Computational methods for propagating phase boundaries

This paper considers numerical methods for computing propagating phase boundaries in solids described by the physical model introduced by Abeyaratne and Knowles. The model under consideration consists of a set of conservation laws supplemented with a kinetic relation and a nucleation criterion. Discontinuities between two different phases are undercompressive crossing waves in the general terminology of nonstrictly hyperbolic systems of conservation laws. This paper studies numerical methods designed for the computation of such crossing waves. We propose a Godunov-type method combining front tracking with a capturing method; we also consider Glimm's random choice scheme. Both methods share the property that the phase boundaries are sharply computed in the sense that there are no numerical interior points for the description of a phase boundary. This properly is well known for the Glimm's scheme; on the other hand, our front tracking algorithm is designed so that it tracks phase boundaries but captures shock waves. Phase boundaries are sensitive to numerical dissipation effects, so the above property is essential to ensure convergence toward the correct entropy weak solution. Convergence of the Godnuov-type method is demonstrated numerically. Extensive numerical experiments show the practical interest of both approaches for computations of undercompressive crossing waves. (C) 1996 Academic Press, Inc.