Abstract
This paper presents a novel A-stable nonlinear method of order of accuracy three, based on optimization for solving the first-order initial value problems. The A-stability is proven and the consistency of order three is investigated as well. The convergence of the method is established under certain assumptions. Additionally, a second-order method is derived based on this approach, and a variable step size formulation is implemented using the second and third-order methods to enhance efficiency. Numerical examples provided in the paper demonstrate the validity of the theory and underscore the advantages of the method compared to three other A-stable methods of order three, including an implicit RK method, a Rosenbrock method, and an explicit (nonstandard) method.
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