Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis
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Abstract
This study presents a pioneering approach to a new type of diagonally implicit Runge-Kutta-type (DIRKT) method developed specifically for solving third-order ODEs of this type. ѵ'''=f(ȶ,ѵ,ѵ'). These methods are called RKTG and are constructed using B series and Tri-colored tree theories to derive the required order conditions. We derive two methods: DRKTG4 and DRKTG5. The first achieves fourth-order accuracy in three stages, while the second reaches fifth order in four stages. Among other things, numerical experiments show these methods are accurate and efficient for solving these problems compared with existing Runge-Kutta methods, which posed these problems as first-order systems for testing.
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