Discretisation central difference

The time derivative is still a central difference but the spatial second derivative now leaves out the central point, substituting for it the mean of the past and future points. Thus, the discretisation is... 

Both the spatial discretisation and the choice of the type of differences (e.g. uplift differences, central differences) have a strong influence on the result. This fuzziness caused by the application of different methods is subsumed as numeric dispersion . 

The derivative is formed in the outward normal direction, and a and can be dependent on time. For the discretisation of (2.253) it is most convenient if the boundary coincides with a grid line, Fig. 2.45, as the boundary temperature which appears in (2.253) can immediately be used in the difference formula. The replacement of the derivative d d/dn by the central difference quotient requires grid points outside the body, namely the temperatures i9k or ()k,, which, in conjunction with the boundary condition, can be eliminated from the difference equations. 

A particularly accurate implicit difference method, which is always stable, has been presented by J. Crank and P. Nicolson [2.65]. In this method the temperatures at the time levels tk and tk+l are used. However the differential equation (2.236) is discretised for the time lying between these two levels tk + At/2. This makes it possible to approximate the derivative (dt>/dt)k+1 2 by means of the accurate central difference quotient... 

If we turn now to the discretisation of Pick s second law, the use of the second-order central difference approximation for the first derivative leads to... 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

Similarly, one could attempt to improve the 3C/3T discretisation (other than by using Runge-Kutta integration). In effect, the Crank-Nicolson scheme does this by specifying a central difference approximation at T+J 8T. The same can be done at T by using the Richardson (1911) formula (denoting time steps by the index k) ... 

At any point with index i, that is at X = iH, the diffusion (1.1) is discretised on the left-hand side in the Euler manner (4.4, or in other words the forward difference formula 3.1) and on the right-hand side with the central three-point approximation (3.41), giving for the iteration going from time T to the next time T + 8T,... 

In this scheme, the temporal derivative is formed by the central (second-order ) difference between the upper and lower points, the second spatial derivative being approximated as usual. This makes the discretisation at the index i in space,... 

In this woric, discretisation of both space and time derivatives was executed, based on either central finite difference (CFD) or orthogonal collocation cm finite elements (OCFE) discretisation in the spatial domain and backward finite difference (BFD) discretisation in the time domain. 

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