Numerical Examples

• Two Circles Example
  
• Moving Grid with Constraint
  
• Adaptive Moving Grid with Moving Boundary
  

Moving Finite Difference

References

About the Authors

UTA MATH Home Page

We could deform the initial (uniform) grid directly to the grid at t+dt. the resulting grid is more precise. It takes up to 4 or 5 runs of the program (solving Poisson and ordinary differential equations (ODES)), instead of a single run as in the previous mode. The diagram is the following:

It is called the static mode since the parameter s in the deformation ODEs is not the real time t in the original PDE.The monitor function f ( x, t ) is constructed as before. Then a new monitor function f ( s ) is constructed which bridges the constant function 1 to f. For instance, let

.

Then f ( 0 ) = 1 and f ( 0.125 ) = f. The deformation ODEs deform the original grid at t = 0 (also s = 0) to the desired grid at t+dt, which corresponds to s = 0.125. In our codes, it takes these 5 steps, s = 0; 0.025; 0.05; 0.075; 0.1; 0.125, to generate the final grid according to f ( x, t ) directly from the initial (uniform) grid.