Numerical approach of thermomechanical coupled problems with moving boundaries in theoretical glaciology

In this work we propose a new numerical method for solving a thermomechanical coupled problem which arises as a mathematical model for the evolution of ice sheets profiles and the corresponding temperatures. From the accumulation-ablation ratio, the atmospheric temperature and the geothermic flux, the ice sheet profile is the solution of a moving boundary problem governed by a nonlinear convection-diffusion equation. Moreover, the mathematical model which governs the temperature distribution involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two-phase Stefan formulation of the poly-thermal regime. The problems are discretized in time with upwind characteristics schemes and in space with finite elements. The nonlinearities are solved either by fixed point methods or by duality techniques. Several numerical simulation examples involving real data sets issued from the Antartic ice sheet are shown.