A singular perturbation approach to solve Burgers–Huxley equation via monotone finite difference scheme on layer-adaptive mesh

This paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient ε. The parameter ε takes any values from the half open interval (0, 1]. At small values of the parameter ε, an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

On the C(R) stability of uncertain singularly perturbed systems

In this paper, a simple criterion for the C(R) stability of uncertain singularly perturbed systems is proposed. Such a criterion can be easily checked by some algebraic inequality. The upper bound of the singular perturbation parameter ε is also given by estimating the unique positive zero of specific function. Finally, a numerical example is provided to illustrate the main result.     

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.     

Un lemme de prolongement holomorphe et ses applications

We prove that an holomorphic function defined in the neighborhood of an algebraic knot f⁻¹(0)∩S(0,ε) in dimension ?3, can be extended along the hypersurface f⁻¹(0)∩B(0,ε). As an application we give a new proof of the Malgrange's singular Frobenius theorem.     

Justification de la théorie non linéaire de Kirchhoff–Love,comme application d'une nouvelle méthode d'inversion singulièreJustification of the nonlinear Kirchhoff–Love theory,as the application of a new singular inverse method

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant–Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff–Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán. We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615–620.     

On the Complexity of Self-Validating Numerical Integration and Approximation of Functions with Singularities

We consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass ofC^k([a, b]\Z), whereZis a finite subset of [a, b]. Using only function values or values of derivatives, we usually cannot guarantee that the costs for obtaining an error less thanεare bounded byO(ε^−1/k) and we may have much higher costs. The situation changes if we also allow “realistic” estimates of ranges of functions or derivatives on intervals as observations. A very simple algorithm now yields an error less thanεwithO(ε^−1/k)-costs and an analogous result is also obtained for uniform approximation with piecewise polynomials. In a practical implementation, estimation of ranges may be done efficiently with interval arithmetic and automatic differentiation. The cost for each such evaluation (also of ranges of derivatives) is bounded by a constant times the cost for a function evaluation. The mentioned techniques reduce the class of integrands, but still allow numerical integration of functions from a wide class withO(ε^−1/k) arithmetical operations and guaranteed precisionε.     

Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter ε, τ=o(ε), ε→0. We establish the following general framework: If there exists an a priori L^∞ bound that is uniformly with respect to ε for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L^∞ bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in L^p to the equilibrium solutions of these systems is established, provided that the relaxation time τ tends to zero faster than ε.     

Bidimensional shallow water model with polynomial dependence on depth through vorticity

In this paper, we obtain a bidimensional shallow water model with polynomial dependence on depth. With this aim, we introduce a small non-dimensional parameter ε and we study three-dimensional Euler equations in a domain depending on ε (in such a way that, when ε becomes small, the domain has small depth). Then, we use asymptotic analysis to study what happens when ε approaches to zero. Asymptotic analysis allows us to obtain a new bidimensional shallow water model that not only computes the average velocity (as the classical model does) but also provides the horizontal velocity at different depths. This represents a significant improvement over the classical model. We must also remark that we obtain the model without making assumptions about velocity or pressure behavior (only the usual ansatz in asymptotic analysis). Finally, we present some numerical results showing that the new model is able to approximate the non-constant in depth solutions to Euler equations, whereas the classical model can only obtain the average velocity.     

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation(NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 ε≤ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(ε~2) and O(1) in time and space,respectively. We begin with the conservative Crank-Nicolson finite difference(CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates.     

Boundedness of para-product operators on spaces of homogeneous type

We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H ^p (X) for 1/(1 + ε) < p ? 1, where X is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ε is the regularity exponent of the kernel of the singular integral operator T. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.     

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form , where (x,y,z)∈Tn×Rn×R2m, ε is a small parameter, and M(ω) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.     

Time scales in linear delayed differential equations

The aim of this paper is to apply and justify the so-called aggregation of variables method for reduction of a complex system of linear delayed differential equations with two time scales: slow and fast. The difference between these time scales makes a parameter ε>0 to appear in the formulation, being a mathematical problem of singular perturbations. The main result of this work consists of demonstrating that, under some hypotheses, the solution to the perturbed problem converges when ε→0 to the solution of an aggregated system whose construction is proposed.     

Two-time scales in spatially structured models of population dynamics: A semigroup approach

The aim of this work is to provide a unified approach to the treatment of a class of spatially structured population dynamics models whose evolution processes occur at two different time scales. In the setting of the C₀-semigroup theory, we will consider a general formulation of some semilinear evolution problems defined on a Banach space in which the two-time scales are represented by a parameter ε>0 small enough, that mathematically gives rise to a singular perturbation problem. Applying the so-called aggregation of variables method, a simplified model called the aggregated model is constructed. A nontrivial mathematical task consists of comparing the asymptotic behaviour of solutions to both problems when ε→₀₊, under the assumption that the aggregated model has a compact attractor. Applications of the method to a class of two-time reaction-diffusion models of spatially structured population dynamics and to models with discrete spatial structure are given.     

A smoothing and regularization Broyden-like method for nonlinear inequalities

In this paper, we consider the smoothing and regularization Broyden-like algorithm for the system of nonlinear inequalities. By constructing a new smoothing function $\phi(\mu,a)=\frac{1}{2}(a+\mu(\ln2+\ln(1+\cosh\frac{a}{\mu})))$ , the problem is approximated via a family of parameterized smooth equations H(μ,ε,x)=0. A smoothing and regularization Broyden-like algorithm with a non-monotone linear search is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The global convergence of the algorithm is established under suitable assumptions. In addition, the smoothing parameter μ and the regularization parameter ε in our algorithm are viewed as two different independent variables. Preliminary numerical results show the efficiency of the algorithm and reveal that the regularization parameter ε in our algorithm plays an important role in numerical improvement, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.