Crank-Nicolson stability

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... 

Crank-Nicolson implicit method This method is a little more complicated but it offers high precision and unconditional stability. Let... 

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... 

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). 

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. 

Analogous stabihty analyses can be executed for the other time-discretization schemes as well. It is important to note here that although the von Neumann stability analysis yields a limiting time-step estimate to keep the roundoff errors bounded, it does not preclude the occurrence of bounded but unphysical solutions. A classical example is the Crank-Nicolson scheme, which from the von Neumann viewpoint is unconditionally stable but can give rise to bounded unphysical solutirms in case all the coefficients of the discretization equation do not happen to be of the same sign [2]. 

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... 

Bieniasz LK, 0sterby O, Britz D (1997) The effect of the discretization of the mixed boundary conditions on the numerical stability of the Crank-Nicolson algorithm of electrochemical kinetic simulations. Comput Chem 21 391 01... 

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