Related papers: A linearly implicit energy-preserving exponential …

In this paper, non-variational systems of differential equations containing small terms are considered, and a consistent approach for deriving approximate conservation laws through the introduction of approximate Lagrange multipliers is…

In this paper we accomplish the development of the fast rank-adaptive solver for tensor-structured symmetric positive definite linear systems in higher dimensions. In [arXiv:1301.6068] this problem is approached by alternating minimization…

This article focuses on an energy-conservation Galerkin finite element method (FEM) for the generalized Klein-Gordon-Zakharov (KGZ) equations. This method combines the bilinear finite element method for spatial discretization with the…

We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range…

We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…

We study the decay of the global energy for the damped Klein-Gordon equation on non-compact manifolds with finitely many cylindrical and subconic ends up to bounded perturbation. We prove that under the Geometric Control Condition, the…

We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to…

We present a new complex non-stationary particle-like solution of the non-linear Klein-Gordon equation with several spatial variables. The construction is based on reduction to an ordinary differential equation.

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible-irreversible coupling). We…

In this paper, a fully implicit Crank-Nicolson discontinuous Galerkin method is proposed for solving the Ginzburg-Landau equation. By leveraging a novel analytical technique, we rigorously establish the unique solvability of the constructed…

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast…

Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…

Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein-Gordon (NLKG), the Korteweg-deVries (KdV) and water wave equations, it is shown that the leading…

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero…

Implicit-Explicit methods have been widely used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the…

We propose an efficient approach for time integration of Klein-Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our…

This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by…

It is shown analytically that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of…

We propose a nonlinear $\sigma$-model in a curved space as a general integrable elliptic model. We construct its exact solutions and obtain energy estimates near the critical point. We consider the Pohlmeyer transformation in Euclidean…

In this paper the stationary Klein-Gordon equation is considered for the Coulomb potential in non-commutative space. The energy shift due to noncommutativeity is obtained via the perturbation theory. Furthermore, we show that the degeneracy…