Crank-Nicolson equation

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. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

Transient is a C-program for solving systems of generally non-linear, parabolic partial differential equations in two variables (that is, space and time), in particular, reaction-diffusion equations within the generalized Crank-Nicolson Finite Difference Method. 

The prior discretization of equation (2.1) uses control volumes with exphcit differences. They are explicit because only the accumulation term contains a concentration at the n -k 1 time step, resulting in an exphcit equation for (equations (E7.1.4), (E7.2.5), (E7.3.4), and (7.25)). Another common option would be fully imphcit (Laasonen) discretization where flux rate terms in equations (7.24) and (7.23) are computed at the n -k 1 time increment, instead of the n increment. Fully implicit is generally preferred over Crank-Nicolson implicit UQ = U Q n + Q.n+i) /P)... 

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 enthusiasm for hopscotch arose from the fact that here was a method with an accuracy thought to be almost comparable with that of Crank-Nicolson, but which was an explicit computation at every step, not requiring the solution of linear systems of equations, as other implicit methods do. It was also stable for all A, thus making it possible to use larger time steps, for example. The convenience of the point-by-point calculation has occasionally led workers to call the method fast [235],... 

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. 

To transfer the Crank-Nicolson [2.65] implicit difference method, which is always stable, over to cylindrical coordinates requires the discretisation of the equation... 

There are a number of numerical algorithms to solve the difference equation representation of the partial differential equations. Implicit algorithms such as Crank-Nicolson scheme where the finite difference representations for the spatial derivatives are averaged over two successive times, t = nAt and t = n + l)At, are frequently used because they are usually unconditionally stable algorithms. Most conservation laws lead to equations of the form... 

This system of coupled equations must be solved by integrating forward in time starting from the initial conditions mk,i(0). Eor stability, the time integration of the diffusion term can be treated implicitly using, for example, the Crank-Nicolson (CN) scheme. If we denote the volume-average moments at time t = n At by , a semi-implicit scheme for... 

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. 

Another approach involves using implicit methods (28, 30, 31) for obtaining/(y, k + 1) [e.g., the Crank-Nicolson (32), the //y implicit finite difference (FIFD) (33), and the alternating-direction implicit (ADI) (34) methods] rather than the explicit solution in (B.1.9). In implicit methods, the equations for calculation of new concentrations depend upon knowledge of the new (rather than the old) concentrations. There are a number of examples of the use of such implicit methods in electrochemical problems, such as in cyclic voltammetry (35) and SECM (36). 

Note that the term accounting for effective transport in the axial direction has been neglected in this model, for the reasons given already in Sec. 11.6. This system of nonlinear second order partial differential equations was integrated by Froment using a Crank-Nicolson procedure [76,77], to simulate a multitubular fixed bed reactor for a reaction involving yield problems. 

A finite difference scheme for discretization in time is used at this stage. In order to reduce the set of ordinary differential equations to algebraic equations, a time weighting coefficient is introduced, that allows to use several schemes explicit, implicit or the Crank-Nicolson scheme. 

The physical transport technique can either be the explicit, the implicit difference method, or the Crank-Nicolson-Scheme (see Kinzelbach, 1986, for instance). The explicit difference method, used in this study, shows the following equation for a two-dimensional, unidirectional aquifer ... 

The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

The first term omitted in the previous formula contains (At) thus, the formula is of second order correct. This approximation is the famous Crank and Nicolson equation (Crank and Nicolson 1947). 

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,. 

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,... 

The first equation conies from Eq. 12.149, the second from Eq. 12.153, and the last is the Crank-Nicolson analog for the first spatial derivative, similar in form to the second spatial derivative Eq. 12.153 derived earlier. 

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. 

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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