Crank-Nicholson scheme

In computer simulations we had chosen the following parameters of the potential h 1, a = 2. With such a choice the coordinates of minima equal Xmin = 1, the barrier height in the absence of driving is A = 1, the critical amplitude Ac is around 1.5, and we have chosen A = 2 to be far enough from Ac. In order to obtain the correlation function K t + x, t] we solved the FPE (2.6) numerically, using the Crank-Nicholson scheme. 

Alternatively, for 0 = 1 /2, i.e., evaluation of the second-order difference as an average of the values at t and t + At, leads to the robust Crank-Nicholson scheme... 

The molecule of the Crank-Nicholson scheme is also shown in Figure 3.15. 

Figure 3.17 Same as Figure 3.16 but for the implicit Crank-Nicholson scheme. 

Solve with the Crank-Nicholson scheme the diffusion problem described in the worked example in Section 3.3.1... 

In the numerical modeling of optical pulse propagation with account of the SS effect, Eq.(2.8) with = 2 = 3 = 0 has been solved. The Crank-Nicholson scheme was used to transform Eq.(2.8) into a finite-difference equation. 

The Crank-Nicholson scheme was used to transform Eq.(2.8) into a finite-difference equation. 

This is the so-called Crank-Nicholson scheme and, formally, it could have been obtained by simply averaging the explicit forward-difference and implicit backward-difference schemes. By conveniently grouping the terms, the following system of linear equations results ... 

Applying the Crank-Nicholson scheme to equation (8-66), relative to a space-time grid characterized by the points ... 

Equations (7) and (8) form a system of six ordinary differential equations in three space dimensions for each individual bubble. This system is integrated in time using a Crank-Nicholson scheme, which provides second order accuracy. Sequential tracking of all bubbles in the system is performed at each time step of the Navier-Stokes solver. 

Fine resolution in the normal direction is necessary around the shear layers, and it gives severe limitation on the time step for numerical stability. Thus, it is preferred to compute the derivatives in the normal direction implicitly, while the derivatives in the streamwise direction are treated explicitly. This leads to a hybrid time-integration scheme with a low-storage third-order RK (RK3) scheme for explicitly treated terms and a second-order Crank-Nicholson scheme for implicitly treated terms. The overall accuracy is thus second order in time. The discretized Navier-Stokes equations have the forms ... 

The algebraic system of equations resulting from the Crank-Nicholson scheme in (25.123) is tridiagonal and can therefore be solved efficiently with specialized routines. 

The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

An implicit time integration scheme sneh as the Crank-Nicholson scheme, which is flee from the eompntational constraint, has an important application to atmospheric... 

The special cases of / = 0, / = 0.5 and / = 1 give rise to the explicit scheme. Crank Nicholson scheme and the implicit scheme, respectively. The integration with respect to the temporal and the spatial variables eventually results in a system of algebraic equations (one for each control volume), which can be solved by standard numerical techniques. 

Analogous stability analyses can be executed for the other time-discretization schemes as well. It is important to note here that although the von Neumann stabihty analysis yields a limiting time-step estimate to keep the round-off errors bounded, it does not preclude the occurrence of an bounded but unphysical solutions. A classical example is the Crank-Nicholson scheme, which from the von Neumann viewpoint is unconditionally stable, but can give rise to bounded unphysical solutions in case all the coefficients... 

Several different propagation methods exist in the market. We will briefly mention two of these a modified Crank-Nicholson scheme, and the split-operator method. 

A Modified Crank Nicholson Scheme. This method is derived by imposing time-reversal symmetry to an approximate time-evolution operator. It is clear that we can obtain the state at time t- - At/2 either by forward propagating the state at t by Z t/2, or by backward propagating the state at t - - At... 

Crank-Nicholson scheme). In accordance with the computing template on Fig. 16.6 moving from left to right ( =2,3,4,5) on the first two layers (z=l), we obtain the following four equations ... 

The advantage of Eq. (160) (representing a parabolic partial differential equation from a mathematical viewpoint) is that it can be solved exactly by standard numerical techniques e.g., by the finite-difference Crank-Nicholson scheme under transient (nonstationary) conditions [37,64]. These calculations showed that the duration of the transient regimes is of the order of seconds, as previously estimated. Under the stationary conditions, Eq. (160) is simplified to the ordinary one-dimensional differential equation which can be solved by standard numerical techniques [18,76,118,119]. 

The fractional step methods have become quite popular. To predict an accurate time history of the flow, higher order discretizations must be employed. Kim and Moin [106], for example, used a second order explicit Adams-Bashforth scheme for the convective terms and a second order implicit Crank-Nicholson scheme for the viscous terms. Boundary conditions for the intermediate velocity fields in timesplitting methods are generally a complex issue [3, 106]. There are many variations of the fractional step methods, due to a vast choice of approaches to time and space discretizations, but they are generally based on the principles described above. 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

The Reynolds equation and the energy equation in the interfacial film and the conduction equation in the seal rings are solved numerically by the finite difference technique. These equations are coupled by the heat exchange conditions on the boundaries of their domains. They are integrated by using the Crank Nicholson scheme. 

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