Related papers: Long time error analysis of the fourth-order compa…
We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to…
We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level…
In this paper, we consider the long time behavior of solution to the quadratic gauge invariant nonlinear Klein-Gordon equation (NLKG) in two space dimensions. For a given asymptotic profile, we construct a solution to (NLKG) which converges…
We prove scattering for the radial nonlinear Klein-Gordon equation $ \partial_{tt} u - \Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) \in H^{s} \times H^{s-1} $, $ 1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4…
We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and…
The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We build up a…
We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th…
This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green…
A scheme stemming from the use of pseudospectral approximations to spatial derivatives followed by a time integrator based on trigonometric polynomials is proposed for the numerical solutions of the coupled nonlinear Klein--Gordon…
In this paper, by introducing two temporal-derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The…
We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at…
We extend the three-dimensional noncommutative relations of the positions and momenta operators to those in the four dimension. Using the Bopp shift technique, we give the Heisenberg representation of these noncommutative algebras and endow…
In this paper we consider the real-valued mass-critical nonlinear Klein-Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the…
In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for…
This paper provides the first provable $\mathcal{O}(N \log N)$ algorithms for the linear system arising from the direct finite element discretization of the fourth-order equation with different boundary conditions on unstructured grids of…
We consider the further development of the formalism of the estimates of higher-order perturbative corrections in the Euclidean region, which is based on the application of the scheme-invariant methods, namely the principle of minimal…
We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ in the nonrelativistic limit, namely as the speed of light $c$ tends to infinity. In particular, we consider an order-$r$ normalized approximation of NLKG (which…
We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time and compact in space (three-point in each space direction) 4th order…
We provide a posteriori error estimates in the $L^\infty(L^2)-$norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schr\"odinger equations up to the critical exponent. In particular for the…
In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions in an open sphere around the origin, where the internal and external damping…