Comparison of shooting method and complementary functions method in linear stability problems

Solution of computationally expensive numerical problems demands hybrid and intelligent algorithms, instead of a single generalized approach. A comparative analysis of well renowned "Shooting method" with another numerical method "Complementary Functions Method" (CFM) is performed on a computationally-cumbersome boundary eigenvalue problem, namely, numerical determination of an eigenvalue (λ) in linear stability problems. Both numerical methods are used for the conversion of a governing boundary value problem into an initial value problem. Shooting method takes a direct approach, starts with the guessing of missing initial condition and lays an iterative scheme based upon the system of equations containing two variables: a missing initial condition and an eigenvalue. Conversely, complementary functions method uses a smart approach, excluding the need of finding the missing initial condition by using general properties of linear differential equations: it produces an iterative scheme of algebraic equations with a single variable, namely, an eigenvalue. Reduction in the number of variables by CFM sets the basis of this comparison between the two numerical methods. Both numerical methods are evaluated in terms of accuracy and efficiency for various cases of linear stability problems: (i) Different cases of Euler's Columns (ii) thermal buckling of a disc restrained along its edges (iii) thermal buckling of a functionally graded (FGM) disc.