Adaptivity with moving grids

Adaptivity with moving grids 113
Example 1. As a first calculation we consider a problem with the initial
and boundary values chosen over the unit square, so that (5.31) has the
exact solution given by
u(x, y, t) = ( 1+e (x+y−t)/2ν) −1 . (5.32)
This solution has a sharp moving front. For the purposes of this example we
consider coupling this system to the PMA algorithm described in Section 3,
to generate a moving mesh which can both compute an approximation to
this solution and follow the front as it evolves over the time interval t ∈
[1/4, 2]. To do this we discretize (5.31) in the Lagrangian form
( 1
u t = ν∆u −
2 (u2 ) x + 1 )
2 (u2 ) y +ẋu x +ẏu y , ν ≪ 1. (5.33)
with all discretizations made in the computational variables. The conservation
form of the equation above is used for this calculation so that, for
example, the advective term u 2 x is rescaled as
u 2 x = 1 J
[
yη u 2 ξ − y ξu 2 η]
,
which is then discretized using a central difference scheme. For this calculation
we also take similar discretizations for the other advective terms. The
equation posed in the computational variables is then coupled to the discretized
form of PMA equation (5.17) with the values of the approximation
U i,j to u(X i ,Y j )andofQ given on the mesh vertices. In these computations
we take an N × N computational mesh (typically N = 40), a viscosity of
ν =0.005, and use the arclength monitor function
√
M = 1+α ( u 2 x + uy) 2 ,
with α = 1. A number of different strategies could be used to evolve this
coupled system forward in time, but in practice a simple scheme which
solved the PMA equation and (5.33) simultaneously using a simple forward
Euler method is effective with a time step ∆t, as given in Zhang and Tang
(2002), determined by the CFL condition. In the PMA equation we used
ɛ =1,γ =0.335 to find the initial mesh (before evolving the solution PDE)
when t =1/4, and then ɛ =0.01,γ = √ max(M) to follow the front up
to t = 2. More details are given in Walsh et al. (2009). We note that
solving the PMA equation coupled to (5.33), using an alternating solution
strategy coupled with an up-winding discretization, has also been shown
to be effective (Sulman 2008). In Figure 5.7 we present the results of a
series of computations using this method, when N = 40, showing both
the solution for (5.31) and the corresponding mesh. In this figure we see
excellent resolution of the solution at the front.