Exponential Time Differencing for the Tracer Equations Appearing in Primitive Equation Ocean Models

TL;DR

This paper introduces an exponential time differencing (ETD) method for the tracer equations in ocean models, allowing larger time steps by treating vertical transport with matrix exponentials, improving computational efficiency especially with multiple tracers.

Contribution

The paper presents a novel ETD solver that combines matrix exponential treatment of vertical terms with explicit horizontal handling, enhancing efficiency over existing semi-implicit methods.

Findings

Significant speed-ups over semi-implicit methods.

Effective handling of multiple tracers.

Improved stability for larger time steps.

Abstract

The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several equations may be added to and coupled to the dynamics system. In many relevant situations, the time-step requirements of explicit methods imposed by the transport and mixing in the vertical direction are more restrictive than those for the horizontal, and this may cause the need to use very small time steps if a fully explicit method is employed. To overcome this issue, we propose an exponential time differencing (ETD) solver where the vertical terms (transport and diffusion) are treated with a matrix exponential, whereas the horizontal terms are dealt with in an explicit way. We investigate numerically the computational speed-ups that can be obtained over other semi-implicit methods, and we analyze the advantages of the method in the case of multiple tracers.