Complex Mapped Matrix Methods in Hydrodynamic Stability Problems

Abstract

The ordinary differential equations governing the linear stability of inviscid flows contain singularities at real or complex points called critical latitudes, which degrade the accuracy of standard numerical schemes. However, the use of a complex mapping prior to the numerical attack offers some respite. This mapping shifts the computational domain to a contour in the complex plane to avoid the critical latitudes. Both quadratic and cubic complex maps are considered in some detail. An analytic result for choosing the optimum quadratic complex map in the case of a single critical latitude is presented. Numerical results are given for two test problems and a barotropic vortex model. A comparison is made between methods with and without these mappings. The results show that the use of complex maps can lead to remarkably accurate solutions.