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
t = f (j x (n + 1) t Uj
Ujn 2 Ujn
(1 ; )f (j x n t Ujn
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
t = f (j x (n + ) t Uj + (1 ; )Uj x ( Uj + (1 ; )Uj ) 4.4. Variable-Coe cient and Nonlinear Problems
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 ];
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
U0n = UJ = 0 (4.4.20c) n
were prescribed, then Un = U1n U2n : : : UJ ;1]T and we would have to solve the nonlinear
n+1 ) = 6 F2 (U ) 7 = 0:
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 )
: : : @F1JnV1 ) 7
k ) = @ Fj (V ) = 6
4 @FJ ;1 (Vk )
J ;1 (
: : : @[email protected];V
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
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) :
k ; V k;1
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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