Characteristics-based marker-in-cell method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties

We have designed a 2D thermal-mechanical code, incorporating both a characteristics based marker-in-cell method and conservative finite-difference (FD) schemes. In this paper we will give a detailed description of this code. The temperature equation is advanced in time with the Lagrangian marker techniques based on the method of characteristics and the temperature solution is interpolated back to an Eulerian grid configuration at each timestep. This marker approach allows for the accurate portrayal of very fine thermal structures. For attaining a high relative accuracy in the solution of the matrix equations associated with both the momentum and temperature equations, we have employed the direct matrix inversion technique, which becomes feasible with the advent of very large shared-memory machines. Our conservative finite-difference schemes allow us to capture sharp variations of the stresses and thermal gradients in problems with a strongly variable viscosity and thermal conductivity. We have tested this code with numerous examples drawn from Rayleigh-Taylor instabilities, the descent of a stiff object into a medium with a lower viscosity, viscous heating and flows with non-Netwonian rheology. We have also benchmarked successfully with variable viscosity convection for lateral viscosity contrast up to 10⁸. We have delineated the regions in thermal problems where the diffusive nature of the temperature equation changes from its parabolic character locally to a non-linear hyperbolic-like equation due to the presence of variable thermal conductivity. Finally we discuss the applicability of this marker-based and finite-difference technique to other evolutionary equations in geophysics.