Moving Finite Difference

Adaptive Grid Generation by the Deformation Method
Dr. Guojun Liao and Dr. Gary dela Pena

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Copyright © 1999 by Dr. Guojun Liao and Dr. Gary dela Pena

An understanble concern over various moving grid methods is the lack of orthogonality of the grid generated. Indeed, while finite element and finite volume methods can be implemented on non-orthogonal grids, finite difference methods are usually implemented on an orthogonal grid.

A moving finite difference method based on the deformation method is formulated. It transforms a time dependent partial differential equation by the grid mapping obtained from the deformation method then simulates the transformed equation on a fixed orthogonal grid in the computational domain.

Incompressible (Unsteady) Navier-Stokes Equation

A solver based on the MAC (Marker and Cell) method is coupled with the moving grid code to solve the flow in a square cavity whose top wall is driven by a lid with uniform velocity. The unsteady solution is solved on a 51 x 51 grid with Re = 1000.

  • Figure 1: 2D contour plot of Streamlines obtained using a moving grid. (You will need a viewer that plays avi files, e.g. windows media player.)
  • Figure 2: Moving grid used to compute numerical solution (You will need a viewer that plays avi files, e.g. windows media player.)
Severe Weather Simulation

  • Figure 5: 3-D contour plot of numerical solution obtained by a moving grid finite-difference method.
  • Figure 6: Moving Grid used to compute numerical solution
  • Figure 7: 3-D contour plot of monitor function based on the gradient of the solution. Note that where the monitor function is large the grid spacing is large and where it is small the grid spacing is small (compare with figure 2)
Magnetic Dynamics (Weiss) Problem

  • Figure 8: 2-D contour plot of numerical solution using a moving grid.
  • Figure 9: Moving Grid used to compute numerical solution
  • Figure 10: 3D contour plot of numerical solution