Crank-Nicolson oscillations

Osterby O., Five ways of reducing the Crank-Nicolson oscillations, Tech. Rep. Daimi PB-558, Dept, of Computer Science, Aarhus University (2002)... 

Implicit methods have the great advantage of being stable, in contrast with the explicit method. It will be seen (and analysed in detail in Chap. 14) that the Laasonen method, a kind of BI, is very stable and responds to sharp transients with smoothly declining (but relatively large) errors, whereas Crank-Nicolson, also nominally stable, responds with error oscillations of declining amplitude, but is highly accurate. The drawbacks of both methods can be overcome, as will be described below. 

Electro chemists first investigated the Saul yev method in 1988 and 1989 [381,382], including GEM, and the incorporation of implicit boundary values was added later [144]. The result of these studies is broadly that the last of Larkin s options above, averaging LR and RL, is the best. This has about the same accuracy as Crank-Nicolson, and could be considered to be easier to program. The third option, alternating LR with RL, produces oscillations. 

This program does the same work as the earlier one, C0TT EX, but uses Crank-Nicolson (with equal intervals). It also includes the choice of M Pearson substeps within the first step, to damp the oscillations, as discussed in Chap. 8, Sect. 8.5.1. 

The difference method of Crank-Nicolson is stable for all M. The size of the time steps is limited by the accuracy requirements. Very large values of M lead to finite oscillations in the numerical solution which only slowly decay with increasing k, cf. [2.57],... 

We aim at the development of fully robust, stable methods and therefore we restrict our attention to implicit methods with Particularly, we shall consider two cases with 6 = Yi and 0 = 1, which correspond to the Crank - Nicolson (CN) and backward Euler (BE) method, respectively. More details can be found e.g. in Quarteroni Valli (1994). Finally, we chose the BE method for its higher stability (the CN scheme can show some local oscillations for large time steps). 

For the pure convective case U2 = 0), the Crank-Nicolson formulation suffered rather severe oscillations, while the centered-difference (which is equivalent to the state-variable formulation for this case) performed well. 

In the pure diffusion case the centered-difference scheme results were encumbered by severe oscillation, while the Crank-Nicolson performed well. The state-variable scheme followed the Crank-Nicolson results well to U2 values of 10 when oscillations became severe. Table 8.3 presents a summary of the pure diffusion results. 

U2 = 0.5 X 10 Concurs with state variable Oscillation Concurs with Crank-Nicolson... 

Blitz D, 0sterby O, Stmtwolf J (2003) Damping of Crank-Nicolson error oscillations. Comput Biol Chem 27 253-263... 

The Peaceman-Rachford ADI method is second-order with respect to time, and performs similarly to Crank-Nicolson. Indeed, Lapidus and Finder write [224, p. 246] ... is a variation of the Crank-Nicolson approximation . It is known to be unconditionally stable [225]. As with CN, ADI may show some error oscillations, as also evidenced by the fact that some habitually use expanding time intervals when employing ADI [226-231], although some of these same workers on occasion also use equal time intervals [232,233]. 

Crank and Nicolson, in their original paper [185], recognised the oscillation problem with their method, writing If 7 [which is their 6t/6x2 J is very large an oscillatory error which only disappears very slowly may arise . The problem is referred to in most texts describing the method. More detail is given in Chap. 14, but the essence of the problem is that CN will oscillate if A > 0.5 ... 

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