Adaptivity with moving grids

80 C. J. Budd, W. Huang and R. D. Russell
avoids the highly nonlinear coupling of the mesh and the physical solution
and preserves many structures such as ellipticity and sparsity in each of
the mesh and physical PDEs. This can lead to significant efficiency gains,
but at the disadvantage of a possible lag in the movement of the mesh and
possible mesh instabilities.
Suppose that the physical solution u n , the mesh x n , and a solution time
step ∆t n are known at a time t n . The alternating solution procedure is
typically implemented as follows.
(1) The monitor function M n (x) =M(t n , u n , x n ) is calculated using u n
and x n .
(2) The MMPDE is discretized in space and then integrated over the time
[t n ,t n +∆t n ]togiveanewmeshx n+1 at the time t n +∆t n ]. The
underlying solution u n is not changed during this calculation. This
calculation can be done by using the (fixed) monitor function M n ,
or by updating it during the integration by using linear interpolation
(Huang 2007).
(3) The physical PDE (3.49) is then discretized in (computational) space
and integrated using an SDIRK or similar method. In this calculation,
the convective terms involving x t use the approximation
ẋ = xn+1 − x n
∆t n .
The mesh used in the discretization is calculated by using linear interpolation
so that
x(t) =x n +(t − t n )ẋ.
(4) It may be necessary to use the new solution u n+1 to update the monitor
function M n and to iterate from step (2) above in order to gain better
control of the grid. In this case repeat these steps until the new mesh
does not change.
(5) This procedure is then repeated from step (1) above.
Ceniceros and Hou (2001) use this method together with the MMPDE (3.34)
(without updating the monitor function between time steps) to solve problems
involving vortex singularities in the Boussinesq equations.
3.4.3. Rezoning in the physical domain
There are various problems associated with solving the Lagrangian form of
the PDE (3.49) in the computational domain. A significant one of these
is that certain properties of the original PDE may be lost when it is put
into the Lagrangian form and coupled to a moving mesh equation. Two
examples of this occurring are the loss of a conservation form of the equation