Adaptivity with moving grids

Adaptivity with moving grids 81
(for example when adaptive methods are used to solve the Euler equations)
or the loss of a Hamiltonian structure when solving problems such as the
KdV or the NLS equations. These problems can be avoided by completely
decoupling the solution of the PDE and the moving mesh equations over
each time step, so that when solving the PDE the mesh is regarded as being
static. The PDE can then be solved over that time step using a method
(for example the finite volume method for problems with a conservation
law) that preserves the significant features of the solution. The rezoning
method can be briefly described as follows.
(1) At time step t n let the solution be u n and the mesh x n . Calculate the
corresponding monitor function M n .
(2) Using this monitor function, solve the moving mesh PDE over the time
interval [t n ,t n +∆t n ] to give a new mesh x n+1 .
(3) Interpolate the solution u n onto the new mesh x n+1 , to give an interpolated
solution û n .
(4) If necessary, repeat steps (2) and (3), updating M until the new mesh
does not change.
(5) Starting from û n solve the original PDE (3.48). Typically, in the physical
domain using high-resolution standard software such as the finite
volume method) over the time interval [t n ,t n +∆t n ], doing all the
calculations on the new mesh x n+1 .
(6) Repeat this from step (1).
This method is of course closely related to the quasi-Lagrangian approach.
To see this in a one-dimensional example, suppose that the mesh velocity
is ẋ and the underlying solution is u with U n i ≈ u(X n i ,tn ); then in step (3)
we have
Ûi n ≈ u(Xi n+1 ,t n )=u(Xi n + δt n ẋ) ≈ Ui n +∆t n u x ẋ.
Thus, if we use a simple forward Euler method to approximate the solution
of (3.48) we obtain
Ui n+1 = Û i n +∆t n f i = Ui n +∆t n u x ẋ +∆t n f i = Ui n (
+ δt n fi + u x ẋ ) ,
which is of course the result of applying the forward Euler method to the
Lagrangian form of the PDE given by (3.49).
The key to the success of this approach is the interpolation step (3),
and an interpolation scheme that preserves some quantities of the solution
is often necessary. We summarize some rezoning methods based on this
approach for solving conservation laws in one and two dimensions, using
the finite volume method, which are described in Tang (2005) and Tang
and Tang (2003).