Related papers: Conjugate-symplecticity properties of Euler--Macla…
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of…
We present a new mixed finite element method for a class of parabolic equations with $p$-Laplacian and nonlinear memory. The applicability, stability and convergence of the method are studied. First, the problem is written in a mixed…
We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
The Euler scheme is up to date the most important numerical method for ordinary differential inclusions, because the use of the available higher-order methods is prohibited by their enormous complexity after spatial discretization.…
In general, the system of $2$nd-order partial differential equations made of the Euler-Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of…
In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic…
We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when…
Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on 2-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and…
We derive the dynamics of several rigid bodies of arbitrary shape in a 2-dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We adopt the idea of Vankerschaver et al. (2009) to derive the…
We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…
The structure of symplectic integrators up to fourth-order can be completely and analytical understood when the factorization (split) coefficents are related linearly but with a uniform nonlinear proportional factor. The analytic form of…
Special exotic class of dynamical systems~ -- the implicit maps~ -- is considered. Such maps, particularly, can appear as a result of using of implicit and semi-implicit iterative numerical methods. In the present work we propose the…
In this paper we will study some interesting properties of modifications of the Euler-Poincar\'e equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism.…
This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the…
We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow…
We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step…
We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as…
We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many…
A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals…