  We describe the application of the phase averaging technique to the nonlinear
rotating shallow water equations on the sphere, discretised using compatible
finite element methods. Phase averaging consists of averaging the nonlinearity
over phase shifts in the exponential of the linear wave operator. Phase
averaging is a form of heterogeneous multiscale method that aims to capture the
slow dynamics in a solutionthat is smoother in time (in transformed variables)
so that larger timesteps may be taken. Following Peddle et al (2019), we
consider finite width phase averaging windows, since the equations have a
finite timescale separation. In a numerical implementation, the averaging
integral is replaced by a Riemann sum, where each term can be evaluated in
parallel. This creates an opportunity for parallelism in the timestepping
method, which we use here to compute our solutions. Here, we focus on the
stability and accuracy of thenumerical solution and an evaluation of the
parallel aspects will follow in later work.
