Question

c) Use von Neumann analysis on (1) to show that the Lax-Wendroff scheme is conditionally stable. What is the condition?

Answer 1

The schemes of the two previous examples are of first-order accuracy, which is generally insufficient for practical purposes. The first second-order scheme for the convection equation with two time levels is due to Lax and Wendroff (1960). The original derivation of Lax and Wendroff was based on a Taylor expansion In the development in time up to the third order such to achieve second-order accuracy (E8.3.8) the second derivative is replaced by (E8.3.9) leading to When this is discretized centrally in mesh point i we obtain As can be seen, the third term, which stabilizes the instability generated by the first two terms, is the discretization of an additional dissipative term of the form (a /2)u. The amplification matrix from the Von Neumann method is (E.8.3. 12) In the complex G-plane this represents an ellipse centred on the real axis at the abscissa (1-σ2) and having a semi-axis length of σ 2 along the real axis and σ along the vertical axis. Hence this ellipse will always be contained in the unit circle if the CFL condition is satisfied (Figure 8.3.3). For ơ-1 the ellipse becomes identical to the unit circle. The stability condition is therefore (E8.3.13) lol ! 309 Im G Region of instability Re G Ơ- (a) 1/2

Im G Region of instability Re G 1/2 (b) d's Figure 8.3.3 Polar representation of the amplification factor for Lax-Wendroff scheme. (a) σ < l and σ'« 1/2; (b) σ < 1 and σ/> 1 /2 310 The dissipation error is given by (E8.3.14) and the phase error by tan-ı [ (o sin φ)(1-a-sin 2φ/2) (E8.3.15) οφ To the lowest order we have (E8.3.16) This relative phase error is mostly lower than one, indicating a dominating lagging phase error. On the high-frequency end the phase angle Ф goes to zero if σ 2 < 1/2 and tends to π if σ 2 > 1/2. These diffusion and dispersion errors are represented in Figure 8.3.4 The phase error is the largest at the high frequencies, hence this will tend to accumulate high-frequency errors (for instance, those generated at a moving Lax-Wendroff scheme 1.0 CFL 25 0.8 0.6 04 CFL 75 02 90 135 180 45 Phase angle Lax-Wendroff scheme 1.0 CFL .75 135 90 180

8.3.3 Extension to three-level schemes The properties of the amplification factor in the previous sections were based on two-level schemes, allowing a straightforward definition of G. However many schemes can be defined which involve more than two time levels, particularly when the time derivatives are discretized with central difference formulas. A general form, generalizing equations (8.2.6), would be (8.3.32) For instance, for the convection equation uau in space we can define a scheme 0 and a central difference (8.3.33) 2Ax which is second-order accurate in space and time. This scheme is known as the leapfrog scheme, because of the particular structure of its computational molecule (Figure 8.3.5) where the nodal value u does not contribute to the computation of u This scheme treats three levels simultaneously and, in order to start the calculation, two time levels n=0 and n= I have to be known. In practical computations this can be obtained by applying another, two-level, scheme for the first time step. The method applied for the determination of the ampli- fication matrix, consists of replacing the multi-level scheme by a two-step i-1 Figure 8.3.5 Computational molecule for the leapfrog scheme 312 system through the introduction of a new variable Z: (8.3.34) Equation (7.3.32) then becomes (8.3.35) and by defining a new variable (8.3.36) the system is rewritten as (8.3.37) and analysed as in the previous cases. Alternatively, the method of introducing an additional variable is fully equivalent to a more direct approach, whereby we write for the amplitudes v" of a single harmonic (8.3.38) and (8.3.39) When this is introduced into the three-level scheme a quadratic equation for G is obtained.

Lax-Wendroff scheme The dissipation of the scheme is of fourth order, since for small φ, from equation (E8.3.13) (8.5.16) showing that the Lax-Wendroff scheme is dissipative to the fourth order Since G(r) 리-2σ2 the Lax-Wendroff scheme is dissipative in the sense of Kreiss for non-zero values of o 8.5.3 Non-linear problems Very little information is available on the stability of general non-linear discretized schemes. Within the framework of the Von Neumann method it can be said that the stability of the linearized equations, with frozen coefficients, is necessary for the stability of the non-linear form but that it is certainly not sufficient. Products of the form u(auax) will generate high- frequency waves which, through a combination of the Fourier modes on a finite mesh, will reappear as low-frequency waves and could deteriorate the solutions. Indeed, a discretization of the form (8.5.17) 2Ax becomes, when the Fourier expansion (8.2.1) is introduced (8.5.18) Σ Σ υ(k)r(h) eKA+ klar sin k.ax Ax The sum (k,+k2)ax can become larger than the maximum value π associated with the (2Ax) wavelength. In this case the corresponding harmonic will behave as a frequency [2r -(kik)Ax] and will therefore appear as a low-frequency contribution. This non-linear phenomenon is called aliasing, and is to be avoided by providing enough disipation in the scheme to damp the high frequencies.