Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. However, using Lagrangian coordinates results in solving a highly nonlinear partial differential equation. The nonlinearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. For linear (spatial) flows we give an explicit formula for the Jacobian. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. Given the formula for the Jacobian, we describe the two situations where the Lagrangian shallow water equations are invalid in Lagrangian coordinates for long time integrations, because the shallow water assumption is violated. On the other hand, in situations where the shallow water assumption is not violated, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.
Mead, Jodi. (2004). "The Shallow Water Equations in Lagrangian Coordinates". Journal of Computational Physics, 200(2), 654-669. http://dx.doi.org/10.1016/j.jcp.2004.04.014