Stability Theorems Concerning High Order Explicit Algorithms for the Linear Advection Equation

摘要

Analysis of the stability of an N-point explicit algorithm for the linear advection equation involves the calculation of the roots of a polynominal of degree N-1. A theorem is proved to show that if the algorithm is of the nth order (n N), then the associated stability polynomial must have a multiple root of order n+2/2 so that, in effect, the analysis need only be carried out for a polynomial of degree N - n + 2/2. This technique for reducing the complexities involved in stability analysis is illustrated for families of 4- and 5- point algorithms. Moreover in the special case of ideal algorithms (for which n = N - 1), a complete stability analysis is carried through, a theorem being proved that such algorithms are always stable in their central range but never outside.