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