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

Abstract

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

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References in corpus (2)

• A characterization of energy-preserving methods and the construction of parallel integrators for Hamiltonian systems
• Symplecticity-preserving continuous-stage Runge-Kutta-Nyström methods

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• Symplectic integration with Jacobi polynomials
• Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems
• Energy-preserving continuous-stage Runge-Kutta-Nyström methods
• Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods