques in
Determining Blow-up Parameters in Solutions.
G. Conner, C. Grant, and S. McKay
Brigham Young University
The solutions to some partial differential equations "blow-up in finite time".
This means that the solution or its derivative take on an infinite value at
some point in time. In certain cases, the blow-up time can be estimated
analytically but these estimates may be quite rough. Standard finite
difference methods can be used to increase the accuracy of these estimates
numerically. As the solution approaches the blow-up time, more accuracy is
needed in the critical region due to the larger gradients of the solution.
This leads to expensive solutions which require refinement of the grid. Since
some solutions behave nicely away from the critical region, irregular grids
may provide a more efficient solution while retaining the accuracy of the
blow-up time and behavior of the solution near blow-up (which are known as
"blow-up parameters").
The fast adaptiv ecomp osite grid method (FAC) is a multigrid-like algorithm
which achieves fast solutions of various boundary value problems by combining
adaptive grid techniques with multi-level solutions. The FAC method can also
be combined with the spectral method (used as a local solver) to yield better
accuracy.
A rescaling algorithm for self-similar blow-up solutions has been proposed by
Berger and Kohn (see [1]). The goal of the current paper is to show how FAC
type algorithms can provide information about blow-up parameters in an
efficient manner which does not rely on the self-similarity of the solution.
References
[1] M. Berger and R. Kohn, A rescaling algorithm for the Numerical Calculation
of Blowing-up Solutions, Comm. Pure and Appl. Math., (XLI) 841-863 (1988).