Unformatted text preview: her-order terms (a = 0) predict absolute stability
when r 1=2. We have just seen that this ensures stability when a 6= 0 however, it does 4.4. Variable-Coe cient and Nonlinear Problems 37 not ensure absolute stability. Ampli cation factors may become larger than unity causing
solutions to grow beyond their initial size. Maintaining no solution growth requires the
additional condition j
jPej ja2 x 1 (4.4.14) where Pe is the cell Peclet or cell Reynolds number. To verify (4.4.14) in the maximum
norm, write (4.4.13) in the usual form Ujn+1 = r(1 ; Pe)Ujn;1 + (1 ; 2r)Ujn + r(1 + Pe)Ujn+1
and invoke the Maximum Principle. All coe cients add to unity and are positive if
r 1=2 and jPej 1. The same condition may be veri ed in L2 by using a von
Neumann analysis. (The von Neumann analysis may also be inferred from the solution
of Problem 3.2.3.)
If is small relative to jaj, maintaining jPej 1 will require a very ne mesh spacing.
We could have anticipated di culties when is small since the forward time-centered
space scheme is not absolutely stable for the kinematic wave equation that results when
= 0. With this view, it may be better to replace the centered-di erence approximation
of aux by forward di erences when a is positive.1 Thus, at
Ujn+1 = Ujn + r(Ujn;1 ; 2Ujn + Ujn+1) + 2 x (Ujn+1 ; Ujn )
or Ujn+1 = rUjn;1 + 1 ; 2r(1 + Pe)]Ujn + r(1 + 2Pe)Ujn+1 (4.4.15) With Pe > 0 for a > 0, use of the Maximum Principle states that there will be no growth
if 1 ; 2r(1 + Pe) < 1 or
2r(1 + Pe) = 2r + 1: (4.4.16) This condition is typically much less restrictive than (4.4.14).
1a is on the opposite side of the equation from our usual form of the kinematic wave equation. 38 Parabolic PDEs The use of forward or backward di erencing for the convective term of the convectiondi usion equation is called upwind di erencing. When a can change sign, we may write
(4.4.15) in the form Ujn+1 = Ujn + 2 (Ujn;1 ; 2Ujn + Ujn+1) + 2 (Ujn+1 ; Ujn;1) (4.4.17a) where
= 2r + j j: (4.4.17b) (The von Neumann analysis of Problem 3.2.3 may again be used to study L2 stability.)
Notice that the \di usion coe cient" =2 = r + j j=2 has increased relative to its value
of r for the forward time-centered space scheme (4.4.13). This di usion is often called
arti cial since it depends on mesh parameters rather than physical parameters.
Example 4.4.4. Let us apply the forward time-centered space (4.4.13) and upwind
(4.4.17) schemes to a convection-di usion problem with a = 1, = 0:01, and the initial
and boundary conditions u(x 0) = (x) =
u(0 t) = 0 0 if 0 x < 1=2
1 if 1=2 x 1 u(1 t) = 1: The initial square pulse propagates from x = 1=2 to x = 0 while di using to form a
steady \boundary layer" at x = 0 where the solution transitions from near unity to zero.
We choose J = 20, so that x = 0:05, and t = 0:025. Thus, = 0:5, r = 0:1,
Pe = 2:5, and = 0:7. With these parameters, the absolute stability condition (4.4.16)
for the upwind scheme is satis ed, but condition (4.4.14) for the centered scheme is not.
The computed solutions for both the centered and upwind schemes are shown in
Figure 4.4.1. The solution with centered di erencing (4.4.13) of the convection term
exhibits spurious oscillations. As noted, the scheme is stable but not absolutely stable.
There is some growth of the initial data beyond unity. The upwind scheme is absolutely
stable but only rst-order accurate. In this example, this is seen through the excess
di usion at the moving front. Thus, with upwind di erencing, the initial square pulse is 4.4. Variable-Coe cient and Nonlinear Problems 39 1.4
1 U 0.8
0.8 0.1 0.6
0 t 0 x 1 0.8 U 0.6 0.4 0.2 0
0.8 0.1 0.6
t 0 0 x Figure 4.4.1: Solutions of Example 4.4.4 obtained by using centered di erencing (4.4.13)
(top) and upwind di erencing (4.4.17) (bottom) of the convection term. 40 Parabolic PDEs di using faster than it should. With the present state of the art, these are, unfortunately,
the two choices:
higher-order schemes introduce spurious oscillations and
rst-order upwind schemes have excess di usion.
Some improvements using higher-order upwind schemes will be discussed in Chapter 6. 4.4.3 Nonlinear Problems
Very few exact solutions of nonlinear problems exist and those that do are far removed
from realistic applications. Numerical techniques such as inite di erence methods often
provide the only means of calculating approximations. Explicit schemes cause very few
computational di culties. Convergence and stability analyses, however, are far more
di cult. Richtmyer and Morton 17] discuss a heuristic approach to stability that is
useful when the coe cients and solution are smooth. Thus, in accord with the discussion
of Section 4.1, they show that a constant-coe cient stability analysis performed with
\frozen" coe cients yields essentially the correct results. We'll subsequently illustrate
this by example.
Implicit di erence schemes lead to nonlinear algebraic systems that must typically
View Full Document
- Spring '14
- The Land, Boundary value problem, Partial differential equation, Dirichlet boundary condition, Neumann boundary condition, di erences