Richtmyer and morton 17 discuss a heuristic approach

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Unformatted text preview: ved by iterative methods. Consider a nonlinear partial di erential equation having the rather general form ut = f (x t u ux uxx): (4.4.18) The trapezoidal-rule form of the weighted average nite-di erence approximation (4.1.7b) of this problem is Ujn+1 ; Ujn Ujn+1 2Ujn+1 n+1 t = f (j x (n + 1) t Uj x x2 )+ Ujn 2 Ujn (1 ; )f (j x n t Ujn (4.4.19a) x x2 ) where 2 0 1]. The average and di erence operators and are de ned in Table 2.1.1. The midpoint-rule form of the weighted average scheme is Ujn+1 ; Ujn n+1 n+1 n n t = f (j x (n + ) t Uj + (1 ; )Uj x ( Uj + (1 ; )Uj ) 4.4. Variable-Coe cient and Nonlinear Problems 2 n+1 n x2 ( Uj + (1 ; )Uj )): 41 (4.4.19b) When = 1=2 both (4.4.19a) and (4.4.19b) are called Crank-Nicolson schemes. Although both are used, the trapezoidal rule form is more common for nonlinear problems. Example 4.4.5 ( 17], p. 201). Consider the nonlinear di usion equation ut = (um)xx (4.4.20a) This di erential equation arises in reaction problems associated with semiconductor device fabrication. Let us discretize it by the weighted average scheme (4.4.19a) to obtain Fj (Un+1 ) Ujn+1 ; t (U n+1 )m ; 2(U n+1 )m + (U n+1 )m ]; j j +1 x2 j;1 Ujn ; (1 ; ) xt2 (Ujn;1)m ; 2(Ujn)m + (Ujn+1)m] = 0: (4.4.20b) As usual, the vector Un+1 represents the vector of unknowns at time level n + 1. If, for example, homogeneous Dirichlet boundary conditions n U0n = UJ = 0 (4.4.20c) n were prescribed, then Un = U1n U2n : : : UJ ;1]T and we would have to solve the nonlinear system 2 3 F1 (Un+1) n+1 7 6 n+1 ) = 6 F2 (U ) 7 = 0: F(U (4.4.20d) 6 7 ... 4 5 FJ ;1(Un+1) With = 0, we obtain the explicit forward time-centered space di erence scheme Ujn+1 = Ujn + xt2 (Ujn;1)m ; 2(Ujn)m + (Ujn+1)m] which causes no computational di culties. We will need an iterative scheme to solve (4.4.19a) or (4.4.19b) whenever 6= 0. Most iterative strategies are variants of Newton's method however, functional ( xedpoint) iteration is also used (cf., e.g., Isaacson and Keller 14], Chapter 3). We'll discuss 42 Parabolic PDEs Newton's method. Thus, suppose that Vk , k 0, is a guess for the solution Un+1 and expand the nonlinear system (4.4.20d) in a Taylor's series about Vk to get 0 = F(Un+1 ) = F(Vk ) + FUn (Vk )(Un+1 ; Vk) + O(kUn+1 ; Vkk2): +1 Neglecting the quadratic terms and denoting the resulting approximation as Vk+1, we obtain Newton's iteration FUn (Vk )(Vk+1 ; Vk ) = ;F(Vk ) +1 k = 0 1 ::: : (4.4.21a) The system (4.4.21a) de nes a set of linear algebraic equations that must be solved at each iterative step for Vk+1. The system matrix is the Jacobian which, for Dirichlet boundary conditions, is 2 @F (Vk ) 3 (k 1 : : : @F1JnV1 ) 7 n+1 +1 @U ; 6 @U1 k 7: ... ... k ) = @ Fj (V ) = 6 ... FUn+1 (V (4.4.21b) 6 7 n+1 4 @FJ ;1 (Vk ) @ Ui k) 5 J ;1 ( : : : @[email protected];V n +1 @U1 +1 1 The Jacobian will be tridiagonal for second-order partial di erential equations with centered di erences. The iteration begins with an initial guess V0, which must be close to the true solution Un+1 of (4.4.20d) for convergence. Fortunately, the solution Un at the previous time step frequently furnishes a good initial approximation. The iteration terminates either upon failure or satisfaction of a criterion such as kVk+1 ; Vk k < kV0k (4.4.21c) for a prescribed tolerance . The converged result is regarded as Un+1 . If p > 0 is the smallest number for which lim kkU n+1; V k kpk = c k!1 U ;V n+1 k+1 with c 6= 0, we say that the convergence rate of the iteration (4.4.21a) is p. Simple Taylor's series arguments show that the convergence rate of Newton's method is two when the Jacobian FUn+1 (Un+1) is not singular ( 14], Chapter 3). This is called quadratic convergence. It implies that the error decreases in proportion to the square of the previous 4.4. Variable-Coe cient and Nonlinear Problems 43 error. When the Jacobian is singular, e.g., at a bifurcation point, the convergence rate of Newton's method is typically linear (p = 1). The work associated with using Newton's method involves (i) calculating the Jacobian and (ii) solving the linear system (4.4.21a) at each iteration. The accuracy of the Jacobian does not a ect the accuracy of the solution, just the convergence rate. Thus, it is not necessary to reevaluate the Jacobian after each iteration and we may choose to evaluate it only at the initial iteration and use FUn (V0)(Vk+1 ; Vk ) = ;F(Vk ) +1 k = 0 1 ::: : This scheme, often called the \chord method," has a linear convergence rate. Although it has a slower convergence rate than Newton's method (4.4.21a), it may still use less computer time when Jacobian evaluations are complex. Along these lines, the partial derivatives within complex Jacobians may be approximated by nite di erences, e.g., @Fj (Vk ) Fj (Vk ) ; Fj (V1k : : : Vik 1 Vik;1 Vik : : : VJk;1) : ; +1 n+1 k ; V k;1 @ Ui Vi i This gives rise to the \secant method," which converges at a rate between linear and quadratic ( 14], Chapter 3). Example 4.4.6. Consider the nonlinear di usion equation (4.4.20...
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