Related papers: Numerical realization of the variational method fo…
We propose a novel approach to the numerical simulation of thin film flows, based on the lattice Boltzmann method. We outline the basic features of the method, show in which limits the expected thin film equations are recovered and perform…
A modified perturbation theory in the strength of the nonlinear term is used to solve the Nonlinear Schroedinger Equation with a random potential. It is demonstrated that in some cases it is more efficient than other methods. Moreover we…
Spurious numerical mixing is a frequent phenomenon in ocean models. In this paper, we present an efficient and robust methodology that defines the vertical grid motion so that this mixing is reduced. This motion is defined as the solution…
We present a new method for the approximate solution of the strongly coupled, nonlinear stress-diffusion problem that appears when modeling hydrogen transport in metals. The most salient feature of the proposed approximation is that it is…
The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schroedinger equation including fractional one- or two-dimensional diffraction…
We investigate in this work the validity of linear stochastic models for nonlinear dynamical systems. We exploit as our basic tool a previously proposed Rayleigh-Ritz approximation for the effective action of nonlinear dynamical systems…
Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrowband in frequency but not necessarily with narrow angular distributions the developed asymptotic…
We present a novel approximation method that can predict the number of solitons asymptotically appearing under arbitrary rapidly decreasing initial wave packets. The number of solitons can be estimated without integration of the original…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
We develop a general form of the Ritz method for trial functions that do not satisfy the essential boundary conditions. The idea is to treat the latter as variational constraints and remove them using the Lagrange multipliers. In…
It is known that the variational methods are the most powerful tool for studying the Coulomb three-body bound state problem. However, they often suffer from loss of stability when the number of basis functions increases. This problem can be…
We derive novel guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable bending stiffness. While the standard finite element Rayleigh-Ritz method automatically yields upper bounds, we obtain lower bounds by…
In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions…
The paper outlines novel variational technique for finding microstructures of optimal multimaterial composites, bounds of composites properties, and multimaterial optimal designs. The translation method that is used for the exact…
We study dynamical reshaping of polychromatic beams due to collective nonlinear self-action of multiple-frequency components in periodic photonic lattices and predict the formation of polychromatic discrete solitons facilitated by…
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian…
Recently, Galley [Phys. Rev. Lett. {\bf 110}, 174301 (2013)] proposed an initial value problem formulation of Hamilton's principle applied to non-conservative systems. Here, we explore this formulation for complex partial differential…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such…
We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the…