This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergence-free for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. L...

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge-Ampere method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis f...

In this paper we introduce a moving mesh method for solving PDEs in two dimensions. It can be viewed as a higher dimensional generalization of the moving mesh PDE (MMPDE) strategy developed in our previous work 24] for one dimensional problems. The MMPDE is derived from a heat ow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps. Geometr...

In this paper we introduce a moving mesh method for solving PDEs in two dimensions. It can be viewed as a higher-dimensional generalization of the moving mesh PDE (MMPDE) strategy developed in our previous work for one-dimensional problems [W. Huang, Y. Ren, and R. D. Russell, SIAM J. Numer. Anal., 31 (1994), pp. 709–730]. The MMPDE is derived from a gradient flow equation which arises using a ...

We present a coupled moving mesh and level set method for computing incom5 pressible two-phase flow with surface tension. This work extends a recent work 6 of Di et al. [(2005). SIAM J. Sci. Comput. 26, 1036–1056] where a moving 7 mesh strategy was proposed to solve the incompressible Navier–Stokes equa8 tions. With the involvement of the level set function and the curvature of the 9 interface,...

Adaptive, unstructured grid methods, in which the mesh is allowed to deform, and grid quality subsequently restored through localized coarsening and refinement, offer the potential of a more rapid, straightforward approach to generalized moving body problems. Automation of this mesh movement and quality correction strategy requires a close coupling with the flow solution process. With parallel ...

The non-equilibrium Richards equation is solved using a moving finite element method in this paper. The governing equation is discretized spatially with a standard finite element method, and temporally with second-order Runge-Kutta schemes. A strategy of the mesh movement is based on the work by Li et al. [R. Li, T. Tang and P. W. Zhang, A moving mesh finite element algorithm for singular probl...

This paper is devoted to the applications of a class of moving mesh nite element methods based on harmonic maps. We review some recent work of the authors on solving PDES, variational inequalities and optimal control problems by use of the moving mesh techniques.