A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

This paper details our note [6] and it is an extension of our previous works  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives.     

A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems

The finite volume element (FVE) methods used currently are essentially low order and unsymmetric. In this paper, by biquadratic elements and multistep methods, we construct a second order FVE scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes. To overcome the numerical oscillation, we discretize the problem along its characteristic direction. The choice of alternating direction strategy is critical in this paper, which guarantees the high efficiency and symmetry of the discrete scheme. Optimal order error estimates in H¹$H^1$-norm are derived and a numerical example is given at the end to confirm the usefulness of the method.     

Existence and non-existence of positive solutions of BVPs for fractional order elastic beam equations with a non-Caratheodory nonlinearity

In this article, the existence and non-existence results on positive solutions of two classes of boundary value problems for nonlinear singular fractional order elastic beam equations is established. Here f depends on x  x^′$x^{\prime }$ and x^″,f$x^{″}\text{,}f$ may be singular at t=0$t=0$ and t=1$t=1$ and f may be a non-Caratheodory function. The analysis relies on the well known Schauder’s fixed point theorem. By applying iterative techniques, results on the existence of positive solutions are obtained and the iterative scheme which starts off with zero function for approximating the solution is established. The iterative scheme obtained is very useful and feasible for computational purpose. Examples and their numerical simulation are presented to illustrate the main theorems. A conclusion section is also given at the end of this paper.     

Note on a Korovkin-type theorem

A new Korovkin-type theorem and its converse theorem are established. We compare our direct result with the Korovkin-type theorem given by Wang [15]. As applications we obtain quantitative estimates for q  -Bernstein-type operators which preserve the functions _e0(x)=1$e_0(x)=1$ and _ej(x)=x^j$e_j(x)=x^j$.     

Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

We study a time dependent eddy current equation for the magnetic field H$\mathit{H}$ accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E$\mathit{E}$ and magnetic H$\mathit{H}$ fields obeys the following power law ν×E=ν×(|H×ν|^α-1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times (|\mathit{H}\times \mathit{\nu }{|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary Γ$\Gamma$ and the initial data _H0${\mathit{H}}_0$. We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.     

Conservative difference methods for the Klein–Gordon–Zakharov equations

Firstly an implicit conservative finite difference scheme is presented for the initial-boundary problem of the one space dimensional Klein–Gordon–Zakharov (KGZ) equations. The existence of the difference solution is proved by Leray–Schauder fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second order convergent for U   in _l∞$l_{\infty }$ norm, and for N   in _l2$l_2$ norm on the basis of the priori estimates. Then an explicit difference scheme is proposed for the KGZ equations, on the basis of priori estimates and two important inequalities about norms, convergence of the difference solutions is proved. Because it is explicit and not coupled it can be computed by a parallel method. Numerical experiments with the two schemes are done for several test cases. Computational results demonstrate that the two schemes are accurate and efficient.     

Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X   be a Banach lattice. We prove that every norm bounded uo-convergent net in X^?$X^{?}$ is w^?$w^{?}$-convergent iff X   has order continuous norm, and that every w^?$w^{?}$-convergent net in X^?$X^{?}$ is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ  -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w^?$w^{?}$-convergent sequence converges weakly/in norm.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

On the complexity of computing the logarithm and square root functions on a complex domain

The problems of computing single-valued, analytic branches of the logarithm and square root functions on a bounded, simply connected domain S   are studied. If the boundary ∂S$\partial S$ of S   is a polynomial-time computable Jordan curve, the complexity of these problems can be characterized by counting classes #P$\#P$, MP (or MidBitP  ), and ⊕P$\oplus P$: The logarithm problem is polynomial-time solvable if and only if FP=#P$\mathit{FP}=\#P$. For the square root problem, it has been shown to have the upper bound P^MP$P^{\mathit{MP}}$ and lower bound P^⊕P$P^{\oplus P}$. That is, if P=MP$P=\mathit{MP}$ then the square root problem is polynomial-time solvable, and if P≠⊕P$P\ne \oplus P$ then the square root problem is not polynomial-time solvable.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.     

Global stability of a delayed SIR epidemic model with density dependent birth and death rates

An SIR   epidemic model with density dependent birth and death rates is formulated. In our model it is assumed that the total number of the population is governed by logistic equation. The transmission of infection is assumed to be of the standard form, namely proportional to I(t-h)/N(t-h)$I(t-h)/N(t-h)$ where N(t)$N(t)$ is the total (variable) population size, I(t)$I(t)$ is the size of the infective population and a time delay h   is a fixed time during which the infectious agents develop in the vector. We consider transmission dynamics for the model. Stability of an endemic equilibrium is investigated. The stability result is stated in terms of a threshold parameter, that is, a basic reproduction number _R0$R_0$.