Numerical analyses of ETD schemes for the solution of ocean and atmospheric models

Sara Calandrini

Department of Scientific Computing, Florida State University

Thursday, Mar 12, 2020, 11:00 am
DSRC Room 1D403

Abstract

Exponential time differencing (ETD) methods, also known as exponential integrators, constitute a class of numerical methods for the time integration of stiff systems of differential equations. The main idea behind exponential integrators is a splitting of the right-hand side term of an equation into a linear part and a remainder, i.e.

∂tθ = F (θ) = Aθ + R(θ),

with an appropriate choice of the linear operator A. Exponential integrators have recently gained attention in the ocean modeling community due to their stability properties that allow time steps considerably larger than those dictated by the CFL condition. In this talk, I will present ETD methods for ocean and atmosphere models.

First, I will present an ETD scheme for the tracer equations appearing in the primitive equa- tion ocean models, where the vertical terms (transport and diffusion) are treated with a matrix exponential, whereas the horizontal terms are dealt with in an explicit way. By treating expo- nentially terms related to fast time-scales, bigger time steps can be taken, and so computational speed-ups can be obtained over explicit methods. Compared to semi-implicit methods, higher accuracy is expected due to an exact treatment of the fast scales. I will investigate numeri- cally the computational speed-ups that can be obtained over other semi-implicit methods, and analyze the advantages of the method in the case of multiple tracers.

Second, I will discuss ETD methods for atmosphere modeling that are capable of handling larger time steps than HEVI ( horizontally explicit, vertically implicit) schemes, and, for a similar accuracy, are less computationally expensive. I will present two different types of exponential integrators, one being a three-stage method with the addition of artificial diffusion, and the second based on a Strang splitting approach, where the implicit step is performed with an exponential integrator.