# Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations
## 2 The Improved Moving Mesh PDEs
### Analysis of [PERSON]'s MMPDEs  
[PERSON] [9] gave the differential forms of the EP:  
\[E_{2}(x(\xi,t),t) \doteq M(x(\xi,t),t)\frac{\partial x(\xi,t)}{\partial\xi}-s(t)=0; \tag{4}\] \[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]  
With the boundary conditions of (2), the Equations (4) and (5) are equivalent to the EP (3). Based on the EP (3) and its derivatives (4), (5) [PERSON] [6, 9] constructed a series MMPDEs, the typical ones of them are as follows:  
\[MMPDE\ 3: \frac{\partial^{2}}{\partial\xi^{2}}(Mx_{t})=-\frac{1}{\tau} \frac{\partial}{\partial\xi}(M\frac{\partial x}{\partial\xi});\] \[MMPDE\ 4: \frac{\partial}{\partial\xi}(M\frac{\partial x_{t}}{\partial \xi})=-\frac{1}{\tau}\frac{\partial}{\partial\xi}(M\frac{\partial x}{\partial \xi});\] \[MMPDE\ 5: x_{t}=-\frac{1}{\tau}\frac{\partial}{\partial\xi}(M\frac{ \partial x}{\partial\xi});\] \[Modified\ MMPDE\ 5: x_{t}=-\frac{1}{\tau M}\frac{\partial}{\partial\xi}(M\frac{ \partial x}{\partial\xi});\] \[MMPDE\ 6: \frac{\partial^{2}x_{t}}{\partial\xi^{2}}=-\frac{1}{\tau}\frac{ \partial}{\partial\xi}(M\frac{\partial x}{\partial\xi}).\]  
Here \(\tau\) is a small parameter that determine the speed of the equidistribution of the mesh. In [PERSON] [9], MMPDE3 and MMPDE4 are obtained from the Taylor expansion of the EP (5) with a time delay by dropping the high order terms. MMPDE5 and MMPDE6 are obtained based on the attraction and repulsion pseudoforces. Numerical and theoretical analysis about these MMPDEs are detailed in [6, 11]. Here we try to propose a simple explanation of the MMPDEs in the following theorem. Compared with the source of the MMPDEs proposed in [6, 9], the new explanation is more rational and direct and many theoretical problems could be settled simultaneously. Theorem 1: _After Euler discretization on time, MMPDE 3-6 could be taken as one step of root searching iteration methods of the EPs (3),(4), or (5)._  
Proof of Theorem 1.: Integrating MMPDE 3,4,6 to \(\xi\) twice, and by the boundary conditions of (2), MMPDE 3-6 becomes:  
\[x_{t} =-\frac{1}{\tau}\frac{E_{1}(x,t)}{E^{\prime}_{1x}(x,t)}, \tag{6}\] \[x_{t} =-\frac{1}{\tau}\frac{E_{1}(x,t)}{E^{\prime}_{1x}(\eta,t)},\] (7) \[x_{t} =-\frac{1}{\tau}E_{3}(x,t),\] (8) \[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]0 (9) \[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]1  
separately, where \(E^{\prime}_{1x}(x,t)=M(x,t)\), \(\eta\in(0,x(\xi,t))\). Taking MMPDE 3 as one example, discreting the above Equation (6) applying the explicit Euler discretization on time, one can get:  
\[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]2  
where \(\triangle t_{k}=t_{k+1}-t_{k}\) is the \(k\)-th time step. In order to interpret MMPDE 3 more precisely and observe its differences with other moving mesh equations, we divide the above equation into two steps:  
\[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]3 \[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]4  
where \(x^{(1)}(\xi,t_{k})\) is an intermediate term. One can conclude that (11a) is one step root searching Newton iteration of EP (3) at time \(t_{k}\) with \(\frac{\triangle t_{k}}{\tau}\) as the relaxation factor. According to the theory of iteration methods, the convergence condition of MMPDE 3 on fixed time \(t_{k}\) is  
\[E_{3}(x(\xi,t),t) \doteq \frac{\partial}{\partial\xi}(M(x(\xi,t),t)\frac{\partial x(\xi,t )}{\partial\xi})=0. \tag{5}\]5  
The source of other MMPDEs could be explained similarly, the theorem is proved.