Lets discretize this equation by the forward time

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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 1.2 1 U 0.8 0.6 0.4 0.2 0 0.2 0.15 1 0.8 0.1 0.6 0.4 0.05 0.2 0 t 0 x 1 0.8 U 0.6 0.4 0.2 0 0.2 0.15 1 0.8 0.1 0.6 0.4 0.05 0.2 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 be sol...
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