Related papers: Stability analysis for localized solutions in PDEs…
In this paper, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth wave speed and internal localized distributed Kelvin-Voigt damping acting faraway from the boundary. Our method is based on the Finite…
We present fast, spatially dispersionless and unconditionally stable high-order solvers for Partial Differential Equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain "Fourier…
We propose a modified local discontinuous Galerkin (LDG) method for second--order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincar\'e--Friedrichs…
We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a…
We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently…
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based…
We present a new Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in which it is possible to use convex optimization to perform stability analysis with little or no conservatism. The first result gives…
In this paper we investigate the $\mathrm{L}^\infty$-stability of fully discrete approximations of abstract linear parabolic partial differential equations. The method under consideration is based on an $hp$-type discontinuous Galerkin time…
Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise…
This paper presents a control-oriented delay-based modeling approach for the exponential stabilization of a scalar neutral functional differential equation, which is then applied to the local exponential stabilization of a one-layer neural…
In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we…
We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We…
We devise an explicit method to integrate $\alpha$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the L\'evy noise, we use a…
We propose a fixed-point-based numerical framework for computing stationary states of nonlocal Fokker-Planck-type equations. Instead of discretising the differential operators directly, we reformulate the stationary problem as a nonlinear…
Firstly, we investigate Euler-Maruyama approximation for solutions of stochastic differential equations (SDEs) driven by a symmetric \alpha\ stable process under Komatsu condition for coefficients. The approximation implies naturally the…
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and…
We prove the existence of exponentially localised and time-periodic solutions in general nonlinear Hamiltonian lattice systems. Like normal modes, these localised solutions are characterised by collective oscillations at the lattice sites…
We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the…
In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the…