A note on continuous-stage Runge-Kutta methods

We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK) methods, can give us a new perspective on RK discretization and it may enlarge the application of RK approximation theory in modern mathematics and engineering fields. A highlighted advantage of investigation of csRK methods is that we do not need to study the tedious solution of multi-variable nonlinear algebraic equations stemming from order conditions. In this note, we will discuss and promote the recently-developed csRK theory. In particular, we will place emphasis on structure-preserving algorithms including symplectic methods, symmetric methods and energy-preserving methods which play a central role in the field of geometric numerical integration

Two types of variational integrators and their equivalence

In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods

Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported

An extended framework of continuous-stage Runge-Kutta methods

We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal polynomials (e.g., Jacobi polynomials, Laguerre polynomials, Hermite polynomials etc.) can be used in the construction of Runge-Kutta-type methods. Particularly, families of Runge-Kutta-type methods with geometric properties can be constructed in this new framework. As examples, some new symplectic integrators by using Legendre polynomials, Laguerre polynomials and Hermite polynomials are constructed.Comment: arXiv admin note: text overlap with arXiv:1806.0338

Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods

As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper, we are concerned with the energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms of the Butcher coefficients for ensuring the energy preservation, symmetry and quadratic-Casimir preservation respectively are presented. With the presented condition and in use of orthogonal expansion techniques, the construction of energy-preserving integrators is examined. A new class of energy-preserving integrators which is symmetric and of order $2m$ is constructed. Some numerical results are reported to verify our theoretical analysis and show the effectiveness of our new methods

Chebyshev symplectic methods based on continuous-stage Runge-Kutta methods

We develop Chebyshev symplectic methods based on Chebyshev orthogonal polynomials of the first and second kind separately in this paper. Such type of symplectic methods can be conveniently constructed with the newly-built theory of weighted continuous-stage Runge-Kutta methods. A few numerical experiments are well performed to verify the efficiency of our new methods.Comment: arXiv admin note: text overlap with arXiv:1805.0995

Energy-preserving continuous-stage Runge-Kutta-Nystr\"om methods

Many practical problems can be described by second-order system $\ddot{q}=-M\nabla U(q)$, in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular momentum, etc. However, conventional numerical integrators for such systems will fail to preserve any of these quantities which may lead to qualitatively incorrect numerical solutions. This paper is concerned with the development of energy-preserving continuous-stage Runge-Kutta-Nystr\"om (csRKN) methods for solving second-order systems. Sufficient conditions for csRKN methods to be energy-preserving are presented and it is proved that all the energy-preserving csRKN methods satisfying these sufficient conditions can be essentially induced by energy-preserving continuous-stage partitioned Runge-Kutta methods. Some illustrative examples are given and relevant numerical results are reported

Symplectic integration with Jacobi polynomials

In this paper, we study symplectic integration of canonical Hamiltonian systems with Jacobi polynomials. The relevant theoretical results of continuous-stage Runge-Kutta methods are revisited firstly and then symplectic methods with Jacobi polynomials will be established. A few numerical experiments are well performed to verify the efficiency of our new methods.Comment: arXiv admin note: text overlap with arXiv:1805.1123

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

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