A Moving Mesh Method Based on the Geometric Conservation Law

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 concentration. The geometric conservation law, an identity satisfied by any nonsingular coordinate transformation, is an important tool which has been used for many years in the engineering community to develop cell-volume-preserving finite-volume schemes. It is used here to transform the algebraic expression specifying the Jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy. It is shown that the resulting method bears a close relation with the Lagrangian method. Advantages of the new approach include the ease of controlling the cell volumes (and therefore mesh adaption) and a theoretical guarantee for existence and nonsingularity of the coordinate transformation. It is shown that the method may suffer from the mesh skewness, a consequence resulting from its close relation with the Lagrangian method. Numerical results are presented to demonstrate various features of the new method.