Hybrid methods for solving higher order ordinary differential equations

In this thesis, a class of numerical integrators for solving special higher order ordinary differential equations (ODEs) is proposed. The methods are multistage and multistep in nature. This class of integrators is called ”hybrid methods”, specifically, hybrid methods for directly solving special third order ODEs denoted by HMTD and for directly solving special fourth order ODEs denoted by HMFD are proposed. B-series approach is developed and used in deriving their algebraic order conditions and analyzing the order of convergence of the methods. Using the algebraic order conditions, a class of explicit HMTD and HMFD are derived. The methods are applied to some test problems alongside some existing integrators in the literature for the purpose of validation. Results obtained show that the proposed methods in this thesis are a better alternatives. To analyze the methods further, convergence analysis is conducted via consistency and zero stability, where the methods are found to be consistent and zero stable, hence, they are convergent. Absolute stability of the methods is also investigated, where stability polynomials of the methods are presented for obtaining intervals and regions of absolute stability. Finally, a set of embedded pairs of two-step hybrid methods for solving special second order ODEs are proposed and investigated. The methods are tested on some model problems using different error tolerances. Results obtained are compared with those of existing embedded methods possessing similar properties. From the comparison, it is found that the new embedded methods possess better accuracy and efficiency.

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