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Introduction

## Publications

Publications (63)

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations. The PINNs have shown impressive performance in solving various differe...

We prove the complete and partial phase synchronization for the Kuramoto oscillators with distributed time-delays and inertia effect. Our results assert that the Kuramoto models incorporated with a small variation of distributed time-delays and inertia effect still exhibit synchronization. This shows the robustness of the synchronization phenomena...

We implement new semi-analytic shooting methods for the stationary viscous Burgers’ equation by modifying the classical time differencing methods. When the viscosity is small, a very stiff boundary layer appears and this boundary layer causes significant difficulties to approximate the solution for Burgers’ equation. To overcome this issue and impr...

We implement our new semi‐analytic time differencing methods, applied to singularly perturbed non‐linear initial value problems. It is well‐known that, concerning the singularly perturbed initial problem, a very stiff layer, called initial layer, appears when the perturbation parameter is small, and this stiff initial layer causes significant diffi...

Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, uϵ, as a viscosity parameter ε tends to zero. The inviscid model, i.e. when ϵ=0, possesses the structure of scalar hyperbolic conservation laws...

In this article, we construct the approximate solutions to the Euler-Poisson system in an annular domain, that arises in the study of dynamics of plasmas. Due to a small parameter (proportional to the square of the Debye length) multiplied to the Laplacian operator, together with unmatched boundary conditions, we find that the solutions exhibit sha...

We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of...

We investigate viscous boundary layers of the plane-parallel flow, governed by the stationary Navier–Stokes equations under a certain symmetry. Following the analysis in Gie et al. (Annales de l’Institut Henri Poincaré C. Analyse Non Linéaire, 2018), we first construct the so-called corrector, which is an analytic approximation of the velocity vect...

We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complet...

Following the approach in Gie and Temam(2010) and Gie and Temam(2015), we construct the Finite Volume (FV) approximations of a class of elliptic equations and perform numerical computations where a 2D domain is discretized by convex quadrilateral meshes. The FV method with Taylor Series Expansion Scheme (TSES), which is properly adjusted from a ver...

In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interaction-corner layers. The boundary layer correctors are determined by a phenomenological study r...

This book is a fairly unique resource regarding the rigorous mathematical treatment of boundary layer problems. The explicit methodology developed in this book extends in many different directions the concept of correctors initially introduced by J. L. Lions, and in particular the lower- and higher-order error estimates of asymptotic expansions are...

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix)...

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerical...

In fluid dynamics, we often study the flow of liquids and gases inside a region enclosed by a rigid boundary or around such a region. Some interesting applications in this field include analyzing, e.g., the motion of air around airplanes or automobiles to increase the efficiency of motion, the flow of atmosphere and oceans to predict the weather, a...

In this chapter, we will study the extension of the results on singular perturbations to higher dimensions. In dimension d ≥ 2, new problems arise related to the geometry of the domain, and in particular whether the domain is sufficiently regular or it has corners. Even if the domain is smooth, some boundary layers occur which are due to the curvat...

The study of Singular Perturbation Problems (SPP) in dimension one has a great importance since the boundary layer problems are generally one-dimensional problems in the direction normal to the boundary and, as we will see throughout the chapters of this book, many higher dimensional problems (in terms of singular perturbations) will be reduced to...

Following the approach introduced in [JT14a, JT11, JT12], we consider in this chapter the convection-diffusion equations in a circular domain where two characteristic points appear. The singular behaviors may occur at these points depending on the behavior of the given data, that is the domain (unit circle D), and f; see (5.1). As explained below,...

In this chapter and in Chapter 5, we investigate the boundary layers of convection-diffusion equations in space dimension one or two, and discuss additional issues to further develop the analysis performed in the previous Chapters 1 and 2

In this chapter, we present some recent progresses, which are based on [GJT16], about the boundary layer analysis in a domain enclosed by a curved boundary.

The Navier-Stokes equations appear as a singular perturbation of the Euler equations in which the small parameter ɛ is the viscosity or inverse of the Reynolds number. In many cases the convergence of the solutions of the Navier-Stokes equations to those of the Euler equations remains an outstanding open problem of mathematical physics. The result...

We investigate the synchronized collective behavior of the Kuramoto oscillators with inertia effect. Both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view. As an application of our general theory, we show the unconditional frequency synchronization for the three-nonidentical-oscillator...

Singularly perturbed stochastic (and deterministic) nonlinear reaction–diffusion equations are considered. We first study the governing problem posed in the channel domain with lateral periodicity and extend the results to general smooth domains. Introducing corrector functions, which correct the boundary values discrepancies, we are able to develo...

In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the sma...

We propose a boundary layer analysis which fits a domain with corners. In particular, we consider nonlinear reaction–diffusion problems posed in a polygonal domain having a small diffusive coefficient . We present the full analysis of the singular behaviours at any orders with respect to the parameter where we use a systematic nonlinear treatment i...

We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the global convergence of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light o...

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates exp...

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions f...

We consider the Dirichlet boundary value problem for the viscous Burgers' equation with a time periodic force on a one dimensional finite interval. Under the boundedness assumption on the external force, we prove the existence of the time-periodic solution by using the Galerkin method and Schaefer's fixed point theorem. Furthermore, we show that th...

The article is devoted to prove the existence and regularity of the solutions of the 3D inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work (Hamouda et al. in Discret Contin Dyn Syst Ser S 6(2):401–422, 2013) which is concerned with the boundary layers generated by the correspond...

In this article we study the boundary layers for the viscous Linearized Primitive Equations (LPEs) when the viscosity is small. The LPEs are considered here in a cube. Besides the usual boundary layers that we analyze here too, corner layers due to the interaction between the different boundary layers are also studied.

In this article, we consider a singularly perturbed nonlinear reaction-diffusion equation whose solutions display thin boundary layers near the boundary of the domain. We fully analyse the singular behaviours of the solutions at any given order with respect to the small parameter ε, with suitable asymptotic expansions consisting of the outer soluti...

In this article, we review recent progresses in boundary layer analysis of some singular perturbation problems. Using the techniques of differential geometry, an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundary is constructed and validated in some suitable functional spaces. In addition, we investigate t...

We study singularly perturbed time dependent convection–diffusion equations in a circular domain. Considering suitable compatibility conditions, we present convergence results and provide as well approximation schemes and error estimates. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct,...

A new semi-analytical time differencing is applied to spectral methods for partial differential equations which involve higher spatial derivatives. This is developed in Jung and Nguyen (J Sci Comput (2015) 63:355-373) based on the classical integrating factor (IF) and exponential time differencing (ETD) methods. The basic idea is approximating anal...

A new adaptive weighted essentially non-oscillatory WENO-$\theta$ scheme in
the context of finite difference is proposed. Depending on the smoothness of
the large stencil used in the reconstruction of the numerical flux, a parameter
$\theta$ is set adaptively to switch the scheme between a 5th-order upwind and
6th-order central discretization. A ne...

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary laye...

A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems. The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter \(\varepsilon \)...

The singularly perturbed problems with a turning point were discussed in [21]. The case where the limit problem is compatible with the given data was fully resolved. However, with limited compatibility conditions on the data, the asymptotic expansions were constructed only up to the order of the level of compatibilities. In this paper, using a smoo...

A hybrid operator splitting method is developed for computations of two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into the random space using the polynomial chaos (PC) projection method, the deterministic and random parts of the solutions are solved separately. There are...

The goal of this article is to study the boundary layers of reaction–diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid asymptotic expansions at any order and devise boundary layer elements to be conveniently used in the fi...

We give an asymptotic expansion, with respect to the viscosity, which is considered here to be small, of the solutions of the 3D linearized primitive equations (PEs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice...

We study the asymptotic behavior at small diffusivity of the solutions,
, to a convection-diffusion equation in a rectangular domain . The diffusive
equation is supplemented with a Dirichlet boundary condition, which is smooth
along the edges and continuous at the corners. To resolve the discrepancy, on , between and the corresponding limit solutio...

We study the asymptotic behavior, at small viscosity ε, of the Navier-Stokes equations in a 2D curved domain. The Navier-Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construc...

We aim to study finite volume approximations which approximate the solutions of convection-dominated problems possessing the so-called interior transition layers. The stiffness of such problems is due to a small parameter multiplied to the highest order derivative which introduces various transition layers at the boundaries and at the interior poin...

We study the boundary layers and singularities generated by a convection-diffusion equation in a circle with noncompatible data. More precisely, the boundary of the circle has two characteristic points where the boundary conditions and the external data f are not compatible. Very complex singular behaviors are observed, and we analyze them systemat...

In this article we aim to study the boundary layer generated by a convection–diffusion equation in a circle. In the model problem that we consider two characteristic points appear. To the best of our knowledge such boundary layer problems have not been studied in a systematic way yet and we indeed know that very complex situations can occur. In the...

We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and determinis...

Continuing an earlier work in space dimension one, the aim of this article is to present, in space dimension two, a novel method to approximate stiff problems using a combination of (relatively easy) analytical methods and finite volume discretization. The stiffness is caused by a small parameter in the equation which introduces ordinary and corner...

In this work, we present a novel method to approximate stiff problems using a finite volume (FV) discretization. The stiffness
is caused by the existence of a small parameter in the equation which introduces a boundary layer. The proposed semi-analytic
method consists in adding in the finite volume space the boundary layer corrector which encompass...

We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random
media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different
media interface randomly in space. We numerically compute the probability distribution and density for...

In this presentation, we are interested in the behavior of the solution
of the primitive equations (PEs) when the viscosity is very small. The
PEs are the central equations for the large scale motion of the ocean
and the atmosphere. Beside the general interest of determining the
behavior at small (realistic) viscosity of the PEs, there is another
m...

We establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension 2. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of...

Our aim in this article is to study the interaction of boundary layers and corner singularities in the context of singularly perturbed convection-diffusion equations. For the problems under consideration, we determine a simplified form of the corner singularities and show how to use it for the numerical approximation of such problems in the context...

It has been demonstrated that the ordinary boundary layer ele-ments play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagona...

In this article, we discuss reaction-diffusion problems which produce ordinary boundary layers and elliptic corner layers. Using the classical polynomial Q1-finite elements spaces enriched with the so-called boundary layer elements which absorb the singularities due to the boundary and corner layers we are able to attain high numerical accuracies....

Turning points occur in many circumstances in fluid mechanics. When the viscosity is small, very complex phenomena can occur near turning points, which are not yet well understood. A model problem, corresponding to a linear convection-diffusion equation (e.g., suitable linearization of the Navier-Stokes or Bénard convection equations) is considered...

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, wh...

Our aim in this article is to show how one can improve the numerical solution of singularly perturbed problems involving boundary layers. By incorporating the structures of boundary layers into finite element spaces, when this structure is available, we can improve the accuracy of approximate solutions and this results ill significant simplificatio...

Our aim in this article is to show how one can improve the numerical solution of singularity perturbed problems involving boundary layers. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. In this article we discuss convection-diffu...

In this article, we consider a convection-diffusion equation with a small diffusion coefficient . It is a version of a simplified The Navier-Stokes equation. By giving the convection coefficient specific conditions in this equation, we can observe a kind of transition layer, called a turning point. It can be observed when flows of two different dir...

Thesis (PhD) - Indiana University, Mathematics, 2006 We demonstrate how one can improve the numerical solution of singularly perturbed problems involving multiple boundary layers by using a combination of analytic and numerical tools. Incorporating the so-called boundary layer elements (BLE), which absorb the singularities due to the boundary layer...