Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients

Linear systems with large differences between the coefficients, called “discontinuous coefficients”, often arise when physical phenomena in heterogeneous media are modeled by partial differential equations (PDEs). Such problems are usually solved by domain decomposition techniques, but these can be difficult to implement when subdomain boundaries are complicated or the grid is unstructured. It is known that for such systems, diagonal scaling can sometimes improve the eigenvalue distribution and the convergence properties of some algorithm/preconditioner combinations. However, there seems to be no study outlining both the usefulness and limitations of this approach. It is shown that L2L2-scaling of the equations is a generally useful preconditioner for such problems when the system matrices are nonsymmetric, but only when the off-diagonal elements are small to moderate. Tests were carried out on several nonsymmetric linear systems with discontinuous coefficients derived from convection–diffusion elliptic PDEs with small to moderate convection terms. It is shown that L2L2-scaling improved the eigenvalue distribution of the system matrix by reducing their concentration around the origin very significantly. Furthermore, such scaling improved the convergence properties of restarted GMRES and Bi-CGSTAB, with and without the ILU(0) preconditioner. Since ILU(0) is theoretically oblivious to diagonal scaling, these results indicate that L2L2-scaling also improves the runtime numerical stability.