  Lagrangian averaging plays an important role in the analysis of
wave--mean-flow interactions and other multiscale fluid phenomena. The
numerical computation of Lagrangian means, e.g. from simulation data, is
however challenging. Typical implementations require tracking a large number of
particles to construct Lagrangian time series which are then averaged using a
low-pass filter. This has drawbacks that include large memory demands, particle
clustering and complications of parallelisation. We develop a novel approach in
which the Lagrangian means of various fields (including particle positions) are
computed by solving partial differential equations (PDEs) that are integrated
over successive averaging time intervals. We propose two strategies,
distinguished by their spatial independent variables. The first, which
generalises the algorithm of Kafiabad (2022, J. Fluid Mech. 940, A2), uses
end-of-interval particle positions; the second directly uses the Lagrangian
mean positions. The PDEs can be discretised in a variety of ways, e.g. using
the same discretisation as that employed for the governing dynamical equations,
and solved on-the-fly to minimise the memory footprint. We illustrate the new
approach with a pseudospectral implementation for the rotating shallow-water
model. Two applications to flows that combine vortical turbulence and Poincare
waves demonstrate the superiority of Lagrangian averaging over Eulerian
averaging for wave--vortex separation.
