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A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial…
We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique…
We consider a free boundary model of epithelial cell migration with logistic growth and nonlinear diffusion induced by mechanical interactions. Using numerical simulations, phase plane and perturbation analysis, we find and analyse…
In this paper we introduce a method for solving linear and nonlinear scattering problems for wave equations using a new hybrid approach. This new approach consists of a reformulation of the governing equations into a form that can be solved…
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant…
We develop a model for steady, laminar boundary layers over small-scale textured surfaces. Although the texture is small relative to the boundary-layer thickness, it modifies the flow via a slip length. We use matched asymptotic expansions…
We derive and analyze well-posed, energy- and entropy-stable boundary conditions (BCs) for the two-dimensional linear and nonlinear rotating shallow water equations (RSWE) in vector invariant form. The focus of the study is on subcritical…
Numerical simulation of wave propagation in an infinite medium is made possible by surrounding a finite region by a perfectly matched layer (PML). Using this approach a generalized three-dimensional (3D) formulation is proposed for…
We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave…
A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously…
It is demonstrated the existence of multidimensional matter-wave solitons in a crossed optical lattice (OL) with linear OL in the $x-$direction and nonlinear OL (NOL) in the $y-$direction, where the NOL can be generated by a periodic…
We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force…
We present an arbitrary-order spectral element method for general-purpose simulation of non-overturning water waves, described by fully nonlinear potential theory. The method can be viewed as a high-order extension of the classical finite…
We describe the results of the two methods we developed to calculate the stationary nonlinear solutions in one-dimensional plasmonic slot waveguides made of a finite-thickness nonlinear dielectric core surrounded by metal regions. These two…
This paper presents a fast iterative solver for Lippmann-Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner and a domain…
The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the…
In anisotropic media, the standard perfectly matched layer (PML) technique suffers irrevocable instability in terminating the unbounded problem domains. It remains an open question whether a stable PML-like absorbing boundary condition…
In this paper, we are concerned with stable solutions , possibly unbounded and sign-changing, of some semi-linear elliptic problem with mixed nonlinear boundary conditions. We establish the nonexistence of stable solutions, the main methods…
We analyze and develop numerical methods for time-harmonic wave scattering in metallic waveguide structures of infinite extent. We show that radiation boundary conditions formulated via projectors onto outgoing modes determine the…
This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier-Stokes-Poisson system in the half line $\mathbb{R}_{+}=:(0,+\infty)$. On account of the…