Crank-Nicolson method

The Crank Nicolson method is a combination of implicit and explicit methods, and the benefit of the method is twofold. First, the method is unconditionaly stable, and second the truncation error is smaller, OQ ) +OQi ). Implementing the Crank-Nicolson method in the heat equation gives 

This section has provided a short introduction to solving PDFs using the finite difference method. But there is much more to explore, and the reader is referred to books on advanced numerical methods to learn more about how to solve PDFs. In most cases the reader will use existing munerical algorithms for solving ODFs and PDFs, either commercial or open-source. Section 6.4 gives a short description of some of these software products. 

Although there is only one explicit formulation for a given problem, the implicit formulation may be written in a number of forms. For example, we may express the righthand side of Eq. (4.72) as 

Reconsider Ex. 4.7. We now wish to determine the transient temperature distribution of the flat plate by using the Crank-Nicolson implicit scheme. 

In view of Eq. (4.76), the proper difference equation for each inner node is 

So far, we have employed three different numerical schemes (explicit, implicit, Crank-Nicolson) to solve a one-dimensional unsteady conduction problem. Pros and cons for these schemes are  

The explicit scheme is very easy to implement in a computer program, must satisfy a restrictive stability criterion, and is first-order accurate in time. 

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Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

The Crank-Nicolson method is popular as a time-step scheme for CFD problems, as it is stable and computationally less expensive than the implicit Euler scheme. 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

The system of Eqs. 13.3-17 and 13.3-18 can be solved for the adiabatic, isothermal, or constant wall flux cases using the Crank-Nicolson method. The thermomechanical and reaction data for such systems were evaluated by Lifsitz, Macosko, and Mussatti (99) at 45°C for a polyester triol and a chain extended 1,6-hexamethylene diisocyanate (HDI) with dibutyltin as a catalyst. Figure 13.46 gives the temperature profiles for the isothermal-wall case. Because of the high heat of polyurethane formation and the low conductivity of... 

Osterby O., The error of the Crank-Nicolson method for linear parabolic equations with a derivative boundary condition, Report PB-534, DAIMI, Aarhus University (1998)... 

The Crank-Nicolson method is a mixture of the implicit and explicit schemes which is less stable but more accurate than the fully implicit scheme (see Britz, 1988). A set of simultaneous equations analogous to those necessary for the fully implicit method must be solved the matrix [Af] has exactly the same structure in either case. 

CN Crank-Nicolson (method for solution of differential equations)... 

In a later paper (31) they report that a number of diflBculties were encountered using the Crank-Nicolson method and the approach was abandoned. More recently, Roth et al. (17) have applied the method of fractional steps to the solution of six equations of the form of (7), four of which are coupled this effort is continuing. 

Table 2.11 Temperatures in the cooling of a steel plate, calculated using the Crank-Nicolson method with M = aAt/Ax2 = 1.

A comparison of the temperatures at time f4 = 960 s with the values shown in Table 2.10, indicates that the Crank-Nicolson method delivers better results than the explicit method, despite the time step being three times larger. 

Numerical methods of solution Hansen (1971) used the orthogonal collocation method to solve both the steady state and transient equations of six different models of the porous particle of increasing complexity. He found that only 8 collocation points were necessary to obtain accurate results. This leads to a considerable saving in computing time compared to the conventional finite difference methods such as the Crank-Nicolson method. 

Numerical Solution. In the numerical formulation of THCC, Equations (2) and (3) are substituted into Equation (1). The resulting set of Nf, partial differential equations is transcribed into Nb finite-difference equations, using central differencing in space and the Crank-Nicolson method to obtain second-order accuracy in time. The set of unknowns consists of i = 1,..., ATft, and Pjt, k = 1,..., A/p, at each finite-difference node. Residue equations for the basis species are formed by algebraically summing all terms in the finite-difference forms of the transport equations. The finite-difference analogs of Equation (1) provide Ni, residue equations at each node the remaining Np residue equations are provided by the solubility products for the reactive solids. 

Note that the backward difference in time method and the Crank-Nicolson method both yield finite difference equations in the form of tridiagonal matrix, but the latter involves computations of three values of y(yi, y/, and y,+, ) of the previous time tj, whereas the former involves only y,. 

Since the time derivative used in the Crank-Nicolson method is second order correct, its step size can be larger and hence more efficient (see Fig. 12.11c). Moreover, like the backward difference method, the Crank-Nicolson is stable in both space and time. 

Another useful point regarding the Crank-Nicolson method is that the forward and backward differences in time are applied successively. To show this, we evaluate the finite difference equation at the ij + 2)th time by using the backward formula that is,... 

Figures 12.11 shows plots of yj = y x = 0.2) as a function of time. Computations from the forward difference scheme are shown in Fig. 12.11a, while those of the backward difference and the Crank-Nicolson schemes are shown in Figs. 12.11h and c, respectively. Time step sizes of 0.01 and 0.05 are used as parameters in these three figures. The exact solution (Eq. 12.129) is also shown in these figures as dashed lines. It is seen that the backward difference and the Crank-Nicolson methods are stable no matter what step size is used, whereas the forward difference scheme becomes unstable when the stability criterion of Eq. 12.139 is violated. With the grid size of Aa = 0.2, the maximum time step size for stability of the forward difference method is At = (Ajc)V2 = 0.02.

Solve Problem 12.9 using the Crank-Nicolson method, and show that the final N finite difference equations have the tridiagonal matrix form. Compute the results and discuss the stability of the simulations. 

Repeat Problem 12.13 for the Crank-Nicolson method to show that A will take the form... 

From Table 6.3(b,i), we seen that the number of points taken is important With 100 points, the value of Hip is close to the value using the Crank-Nicolson method with six points [Table 6.2(b)]. From Table 6.3(j), we note that if D is changed to 1.45, the value of Hip may become greater than 1, which means that it is safer to... 

The Crank-Nicolson method for the numerical solution of the boundary value problem defined by Eqs.(6)-(8) is valid for the case of discontinuous coefficients, k(T) and pCp(T), as obtains, or nearly obtains, at Tg. This method provides the temperature distribution at time t+ot, given the distribution at time t, as the solution of a certain non-linear system of algebraic equations, which are solved with the use of the Newton-Raphson method and the Thomas Algorithm. The Crank-Nicolson method is more easily (and more generally) applied to the heat equation with boundary conditions of the kind given by Eq.(8), rather than by Eq.(7). For the numerical solutions by the Crank-Nicolson method discussed below, then, the boundary condition. 

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