Discretisation backward difference

The restriction on the step size (2.304) due to the stability condition for the explicit difference method can be avoided by using an implicit method. This means that (2.298) is discretised at time tk+1 and the backward difference quotient is used to replace the time derivative. With... 

The Galerkin weighted residual method is employed to formulate the finite element discretisation. An implicit mid-interval backward difference algorithm is implemented to achieve temporal discretisation. With appropriate initial and boundary conditions the set of non-linear coupled governing differential equations can be solved. 

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. 

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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