# A Moving Mesh Method Based on the Geometric Conservation Law

@article{Cao2002AMM, title={A Moving Mesh Method Based on the Geometric Conservation Law}, author={Weiming Cao and Weizhang Huang and Robert D. Russell}, journal={SIAM J. Sci. Comput.}, year={2002}, volume={24}, pages={118-142} }

A new adaptive mesh movement strategy is presented, which, unlike many existing moving mesh methods, targets the mesh velocities rather than the mesh coordinates. The mesh velocities are determined in a least squares framework by using the geometric conservation law, specifying a form for the Jacobian determinant of the coordinate transformation defining the mesh, and employing a curl condition. By relating the Jacobian to a monitor function, one is able to directly control the mesh…

## 89 Citations

A Note on Geometric Conservation Law

- Computer Science
- 2012

This work treats the geometric conservation law (GCL) as a special case of the transport formula in fluid dynamics and derives a particularly effective method for moving mesh generation - the deformation method from the same transport formula.

MOVING MESH GENERATION USING THE PARABOLIC MONGE–AMPÈRE EQUATION∗

- Computer Science, Mathematics
- 2008

This method gives a new technique for performing r-adaptivity 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).

Moving Mesh Generation Using the Parabolic Monge--Amp[e-grave]re Equation

- Computer ScienceSIAM J. Sci. Comput.
- 2009

This method gives a new technique for performing $r-adaptivity 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).

An investigation of conservative moving-mesh methods for conservation laws

- Computer Science, Environmental Science
- 2016

The notion of the reference space is used as a mathematical tool to analyse the moving mesh methods allowing us to show the accuracy, stability conditions and convergence.

Interpreting moving mesh methods and transformation techniques from various applications in terms of the mass transport problem

- Computer Science
- 2003

The link between moving mesh theory and fluid dynamics, meteorology, computational anatomy and image registration is investigated by establishing a relationship between the coordinate transformations used in their formulations.

A finite difference moving mesh method based on conservation for moving boundary problems

- Computer ScienceJ. Comput. Appl. Math.
- 2015

A FEM-ALE APPROACH FOR CONTINUOUS DOMAINS WITH ARBITRARY MOVING BOUNDARIES

- Computer Science
- 2004

To determine mesh velocity using the nonstationary Laplace equation, the arbitrary Lagrangian-Eulerian (ALE) formulation is used to transform the integral master balance law into an equivalent PDE and mesh velocity is introduced as additional degree of freedom.

Adaptive grid based on geometric conservation law level set method for time dependent PDE

- Computer Science
- 2009

A new method for mesh generation is formulated based on the level set functions, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values…

A study on moving mesh finite element solution of the porous medium equation

- Computer ScienceJ. Comput. Phys.
- 2017

Numerical Modelling of Two-Phase Flow with Moving Boundary Fitted Meshes

- Computer Science, Physics
- 2018

The presented approach is shown to provide an accurate description of different two-phase flow phenomena, including phase change, and to handle cases with large material property ratios.

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