# Moving mesh methods for problems in meteorology

• Emily Jane Walsh

Student thesis: Doctoral ThesisPhD

### Abstract

This thesis considers the development and implementation of a moving mesh strategy which is suitable for the numerical solution of partial differential equations (PDEs) that arise in problems relevant to meteorology. We concentrate primarily on developing the Parabolic Monge-Ampère (PMA) moving mesh method. This is an r-adaptive method which is based on ideas from optimal transportation combined with the equidistribution principle applied to a (time varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method is straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension. Furthermore it produces meshes of provable regularity and smoothness. The mesh equation is augmented with suitable Neumann or periodic boundary conditions, with adaptivity along the boundaries handled automatically. Examples are presented of periodic and non-periodic meshes generated for a prescribed monitor function. The PMA mesh equation is then successfully coupled to a number of convection dominated PDEs in 1D and 2D and the relative merits of solving the resultant systems, using a simultaneous or an alternate solution procedure, are explored. The main test problem considered is the two-dimensional Eady problem, a meteorological problem which models the development of cyclones at mid-latitudes. Numerical solutions obtained on an adaptive grid using PMA are presented. The results show improved resolution of the front when compared to uniform grid solutions with an equivalent number of mesh points and computed with the same time step. A pressure-correction method is implemented on a semi-staggered adaptive grid that also conserves important physical properties of the solution. All numerical solutions presented involve discretising the underlying equations in the computational domain, which is fixed and uniform, using a finite difference scheme. An alternating strategy is implemented whereby the Eady equations are integrated first and then the mesh is updated. A conservative interpolation scheme is used to interpolate the updated solution from the old grid onto the new grid.
Date of Award 18 May 2011 English University of Bath Chris Budd (Supervisor)

'