Solving the Crank-Nicolson system

The system of equations 5.32 is a tridiagonal matrix equation and could be solved by the subroutines commonly available at computer installations. The number of variables must be kept down to m and this is achieved by noting that is equal, at all times, to the bulk 

Start with the last equation in system 5.32 and move the known to 

This process can be carried out repeatedly, going backwards, evaluating all a and b by the recursive relations 

Assuming now that we have c, the boundary value (more on this in Sect. 5.2.2), we can use this to start a recursive forward scan, which yields all c explicitly for all i (i = l..m). 

All this is not too bad and is most conveniently programmed as a subroutine with input parameters the old c values (including Cq) and the new values c and along with the obvious parameters such as X and 

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Assume that the first step of the procedure for solving the Crank-Nicolson system 5.32 has been carried out that is, that system 5.42 is established and thus all a and b are known. Combining Eq. 5.50 with the first equation of the system 5.42,... 

TO SOLVE THE CRANK-NICOLSON SYSTEM, BY THE BACKWARDS/FORWARDS... 

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... 

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... 

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. 

The computational way to solve the problem of diffusion by the Crank-Nicolson method is quite different from that followed in the explicit schemes. Once the old concentration profile is known, three unknowns are still present in the equation written above. If n+1 points are required to describe the concentration profile in the diffusion layer, n equations similar to that written above will constitute a system with n+2 unknowns (aq, aj, 2 %1 /+1 n+l)- boundary conditions... 

In the Crank-Nicolson technique we have an extra round-off source. Recall (Sect. 5.2) that, in solving the system of equations for CN, we recursively generate an array of a and b coefficients and then, having determined c, recursively generate all other c. This will often involve several hundred concentration points or something like 100-1000 recursive calculations. From the simple calculation above, this will... 

SOLVES THE I-DEPENDENT CRANK-NICOLSON SYSTEM, FOR UNEQUAL INTERVALS... 

TO SOLVE THE UNEQUAL-INTERVALS CRANK-NICOLSON SYSTEM, WITH 7-POINT... 

Equation (4.84), can now be solved using the Crank-Nicolson [19, 20] method by transforming to a logarithm time and space coordinate system, with nx = Ux and In T = Fr to make the solution tractable. Rewriting Eq. (4.84) in the logarithmic scale then gives... 

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. 

It was soon realised that at least unequal intervals, crowded closely around the UMDE edge, might help with accuracy, and Heinze was the first to use these in 1986 [300], as well as Bard and coworkers [71] in the same year. Taylor followed in 1990 [545]. Real Crank-Nicolson was used in 1996 [138], in a brute force manner, meaning that the linear system was simply solved by LU decomposition, ignoring the sparse nature of the system. More on this below. The ultimate unequal intervals technique is adaptive FEM, and this too has been tried, beginning with Nann [407] and Nann and Heinze [408,409], and followed more recently by a series of papers by Harriman et al. [287,288,289, 290,291,292,293], some of which studies concern microband electrodes and recessed UMDEs. One might think that FEM would make possible the use of very few sample points in the simulation space however, as an example, Harriman et al. [292] used up to about 2000 nodes in their work. This is similar to the number of points one needs to use with conformal mapping and multi-point approximations in finite difference methods, for similar accuracy. 

Crank-Nicolson 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. 

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,... 

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