Related papers: Simple bespoke preservation of two conservation la…
We show how to descritize the Korteweg-de Vries equation in such a way as to preserve all the Lie point symmetries of the continuous differential equation. It is shown that, for a centered implicit scheme, there are at least two possible…
The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the…
We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the…
We construct stochastic multisymplectic systems by considering a stochastic extension to the variational formulation of multisymplectic partial differential equations proposed in [Hydon, {\it Proc. R. Soc. A}, 461, 1627--1637, 2005]. The…
We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs…
This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for…
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such…
We establish a version of the first Noether Theorem, according to which the (equivalence classes of) conserved quantities of given Euler-Lagrange equations in several independent variables are in one-to-one correspondence with the…
We consider the calculation of Euler--Lagrange systems of ordinary difference equations, including the difference Noether's Theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We…
The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem…
This paper deals with the numerical solution of conservation laws in the two dimensional case using a novel compact implicit time discretization that enables applications of fast algebraic solvers. We present details for the second order…
Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric…
We are interested in a class of numerical schemes for the optimization of nonlinear hyperbolic partial differential equations. We present continuous and discretized relaxation schemes for scalar, one-- conservation laws. We present…
In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of…
Discrete maximum principles in the approximation of partial differential equations are crucial for the preservation of qualitative properties of physical models. In this work we enforce the discrete maximum principle by performing a simple…
For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation…
The rotation-two-component Camassa--Holm system, which possesses strongly nonlinear coupled terms and high-order differential terms, tends to have continuous nonsmooth solitary wave solutions, such as peakons, stumpons, composite waves and…
There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for…
We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces,…
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference…