Preserving poisson structure and orthogonality in numerical integration of differential equations

Preserving Poisson Structure and Orthogonality in Numerical Integration of Differential Equations

We consider the numerical integration of two types of systems of differential equations. We first consider Hamiltonian systems of differential equations with a Poisson structure. We show that symplectic Runge-Kutta methods preserve this structure when the Poisson tensor is constant. Using nonlinear changes of coordinates this structure can also be preserved for non-constant Poisson tensors, as ...

Numerical solution and simulation of random differential equations with Wiener and compound Poisson Processes

Ordinary differential equations(ODEs) with stochastic processes in their vector field, have lots of applications in science and engineering. The main purpose of this article is to investigate the numerical methods for ODEs with Wiener and Compound Poisson processes in more than one dimension. Ordinary differential equations with Ito diffusion which is a solution of an Ito stochastic differentia...

Multiple Orthogonality and Applications in Numerical Integration

In this paper a brief survey of multiple orthogonal polynomials defined using orthogonality conditions spread out over r different measures are given. We consider multiple orthogonal polynomials on the real line, as well as on the unit semicircle in the complex plane. Such polynomials satisfy a linear recurrence relation of order r+1, which is a generalization of the well known three-term recur...

Geometric Numerical Integration of Differential Equations

Geometric Numerical Integration of Differential Equations

What is geometric integration? ‘Geometric integration’ is the term used to describe numerical methods for computing the solution of differential equations, while preserving one or more physical/mathematical properties of the system exactly (i.e. up to round–off error)1. What properties can be preserved in this way? A first aspect of a dynamical system that is important to preserve is its phase ...