Complex Boundary Integral Equation Formulation and Stability Analysis of a Maxwell Model and of an Elastic Model of Solid-Solid Phase Transformation

We study a viscoelastic model of the solid-solid phase change of olivine to its denser $beta$-spinel state at high pressures and temperatures reachable in laboratory experiments matching conditions typical of Earth's mantle. Using a previously unknown technique, the equations are transformed to the problem of finding two complex analytic functions in the sample satisfying certain conditions on the outer boundary. The Sherman-Lauricella boundary integral equation is used in a numerical algorithm that eliminates the bottleneck of having to solve a large matrix equation at every timestep. The method is implemented and used to compute the solution of a number of non-axisymmetric test problems, some static and some dynamic in time. Next we develop an alternative formulation in which the Lam'e equations of linearized elasticity are used to model the deformation of the two phases, and we allow for compressibility. The formulation is novel in that separate reference configurations are maintained for the core and shell regions of the sample that grow or shrink in time by accretion or removal at the boundary, one at the expense of the other. We then compare the behavior of the evolution of this system to the incompressible viscoelastic case and to an alternative elastic model. Finally, we study the stability of circular interfaces with axisymmetric initial data under the evolution equations. For various parameter values of the circular interface evolution, we find families of small perturbations of the circular interface and radial interface velocity jump that either grow or decay exponentially in time. In unstable cases, the growth rate increases without bound as the wave number of the perturbation increases. In stable cases, the evolution equations are well-posed until the interface leaves the stability regime, at which point the numerical solutions blow up in an oscillatory manner. Examples of stable and unstable behavior are presented.