# Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations
## 4 Experiments
### Advection-Diffusion Equation  
The first example is the advection-diffusion equation  
\[u_{t}=\epsilon u_{xx}-V\frac{\partial u}{\partial x},x\in(0,1), \tag{22}\]  
where \(\epsilon>0\) is the diffusion coefficient, and V is the flow velocity. Let \(\epsilon=10^{-5}\), \(V=1\) and Dirichlet boundary conditions at the endpoints \(x=0\) and \(x=1\) are chosen such that the exact solution is:  
\[u(x,t)=\frac{1}{2}erfc(\frac{x-t}{\sqrt{4\epsilon t}})+\frac{1}{2}exp(\frac{x }{\epsilon})erfc(\frac{x+t}{\sqrt{4\epsilon t}}),\]  
subject to the boundary conditions:  
\[u(0,t)=u(1,t)=0, \tag{24}\]  
and initial conditions:  
\[u(x,0)=n\sin(2\pi x)+0.5n\sin(\pi x), \tag{25}\]  
where \(n\) is a constant and \(\epsilon=10^{-5}\), the solution has a smooth initial profile and develops a steep front. For different \(n\), the front grows to \(n\) times higher and propagates toward the right end and eventually dies out at \(t=\frac{1}{n},x=1\) which is \(n\) times quicker than \(n=1\). The difficulty with the numerical solution of the problem is the resolution of this propagating steep front.  
For Burgers Equations (23)-(25) with \(n=1\), Figure 3 shows the numerical solutions and the trajectories applying [PERSON]'s methods (modified MMPDE5) and the proposed method. Both the two methods simulate the Burgers Equation precisely in a short time and the trajectories are smooth. The advantage of the new method is not valid. Figure 4 shows the time step sizes of the two methods. It is possible to determine from this that, for the same accuracy, the time step sizes of the proposed method are larger than [PERSON]'s methods, which means that the proposed method consumes less time and iterations when simulating the Burgers equation.  
Figure 4: The time step sizes for Burgers equation with \(n=1\) applying [PERSON]’s method (**a**) and the proposed method (**b**).  
Figure 3: Numerical solutions and trajectories of the Burgers equation with \(n=1\) applying [PERSON]’s method (**a**,**b**) and the proposed method (**c**,**d**).  
When \(n=30\), according to the above analysis, the solution grows to 30 times of \(n=1\) and the shock wave propagates to the end at time \(t=\frac{1}{30}\) (about 0.033). As is shown in Figure 5, from the solutions applying the [PERSON]'s method bad nodes emerge, and the trajectories begin to be rough and unstable. Figure 6 shows that, for the same accuracy, the time step sizes of [PERSON]'s method are smaller than the proposed method. As a result, [PERSON]'s method consumes more time and iterations. However, the proposed method could simulate the PDE more efficiently and stably.  
Table 1 shows the computational time and iteration times of the Burgers equation applying 21 knots with different \(n\). One can observe that when \(n\) is small, the two methods' computational time and iterations are similar, as \(n\) grows, the modified MMPDE 5 spends more time and yields much more time iterations than the proposed methods. When \(n=30\), the Computational time and iteration times of the proposed method is no more than \(\frac{1}{4}\) of [PERSON]'s method. From the figures and table above, the knots move steadily applying the new method, this corresponds to the analysis of Section 2.  
Figure 5: Numerical solutions and trajectories of Burgers equation with \(n=30\) applying [PERSON]’s method (**a**,**b**) and proposed method (**c**,**d**).  
Figure 6: The time step sizes for Burgers equation with \(n=30\) applying [PERSON]’s method (**a**) and the proposed method (**b**).