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We develop continuous-stage Runge-Kutta methods based on weighted orthogonalpolynomials in this paper. There are two main highlighted merits for developingsuch methods: Firstly, we do not need to study the tedious solution ofmulti-variable nonlinear algebraic equations associated with order conditions;Secondly, the well-known weighted interpolatory quadrature theory appeared inevery numerical analysis textbook can be directly and conveniently used. Byintroducing weight function, various orthogonal polynomials can be used in theconstruction of Runge-Kutta-type methods. It turns out that new families ofRunge-Kutta-type methods with special properties (e.g., symplectic, symmetricetc.) can be constructed in batches, and hopefully it may produce newapplications in numerical ordinary differential equations.

Continuous-stage Runge-Kutta methods based on weighted orthogonal polynomials