High-order boundary conditions for linearized shallow water equations with stratification, dispersion and advection

The two-dimensional linearized shallow water equations are considered in unbounded domains with density stratifications. Wave dispersion and advection effects are slso taken into account. The infinite domain is truncated via a rectangular artificial boundary B, and a high-order Open Boundary Condition (OBC) is imposed on B. Then the problem is solved numerically in the finite domain bounded by B. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBC's originaly proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis on the effect of stratification

A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations

International audienceThe aim of this paper is to describe concisely the recent theoretical and numerical developments concerning absorbing boundary conditions and perfectly matched layers for solving classical and relativistic quantum waves problems. The equations considered in this paper are the Schrödinger, Klein-Gordon and Dirac equations

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Energetic Boundary Element Method for accurate solution of damped waves hard scattering problems

AbstractThe paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains

Time-Dependent Attractor for the Oscillon Equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Klein-Gordon equation in an expanding background, in one space dimension with periodic boundary conditions, with a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular, time-dependent global attractor. The sections of the attractor at given times have finite fractal dimension.Comment: to appear in Discrete and Continuous Dynamical System