A Quantitative Look at the Overshoot Problem in Mantle Convection Simulation

In numerical models of mantle convection, overshoots of the computed temperature or compositional field will arise, for example, in a model of a rising plume or sinking subducted slab. The overshoot or undershoot will occur at the steep gradient of a scalar quantity such as temperature, and occurs in models in which advection dominates over diffusion. The overshoot and undershoot are caused when a second-order or higher method is used to model advection, causing an overestimation of fluxes along a steep gradient compared to lower-order method. Several methods have been developed for countering this effect, including streamline-upwind Petrov-Galerkin (SUPG), filtering techniques, and artificially increased diffusivity. As geodynamics research moves towards using numerical modeling to address multiscale physics, a small overshoot or undershoot can significantly alter quantities of interest including viscosity (which is exponentially temperature dependent) or the predicted location and quantity of melting. In our analysis, we find that the overshoot causes physically impossible oscillations in the calculated temperature solution, as well as incorrect plume formation and abnormal deformations in the pressure and velocity fields. We carry out a suite of models to characterize the overshoot-undershoot behavior of a finite element code in order to investigate the response to various methods for reducing this phenomenon.