Optimizing Advection of Tracer Particles in Fluid Dynamic Processes with Variable Numerical Integration Methods

Mantle convection simulations and other fluid dynamic processes often use tracer particles to characterize the flow behavior of different materials. A single high-accuracy method with an adaptive stepping scheme is typically chosen to ensure the reliability of results. The strong heterogeneity of flow fields, such as that found in mantle simulations, provides the possibility of improving performance by using a cheap, low-accuracy method in vector fields that are relatively linear without incurring the expected large error. In this work we propose a hybrid particle integration method applicable to mantle convection or other fluid dynamics simulations. This method combines low and high accuracy methods depending on the local characteristics of the flow field. Our hybrid method allows for faster computation, while retaining low total simulation error. The optimal combination of methods is based on a measure of flow curvature in a given vector field, such as curl or strain rate. Previous research demonstrated using only the average angle between two vectors was sufficient, but further improvements are made with more sophisticated hybrid scheme. Tracer particles experiencing small flow curvature may be advanced with Euler's method of integration, while large flow curvature would require a more sophisticated scheme such as second or fourth order Runge-Kutta. The effectiveness of this hybrid numerical integration scheme was implemented and tested in the mantle convection simulation code ASPECT. We find a notable increase in performance speed while maintaining user-accepted levels of accuracy.