Pentadiagonal system

Extension to the other Less trivial cases appears straightforward. Most boundary conditions will put up to 2n elements into the first and second rows. For those cases involving coupled equations, the rows after the second will contain five elements. It can be seen from the first row of (6.55), that there needs to be a zero inserted after the C 0 t, for the nonexistent C R,i-1. but not a similar insertion after the last element, or a total of five. Thus, if one eliminates the excess elements in the first two rows, one can then use a solver for pentadiagonal systems, which is also quite feasible. 

An obvious extension of the above method to higher order formulas is to use more points in the approximation of the spatial derivatives. These ideas lead to the well-known five-point discretization of fourth order. Unfortunately, for implicit integration a block-pentadiagonal system has to be solved now. In addition, the boundaries need a special treatment that leads to an even larger bandwidth of the matrices or to a decrease of the approximation order at the boundaries. 

A simple preliminary elimination step in the equation system is necessary to reduce the Jacobian matrix to pentadiagonal form (that is, to remove the term ) ... 

These solutions are rather formal statements, and are rarely used as such, because the matrices involved are almost always either tridiagonal or pentadiagonal, making such direct solutions wasteful. It has been done in some cases [31, 32], without any attempt at optimisation. It is possible to use solution methods that recognise the sparse nature of these systems and many professional program packages are available. One of these will be mentioned below. For methods for pdes corresponding to BI, trapezium and BDF, there are more efficient procedures for the solution, to be described in a later chapter. 

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