Related papers: A Numerical Approach to Scalar Nonlocal Conservati…
Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important…
A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion-reaction processes in $n\geq 1$ dimensions, where the diffusivity depends on the gradient of the flow. For this class, all…
We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the…
Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be…
We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of…
We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the…
We develop a numerical algorithm for identifying approximately conserved quantities in models perturbed away from integrability. In the long-time regime, these quantities fully determine correlation functions of local observables. Applying…
We apply the semi-discrete method, c.f. \emph{N. Halidias and I.S. Stamatiou (2016), On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method, Computational Methods in Applied…
Conservation laws are of great theoretical and practical interest. We describe a novel approach to machine learning conservation laws of finite-dimensional dynamical systems using trajectory data. It is the first such approach based on…
Any symmetry reduces a second-order differential equation to a first-order equation: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion…
Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed…
We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We…
Nonlinear conservation laws subject to random initial conditions pose fundamental problems in the evolution and interactions of shocks and rarefactions. Using a discrete set of values for the solution, we derive a hierarchy of equations in…
Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the…
The relation between symmetries and local conservation laws, known as Noether's theorem, plays an important role in modern theoretical physics. As a discrete analog of the differentiable physical system, a good numerical scheme should admit…
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…
A particle scheme for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities. Particle interaction is resolved locally, satisfying exact…
We study a nonlocal balance equation that describes the evolution of a system consisting of infinitely many identical particles those move along a deterministic dynamics and can also either disappear or give a spring. In this case, the…
In an earlier work by a subset of the present authors, the method of the so-called neural deflation was introduced towards identifying a complete set of functionally independent conservation laws of a nonlinear dynamical system. Here, we…