Numerical Examples

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

Moving Finite Difference

References

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In both 2D and 3D, the method has the following properties:

Nonfolding, precise volume control, good smoothness, some indirect control over orthogonality. It is easy to implement since the main ingredients are

1. Solving a Poisson equation on D.
2. Solving n ordinary differential equations for each node of the initial grid.

Thus it works well for general domains of higher dimensions.

How does the Deformation Method Generate an Adaptive Grid?

At time , evaluate u = u ( x, t ) by a solver on an existing grid. Use the calculated values of u to construct a monitor function. Solve for a real valued function a ( x, t ) at each fixed from a Poisson equation with Neumann boundary condition.

Let v be the vector field defined by . Then solve for , the new position at t of each node x of the initial grid, from a system of deformation ODEs.

Solve for u ( x, t ) from the PDE on the new grid . Repeat the above procedures. It is illustrated in the following diagram:

The mathematical foundation of the deformation method is provided by the following result:

The Jacobian determinant of a mapping from D1 to D2 in :

where dA' is the image of an area element dA at x as shown in the figure:

Thus, the theorem assures precise control over the cell size dA' relative to that of the fixed initial grid.