Conclusions

From the above four examples, we have observed the great advantages of moving mesh methods:

• they help the central difference to work for some hyperbolic systems with discontinuities;
• they reduce the time variation to allow larger time step;
• they do not need interpolations between regriddings;
• they can automatically detect, resolve, and track steep wave fronts and moving boundaries.

The disadvantages of the moving mesh methods include:

• moving mesh increases the stiffness of the system and require implicit time integration to work efficiently;
• choosing time scale $\tau$ in equidistributing moving mesh methods is highly heuristic;
• moving mesh does not resolve the contact discontinuity as well as the shock;

From the numerical results, we have the following conclusions on the implementation of equidistributing moving mesh methods:

1.

Dorfi and Drury method (DD) is faster than local smoothing method (LSM) with the same number of nodes. However, DD has a more stringent requirement on the minimum number of nodes than LSM. LSM can work well with fewer nodes than DD.

2.

The optimal $\tau$ is proportional to the average time stepsize. For problems with a problem time scale of O(1), we found the optimal $\tau =10^{-3}$ for DD. For other problem time scales, the optimal $\tau$ must be scaled accordingly. For example, for the scalar combust model with a time scale O(10^-2), the optimal $\tau$ is 10^-5.

3.

The arclength monitor is the most efficient one among all of our choices. It yields good results with Dorfi and Drury method for most problems. However, for some problems involving many corners (greater than 2) in the solutions, the modified curvature monitor (5) has better accuracy and comparable efficiency.

The equidistributing moving mesh methods are more robust but slower than time variation approach moving mesh method. In [9], we provide some techniques to increase the robustness of the time variation approach while keeping its efficiency.