Numerical methods 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. 

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. 

In all cases, whether an analytical solution exists or not, the numerical treatment is feasible. The thickness of the layer is divided into finite increments of space Ax and the increment of time At is considered [3, 4]. The Crank-Nicolson method is used for calculation [10]. 

Bieniasz LK, 0steiby O, Blitz D (1995) Numerical stability of finite difference algorithms for electrochemical kinetic simulations matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundtiry conditions. Comput Chem 19 121-136... 

Generally, the Crank-Nicolson method is used for numerical solution of heat conduction problems that are parabolic in nature. This method solves a mesh with mesh size h in x-direction and mesh size k in y-direction (time direction). It calculates the values of u at six points as shown in Figure 1.17. 

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... 

Neither analytical solution [1 or 2] successfully describes the sodium data shown on Figures 3 and 4, which indicate that sodium diffusion is both parabolic and dependent on the aqueous sodium concentration. Therefore, a numerical solution was developed (29) with boundary conditions for the range > R > 0 and > P > 1 using the Crank-Nicolson implicit method (31). The initial conditions at time, t = 0, assume that the concentration of sodium in the glass is homogeneous and equal to the analytical concentration. The mass of sodium in the aqueous solution is equal to, times the total surface area of glass. At t > 0,... 

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],... 

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. 

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). 

Since the external potentials are highly nonlinear functionals of the density fields (they cannot even be inverted analytically), the partial differential equations we have to solve numerically are in themselves highly nonlinear. We should be very careful in choosing a method, since there are apparent risks of introducing numerical errors (for instance by taking the time steps too large). The Crank-Nicolson (CN) method, that aims to solve differential equations by mixing implicit and explicit parts, is known to be rather stable for this kind of problem. We use periodic boundary conditions. 

The numerical method uses centered finite differences for spatial derivatives and time integrations are performed using the ADI method. The ADI scheme splits each time step into two and a semi-implicit Crank-Nicolson scheme is used treating implicitly the r-direction over half a time step and then the -direction over the second half. In addition, a pseudo-unsteady system is solved which includes a term d tl ldt on the left hand side of (121) and integrating forward to steady state (see Peyret and Taylor [55]). The physical domain is mapped onto a rectangular computational domain by the transformation r = 0 = Try,... 

The basic idea of the finite differences method for solving partial differential equations is to replace spatial and time derivatives by suitable approximations, then to solve numerically the resulting difference equations. In other words, instead of solving for C(x,t) with x and t, it is solved for Cjj = C(Xi, tj), where Xj s iAx, tj = jAt. Thus, the concentrations,, of the diffusing species for the location step, i, and the time step, j -H1, are calculated from the neighbouring concentrations according to the implicit Crank-Nicolson solution for the diffusion differential equation (29.2) ... 

J. Crank and P. Nicolson, A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-conducting Type, Proc. Cambridge. Philos. Soc., 43,50-67 (1947). 

Crank, J., and Nicolson, P. A practical method for numerical evaluation of solutions of partial differential equations of the heat type, Proc. Cambridge Phil. Soc. 43, 50-67... 

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