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31 coe cients of backward di erence formuas 4311 11

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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 Higher-order 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 second-order 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 high-order methods in regions where the solution is smooth and low-order methods with ne meshes near singularities. Adjerid et al. 1] consider multi-dimensional problems with automatic mesh re nement, method-order 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 Buckley-Leverett 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 centered-di 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 solution-dependent 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 second-order 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 30-element 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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