Exponential time differencing for mimetic multilayer ocean models

Abstract

A framework for exponential time discretization of the multilayer rotating shallow water equations is developed in combination with a mimetic discretization in space. The method is based on a combination of existing exponential time differencing (ETD) methods and a careful choice of approximate Jacobians. The discrete Hamiltonian structure and conservation properties of the model are taken into account, in order to ensure stability of the method for large time steps and simulation horizons. In the case of many layers, further efficiency can be gained by a layer reduction which is based on the vertical structure of fast and slow modes. Numerical experiments on the example of a mid-latitude regional ocean model confirm long term stability for time steps increased by an order of magnitude over the explicit CFL, while maintaining accuracy for key statistical quantities.

Localized Exponential Time DifferencingMethod for Shallow Water Equations: Algorithms and Numerical Study

“…The multilayer shallow water model describes the motion of a stack of fluids with distinct densities [36][37][38]. It is able to provide more accurate vertical profiles of the fluid velocity and depth, compared with the single-layer model.…”

“…In this work, we focus on efficient time-stepping methods for the nonlinear shallow water equations discretized in space by the TRiSK scheme; specifically, we propose a localized ETD method that combines the ETD method with the overlapping domain decomposition. The ETD methods have been increasingly used as the time integration scheme for the shallow water equations on the sphere of earth [15,19,34,37]. Among them, in [37], different ETD methods have been proposed and analyzed for the multilayer rotating shallow water equations with TRiSK discretization.…”

“…The ETD methods have been increasingly used as the time integration scheme for the shallow water equations on the sphere of earth [15,19,34,37]. Among them, in [37], different ETD methods have been proposed and analyzed for the multilayer rotating shallow water equations with TRiSK discretization. These schemes are efficient in the sense that considerably larger time steps over an explicit integrator can be taken while stability and sufficient accuracy are still maintained; moreover, numerical results show that significant cost reductions are achieved with the ETD method over an explicit time discretization scheme (RK4).…”

Exponential time differencing for the tracer equations appearing in primitive equation ocean models

“…This section describes exponential time differencing methods, that later will be used to solve the tracer equation. ETD methods have already been employed to solve the single layer ( [21,22,23]) and multi-layer ( [8]) shallow water equations.…”

“…More recently, exponential time differencing (ETD) methods, also known as exponential integrators, have 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 [8], an ETD scheme has been developed for the rotating shallow water equations with multiple horizontal layers, which correspond to a vertical discretization of the primitive equations dynamics in an isopycnal vertical coordinate system. 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.…”