This is the case for the lax friedrichs scheme even

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Unformatted text preview: .3.3. The Lax-Wendro scheme for the kinematic wave equation ut + aux = 0 is obtained from the Taylor's series in time 2 un+1 = un + t(ut)n + 2t (utt )n + : : : : j j j j Time derivatives are replaced by spatial derivatives that are obtained from the partial di erential equation. For simplicity, let us suppose that a is a constant, then ut = ;aux utt = ;auxt = ;a(ut )x = a2uxx ::: 3.3. Matrix Stability Analysis 25 (In fact, the Lax-Wendro scheme is rather di cult to implement when is not constant. We'll discuss an alternative in Chapter 6 that is superior to this approach in this case.) The Taylor's series is truncated and the spatial derivatives are approximated by centered di erences (2.1.7, 2.1.9). Retaining O( t2 ) terms leads to the second-order LaxWendro scheme 2 or Ujn+1 = Ujn ; 2 (Ujn+1 ; Ujn;1) + 2 (Ujn+1 ; 2Ujn + Ujn;1) Ujn+1 = ( 2+ 1) Ujn;1 + (1 ; 2)Ujn + ( 2; 1) Ujn+1: (3.3.8) where, as usual, is the Courant number. Taylor's series arguments immediately show that the local error is O( t2) + O( x2 ). The coe cients of this Lax-Wendro scheme are not all positive. For example, if 0 < < 1 then the coe cient of Ujn+1 will be negative. Likewise, the coe cient of Ujn;1 is negative when is negative. Thus, Theorem 3.1.1 cannot be used to establish stability in the maximum norm. As expected, the Lax-Wendro scheme gives the exact solution of the kinematic wave equation when j j = 1. We'll establish the stability of a periodic initial value problem in the Euclidean norm when j j 1 using de nition (3.1.14b). The von Neumann method could also be used in this case (cf. Problem 3.2.3). To begin, we square (3.3.8) to obtain (Ujn+1 )2 = ( + 1) Ujn;1 + (1 ; 2 )Ujn + ( ; 1) Ujn+1]2: 2 2 If j j 1 then 2 (1 ; 2 ) Ujn;1 ; 2Ujn + Ujn+1]2 0: 4 Add this term to the right side of the previous expression and sum over a period to get J ;1 J ;1 2 X n+1 2 X 2(1 + ) n 2 (Uj;1) + (1 ; 2)(Ujn )2 + (12; ) (Ujn+1)2 (Uj ) 2 j =0 j =0 ; (1 ; 2)(Ujn+1Ujn ; Ujn Ujn;1)]: This expression can be simpli ed by reindexing the summations, e.g., J ;1 X j =0 (U ;1 n j )2 = J ;2 X k (U =;1 n k J ;1 2 = X(U n )2 + (U n )2 ; (U n )2 : ) k ;1 J ;1 k =0 26 Basic Theoretical Concepts n n The solution is periodic so U;1 = UJ ;1 hence, J ;1 X j =0 (Ujn;1)2 = Similar reindexing of other terms yields J ;1 X J ;1 2 = X(U n )2 (U +1) k j =0 k =0 n j J ;1 X k =0 J ;1 X j =0 (Ukn)2 : U U ;1 = n j n j J ;1 X k =0 Ukn+1Ukn : Thus, the summation simpli es to J ;1 X j =0 (U n j +1 )2 J ;1 J ;1 2 X 2(1 + ) 2 ) + (1 ; ) ](U n )2 = X(U n )2 : + (1 ; j j 2 2 j =0 j =0 Therefore, kUn+1 k2 kUnk2 and the Lax-Wendro scheme is stable when the Courant, Friedrichs, Lewy Theorem (j j 1) is satis ed Centered schemes like the Lax-Wendro (3.3.8) and Lax-Friedrichs (3.2.7) methods may require arti cial boundary conditions for initial-boundary value problems. For instance, suppose the kinematic wave equation with a positive wave speed a is to be solved on 0 < x < 1. This problem only requires a boundary condition at x = 0 (Section 1.3). A numerical solution at the right-most point j = J , however, cannot be computed by either n the Lax-Wendro or Lax-Friedrichs schemes. Another method is needed to compute UJ . As we'll learn in Chapter 6, a poor choice could a ect the stability of the method. Problems 1. Write a computer program for the di erence scheme (3.2.8). Implement it with = 2, which corresponds to the Lax-Wendro scheme. Assume that the initial data u(x 0) = (x) is periodic in x with period 2. The problem should be solved on ;1 0 < t T . Use J (= 2= x), , and N (= T= t) as input parameters. 1.1. Execute your program when a = 1 and (x) has the form 8 < x + x if ; x x < 0 (x) = : x ; x if 0 x < x : 0 elsewhere for x 2 (;1 1) x 1, 3.4. The Lax Equivalence Theorem 27 1.2. This data is an attempt to simulate the e ects of a small error introduced by, say, round o . Execute your program for about 20 time steps with J = 10 and = 0.5, 0.999, 1.1 (more if you like). Plot the numerical and exact solutions as functions of x for a few times. Comment on the solutions for each value of . Which choice of would be preferred if the initial data actually did correspond to a rounding error? 1.3. Solve a problem...
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