Tri-diagonal matrix algorithm

Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm... 

SOM Second Order Moments SOR Successive Over-Relaxation SUPERBEE SUPERBEE function TDM A Tri-Diagonal Matrix Algorithm TVD Total Variation Diminishing UDS Upstream Differencing Scheme... 

The above equation may be applied to each of the kmax control volumes Which comprise a column of fluid having a height equal to the local film thickness. By including the no slip velocity conditions which occur at each fluid-solid interface, it is possible to express the kmax unknown Couette velocities in terms of kmax equations. This represents a system of equations which can be solved easily and efficiently by the tri-diagonal matrix algorithm (TDNA). 

For implicit schemes, we will obtain a system of linear algebraic equations that must be solved. As mentioned in Example 8.1, one-dimensional diffusion problems generate tri-diagonal matrices, that can be solved for using the Thomas algorithm or other fast matrix routines. Equation (8.83) can be written as... 

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