The Tri-Diagonal Matrix Algorithm

For convenience in presenting the TDMA algorithm, a somewhat different nomenclature will be used [164], Suppose the grid points were numbered 1,2,3,. Af, with points 1 and N denoting the boundary points. Consider a system of equations that has a tri-diagonal form  

These equations can be solved by forward elimination and back-substitution. In the forward elimination process we seek a relation  

For the back-substitution we use the general form of the recurrence relationship (12.535). 

The iterative procedure is considered to give a converged solution if the absolute normalized residuals for all the variables as well as the mass source b of the pressure correction equation are less than a prescribed small value, denoting the convergence criterion. The absolute normalized residual is defined as  

The Fin denotes the total inflow of the property tp into the calculation domain and Min,mass represents the total inflow of mass. The prescribed small threshold value used to define the convergence criterion is problem dependent and may vary with grid resolution. Nevertheless, the iteration is generally aborted when the normalized residuals for all the variables fall below 10 .  

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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... 

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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