A New Monotonicity-Preserving Numerical Method for Approximating Solutions to the Rayleigh-Benard Equations

Abstract:

In many modern computer models of convection in the Earth's mantle, second and higher order accurate finite element or finite difference methods are used to approximate solutions to an advection-diffusion equation for temperature. However, high-order accurate methods can introduce physically incorrect overshoot and undershoot of the computed solution at steep gradients. Values of the temperature that are physically too large or too small in turn can lead to significant inaccuracies in other quantities, such as the viscosity, which is exponentially proportional to temperature.

We have developed a numerical method for approximating solutions to the Rayleigh-Benard equations (i.e., the incompressible Stokes equations, an advection-diffusion equation for the temperature, and a Boussinesq approximation for the density) in which solutions to the advection-diffusion equation are approximated with a monotone finite difference method. This monotone finite difference method is based on a second-order accurate finite difference method for approximating solutions to the Navier-Stokes equations, originally developed by Bell, Colella, and Glaz, with appropriate modifications to ensure stability of a solution due to Minion. We present examples demonstrating that our new method preserves the monotonicity of the temperature field in problems for which widely used mantle convection codes do not.

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.