# A robust layer-resolving numerical method for a free
convection problem

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0003.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**A robust layer-resolving numerical method for a free
convection problem**
We consider free convection near a
semi-infinite vertical flat plate. This problem is singularly
perturbed with perturbation parameter *Gr*, the Grashof number.
Our aim is to find numerical approximations of the solution in a
bounded domain, which does not include the leading edge of the
plate, for arbitrary values of *Gr*, = 1 or > 1.
Thus, we need to determine values of the velocity components
and temperature with errors that are *Gr*-independent. We use
the Blasius approach to reformulate the problem in terms of two
coupled non-linear ordinary differential equations on a semi--
infinite interval. A novel iterative numerical method for the
solution of the transformed problem is described and numerical
approximations are obtained for the Blasius solution functions,
their derivatives and the corresponding physical velocities and
temperature. The numerical method is *Gr*-uniform in the sense
that
error bounds of the form *Cp N-p*, where
*Cp* and *p*
are independent of the *Gr*, are valid for the interpolated
numerical solutions. The numerical approximations are therefore of
controllable accuracy.