Numerical Examples

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

Simulate u, where u is a scalar or vector function, satisfying

where L is a differential operator, on a domain D in .

An Initial (uniform) grid on D is deformed in real time to an adaptive grid according to a monitor function f ( x, t ).

Equidistribution Principle is used to construct the monitor function. A posteriori error estimates, residues and truncation errors, etc. are redistributed evenly over the whole domain. In most cases, we want to put refined grids in the regions where u changes rapidly.

For flow patterns with shock waves, we can take

where p is the pressure. In general, terms involving the values of u and the second derivatives of u (or curvatures) can also be included.

For interface resolution, we could construct f by using the level set function d as follows: Let f be piecewise linear such that

Required properties of grids: Non-folding, volume (cell size), smoothness, orthogonality and aspect ratios.

Evaluation of grid generation methods: Higher dimensions? Complex Geometry? Easy to implement? Work well with various numerical methods such as finite difference/volume, finite elements, the method of lines, particle methods, etc.? Work well for general equation(s), not problem dependent?