A nonstandard finite difference scheme for a strain-gradient theory

Abstract

Strain-gradient theories have proved very useful in describing computational aspects of phase transforming materials such as shape memory alloys. A significant feature implicit to these theories is a relation between the driving force acting on a phase boundary and its velocity. Numerical calculation of the kinetic relation using standard finite difference methods shows significant quantitative and qualitative departures from the analytical kinetic relation. In this paper we derive a nonstandard finite difference scheme and show that the kinetic relation evaluated using this scheme displays the correct qualitative behaviour and matches the analytical solution significantly better quantitatively. In particular the nonstandard finite difference scheme eliminates spurious lattice trapping in the kinetic relation. © Springer-Verlag 2004.