Related papers: On an integrable discretization of the modified Ko…
The main result of this paper is the discretization of Hamiltonian systems of the form $\ddot x = -K \nabla W(x)$, where $K$ is a constant symmetric matrix and $W\colon\mathbb{R}^n\to \mathbb{R}$ is a polynomial of degree $d\le 4$ in any…
This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two…
We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the…
Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of…
Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness…
We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of…
In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization…
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to…
We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…
We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows…
We consider time discretizations of the two-dimensional Euler equation written in vorticity form. The discretization method uses a Crouch-Grossman integrator that proceeds in two stages: first freezing the velocity vector field at the…
We show how to reduce the computational time of the practical implementation of the Raviart-Thomas mixed method for second-order elliptic problems. The implementation takes advantage of a recent result which states that certain local…
We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time…
We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at…
We prove that weak solutions obtained as limits of certain numerical space-time discretizations are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, in the space-periodic setting, we consider a full…
We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in…
A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus…
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be…
The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that…
A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting…