points (such as corners), we can set the velocity to be zero. For those on a boundary edge or surface, the mesh velocities should be modified such that they do not move out of the domain.

The mesh equation (\[mmpde-2\]) (with proper modifications for boundary mesh points) is integrated from $t=t_n$ to $t_{n+1}$ starting with $\cTco$ as the initial mesh. In our computation, the Matlab ODE solver [*ode15s*]{} (an implicit scheme) is used to integrate (\[mmpde-2\]).

Equation (\[mmpde-2\]) is called the ${\boldsymbol{\xi}}$-formulation of the MMPDE moving mesh method since it has been formulated in terms of the derivatives of $I_h$ with respect to ${\boldsymbol{\xi}}_j$ and the velocities for the computational coordinates. We can obtain an ${\boldsymbol{x}}$-formulation by directly differentiating $I_h$ with respect to ${\boldsymbol{x}}_j$ (with $\cTc$ being taken as $\cTco$ and fixed) and the new physical mesh $\cTh^{n+1}$ by directly integrating this formulation. The main disadvantage of this formulation is that its formula is more complicated than that of the ${\boldsymbol{\xi}}$-formulation and the metric tensor, which is defined on $\cTh^{n}$, needs to be updated every time the physical mesh is changed during the time integration of the mesh equation for $\cTh^{n+1}$. It is analytically shown in [@HK2015] that the mesh 