10. Introduction to solution methods for linear algebraic equation systems in CFD

This lecture deals with solution methods for linear algebraic equation systems. Although direct solution methods are seldom used for this purpose in CFD, I first describe two methods because one of them serves as the basis for some iterative solution methods.
The next topic are iterative solution methods, which are almost exclusively used to solve algebraic equation systems in CFD. Because there are many of them, I first describe the principles of iterative solution approach and then present the simplest methods: Jacobi, Gauß-Seidel and over-relaxation solvers.
Line-by-line application of the direct solver for one-dimensional problems, the tri-diagonal matrix algorithm or TDMA, is described next. More complex are methods based on incomplete lower-upper factorization of the coefficient matrix. The basic algorithm is seldom used, but a version specially designed for the solution of partial-differential equations, published by Stone in 1968, is efficient and widely used. It is therefore described in detail.
Finally, iteration errors and their estimation are discussed. More details will come in lecture 12, where the performance of different solvers will be compared and analyzed in detail.
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[@digguscience]

Very relaxed explanation but easy to understand