By Dmitri Kuzmin

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**Additional resources for A Guide to Numerical Methods for Transport Equations**

**Example text**

If this is not the case, the code is likely to contain a pernicious bug. 2 Zero Row Sums The second basic rule is: if a continuous operator produces zero when applied to a constant, so should its discrete counterpart. 13) with s ≡ 0, this equation remains valid if the solution and its boundary values are increased by an arbitrary constant. For the solution of the discrete problem to possess the same property, it is sufficient to require that the matrices C and D have zero row sums ∑ ci j = 0, j ∑ di j = 0.

The iterative process is terminated when certain stopping criteria are satisfied. As a rule, these criteria amount to monitoring the differences between two successive iterates and/or the residuals measured in a suitably chosen norm. Stopping too early gives rise to large iteration errors; stopping too late results in a waste of CPU time. Iterative solvers use a rather small amount of computer memory. Many of them do not even require that the matrix A be available. All they need is a subroutine that evaluates the residual of the linear system for a given tentative solution.

Stopping too early gives rise to large iteration errors; stopping too late results in a waste of CPU time. Iterative solvers use a rather small amount of computer memory. Many of them do not even require that the matrix A be available. All they need is a subroutine that evaluates the residual of the linear system for a given tentative solution. The most efficient iterative algorithms are based on multigrid methods [131, 345]. If properly configured, they can solve a system of N equations using as few as O(N) arithmetic operations, as compared to O(N 3 ) for direct solvers based on Gaussian elimination and O(N 2 ) for forward/backward substitution, given a precomputed LU factorization.