Unformatted text preview: n + tVn+1:
Use of this method with the centered space di erences of Example 4.3.1 yields the backward Euler method (4.1.4), i.e.,
!
Vjn+1 ; 2Vjn+1 + Vjn+1
;1
+1 :
Vjn+1 = Vjn + t
x2
Higherorder backward di erence formulas have the form Vn+1 = S ;1
X
s=0 kV n;s + 0 _
tVn+1 (4.3.11) where the coe cients s, s = 0 1 : : : S , and appear in Table 4.3.1 for methods of
orders S = 1 through 6 11]. Use of the secondorder formula with the centered spatial
di erences of Example 4.3.1 gives the scheme
n+1
n+1
n+1 !
n+1 = 4 V n ; 1 V n;1 + 2 t Vj ;1 ; 2Vj + Vj +1 :
Vj
3j 3j
3
x2
Berzins and Furzeland 6] developed a system called SPRINT that utilizes method
of lines integration with a variety of spatial discretization schemes. Modern method of
lines software can also adjust the spatial mesh and order to satisfy prescribed spatial 24 Parabolic PDEs accuracy criteria in an optimal manner. Adjerid and Flaherty 2, 3] describe a method
that automatically moves the mesh in time while adding and deleting computational cells.
Adjerid et al. 4] and Flaherty and Moore 9] additionally vary the spatial order of the
method to achieve further gains in e ciency. Roughly speaking, they utilize highorder
methods in regions where the solution is smooth and loworder methods with ne meshes
near singularities. Adjerid et al. 1] consider multidimensional problems with automatic
mesh re nement, methodorder variation, and mesh motion for arbitrarily shaped regions.
Recent developments are summarized in Fairweather and Gladwell 8]. Many of the
articles in these proceedings describe mesh motion techniques that concentrate the mesh
in regions of high solution activity. The following example illustrates such a procedure.
Example 4.3.2. Adjerid and Flaherty 3] consider the BuckleyLeverett system ut + f (u)x = uxx 0<x<1 u(x 0) = 10x1+ 1
u(0 t) = 1 t>0 0x1 ux(1 t) = 0 t>0 2
f (u) = u2 + au ; u2) :
(1 (4.3.12a)
(4.3.12b)
(4.3.12c)
(4.3.12d) This model has been used to describe the simultaneous ow of two immiscible uids
through a porous medium neglecting capillary pressure and gravitational forces. The solution u(x t) represents the concentration of one of the species, is a viscosity parameter
taken as 10;3 and a = 1=2.
Adjerid and Flaherty 3] solved this problem on a moving mesh using piecewise linear
nite element approximations. In this case, the nite element technique yields approximately the same discrete system as centered di erences. The problem is nonlinear and
centereddi erence approximations of such systems is discussed in Section 4.4. Despite
these uncertainties, let us concentrate on their mesh moving scheme xj ; xj;1 = ; (Wj ; W )
__ j = 1 2 ::: J ; 1 (4.3.13a) 4.3. Multilevel Schemes 25 where a superimposed dot denotes time di erentiation, xj (t) is the position of the jth
mesh point at time t, > 0 is a parameter to be chosen, Wj is a mesh motion indicator
on the subinterval (xj;1 xj ), and W is the average of Wj , j = 1 2 : : : J .
If Wj > W , the right side of (4.3.13a) is negative and the mesh points xj and xj;1
move closer to each other. The opposite is true when Wj < W . Neglecting the \boundary
conditions" x0 = 0 xJ = 1 (4.3.13b) the nodes tend to an \equidistributing mesh" where Wj = W , j = 1 2 : : : J , as t ! 1.
The parameter controls the relaxation time to equidistribution. Large values of
produce shorter relaxation times but introduce sti ness into the system (4.3.13a). Smaller
values of produce a less sti system that may not be able to follow some rapid dynamic
phenomena. Adjerid and Flaherty 3] discuss an automatic procedure for selecting that
balances sti ness and responsiveness.
The motion indicator is a solutiondependent parameter that should be large where
the mesh should be ne and small where it should be coarse. It could be chosen in
proportional to a solution derivative, e.g.,
jU n ; U n j
Wj (tn ) = x2(maxx ) jux(x tn)j x (t j) ; x j;1(t ) :
xj ;1 j
jn
j ;1 n
Choosing Wj proportional to the second derivative of u is also popular and, in most
cases, preferable. When using secondorder schemes, the gradient could be large while
the actual discretization error is small. A large second derivative is a better indicator
of large error. Adjerid and Flaherty 2] used an estimate of the spatial discretization
error as a mesh motion indicator. In particular, the results shown in Figure 4.3.3 were
obtained with
Z xj
Wj (t) =
jE (x t)jdx
xj ;1 where E (x t) is an estimate of the spatial discretization error. We will discuss methods
for obtaining such errors later in this section.
The di erential equations (4.3.13a) for the mesh motion can be solved simultaneously
with the spatially discrete version of (4.3.12) using the ordinary di erential equations soft 26 Parabolic PDEs Figure 4.3.3: Method of lines solution of (4.3.12) (bottom) on a moving 30element mesh
(top) 2]. 4.3. Multilevel Schemes 27 ware. The global parameter W can be eliminated from (4.3.13a) by combining equations
on two successive in...
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 Spring '14
 JosephE.Flaherty
 The Land, Boundary value problem, Partial differential equation, Dirichlet boundary condition, Neumann boundary condition, di erences

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