Mathematical model
Injection of individual dike leads to displacement of elastic host rocks, heat transfer, rock melting and magma solidification. We model individual dike as an ellipsoid with semi-axes a and b and use analytical solution \cite{Muskhelishvili_1977} in order to calculate host rock displacement. Volume of the individual dike and frequency of emplacement is controlled by the specified feeding rate of the magma Qin (km3/y). We assume that the emplacement occurs in 2D plain geometry and the third spacial dimension is specified and constant. This situation is possible in the extensional tectonic environment, where the local stress field leads to preferentially parallel dyke orientation. We allow random anglo of an individual dike emplacement or change in the dike orientation from vertical at depth to horizontal near the surface.
\(\begin{equation}
\label{eq:eq1}
\rho C\left( {\frac{{\partial T}}{{\partial t}} + \vec Vgrad\left( T \right)} \right) = div\left( {k\,grad\left( T \right)} \right) + \rho L\frac{{d\beta }}{{dt}}\\
\frac{{\partial \alpha }}{{\partial t}} + \vec Vgrad\left( \alpha \right) = 0\\
\rho C = {\rho _r}{C_r}\left( {1 - \alpha } \right) + {\rho _m}{C_m}\alpha \\
k = {k_r}\left( {1 - \alpha } \right) + {k_m}\alpha \\
{\beta _r} = {\beta _r}\left( T \right),\,{\beta _m} = {\beta _m}\left( T \right).
\end{equation}\)
Rock-magma temperature evolution \(T\) is governed by heat conduction equation (\ref{eqn:eq1}) that accounts (\ref{eqn:eq1}) for advection due to rock displacement, latent heat of crystallization and heat conduction. Melt fraction depends on temperature according to Annen et al 2006. Injection rate is 0.015 km3/y (0.5 m3/s), initial magma temperature is 950 °C, typical for an island arc system. In order to minimize numerical diffusion PIC/FLIP hybrid method which mixes the perspectives of solving the system from a particle point of view (Lagrangian) and solving the system from a grid point of view (Eulerian) is used. Particle displacement is calculated after each individual dike injection, while heat conduction is solved on a fixed grid.