At each iteration step, Flint will solve the governing equations pertaining to all the relevant field variables one at a time by calling the matrix solver. The matrix solver uses a line-by-line method which is a combination of the Gauss-Seidel method combined with the TDMA (TriDiagonal Matrix Algorithm) method. This is known as the Line Gauss-Seidel (LGS) Method.
TDMA method can be used for the solution of one dimensional flow problems, where the field equations yield a tri-diagonal coefficients matrix thus making it extremely easy to solve the problem directly by a forward and backwards substitution method.
Unfortunately no such technique can be applied to the direct solution of two and three dimensional systems as the coefficients matrix no longer retains its tri-diagonal shape. Never the less, it is possible to apply the TDMA techniques to a line of cells at a time, thus reducing the problem to N*M systems of one dimensional TDMA problems. Where; N and M are the number of planes of cells in the second and third dimensions.
The gauss-seidell method comes into effect when the latest values of the field variable along the two adjacent lines of cells are used while calculating the TDMA coefficients of a line of cells, which is sandwiched between them.
The matrix solver carries out the TDMA technique on all the lines of cells along a predefined direction, known as the SWEEP DIRECTION . When the sweep direction is 1 all the cells along the I direction form a TDMA line. I.e. along each line J is constant. Similarly when the sweep direction is 2 all the cells along the J direction form a TDMA line. It is recommended that the sweep direction should be choosen to be orthogonal to the primary flow direction.

Sweeps of solver for each of the field variables define, how many times, during each iteration step, the matrix solver is called to recalculate the values of that variable. This is normally once per variable, except pressure which is five times. This is because the pressure equation is usually harder to converge.