A High-Order Monotonicity-Preserving Algorithm for Modeling Scalar Advection in the Earth's Mantle

Abstract:

Modern computer models of convection in the Earth's mantle are typically based on high-order accurate finite element or finite difference methods for approximating solutions to the Rayleigh-Benard equations; i.e., the incompressible Stokes equations coupled to an advection-diffusion equation for the temperature together with a Boussinesq approximation to the density as a function of the temperature. Many problems of interest involve the advection of some scalar quantity (e.g., bulk composition, trace elements, melt fraction, water content, grain size). However, it is well-known that in the absence of limiters or some other numerical technique, high-order accurate advection methods will introduce a physically incorrect 'overshoot' and 'undershoot' of the advected quantity in regions where the computed solution has steep gradients. We have developed a finite difference method for approximating solutions to the Rayleigh-Benard equations coupled to a scalar advection equation in which the solution to this advection equation are approximated with a monotone finite difference method. This method is based on second-order accurate finite difference methods with slope limiters that are a proven and mature methodology for modeling solutions of the equations of gas dynamics, advection in meterological flows, etc. We test our method by computing the classic falling block benchmark originally introduced by Gerya and Yuen in 2003 in which the viscosity is the scalar quantity which jumps by up to six orders of magnitude between the block and the background flow.

 The long-term goal of this project is to implement monotonicity preserving numerical methods in modern codes for modeling convection in the Earth's mantle.