Discrete Norm Research Articles

A high-order fully conservative block-centered finite difference method for the time-fractional advection–dispersion equation

Based on the weighted and shifted Grünwald–Letnikov difference operator, a new high-order block-centered finite difference method is derived for the time-fractional advection–dispersion equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δt3+h2+k2) both for solute concentration and the auxiliary flux variable are established on non-uniform rectangular grids, where Δt, h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Consistent convergence rate estimates in the grid W 2,0 2 (ω) norm for difference schemes approximating nonlinear elliptic equations with mixed derivatives and solutions from W 2,0 m (Ω), 3 < m ≤ 4

The Dirichlet boundary value problem for nonlinear elliptic equations with mixed derivatives and unbounded nonlinearity is considered. A difference scheme for solving this class of problems and an implementing iterative process are constructed and investigated. The convergence of the iterative process is rigorously analyzed. This process is used to prove the existence and uniqueness of a solution to the nonlinear difference scheme approximating the original differential problem. Consistent with the smoothness of the desired solution, convergence rate estimates in the discrete norm of W 2,0 2 (ω) for difference schemes approximating the nonlinear equation with unbounded nonlinearity are established.

Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

ABSTRACTIn this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete norm. The proposed scheme is shown to be convergent with the convergence order of in the discrete norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experiments are conducted to verify the theoretical analysis.

A block‐centered finite difference method for fractional Cattaneo equation

In this article, a block‐centered finite difference method for fractional Cattaneo equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norm with optimal order of convergence both for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Characteristic block-centered finite difference method for compressible miscible displacement in porous media

In this paper, the characteristic block-centered finite difference method is introduced and analyzed to solve compressible miscible displacement in porous media. Error estimates for the pressure, velocity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Block-Centered Finite Difference Method for Simulating Compressible Wormhole Propagation

In this paper, the block-centered finite difference method is introduced and analyzed to solve the compressible wormhole propagation. The coupled analysis approach to deal with the fully coupling relation of multivariables is employed. By this, stability analysis and error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the given scheme.

Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems

In this work, we prove optimal [Formula: see text]-approximation estimates (with [Formula: see text]) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an [Formula: see text]-boundedness result for [Formula: see text]-orthogonal projectors on polynomial subspaces. The [Formula: see text]-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these [Formula: see text]-estimates to derive novel error estimates for a Hybrid High-Order (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in [Formula: see text] for some [Formula: see text]. This kind of problems appears, e.g. in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by [Formula: see text] the meshsize, we prove that the approximation error measured in a [Formula: see text]-like discrete norm scales as [Formula: see text] when [Formula: see text] and as [Formula: see text] when [Formula: see text].

Characteristic block-centered finite difference method for simulating incompressible wormhole propagation

In this paper, the characteristic block-centered finite difference method is introduced and analyzed to solve the incompressible wormhole propagation. Error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

The Usual and Discrete Norms of Some Polynomials with Coefficients on the Unit Circle

The relation between the usual and discrete norms of polynomials was first introduced by Zygmund. In this paper, we present some behaviors of both norms on different subclasses of the polynomials with coefficients on the unit circle.

Sampling inequalities for sparse grids

Sampling inequalities play an important role in deriving error estimates for various reconstruction processes. They provide quantitative estimates on a Sobolev norm of a function, defined on a bounded domain, in terms of a discrete norm of the function's sampled values and a smoothness term which vanishes if the sampling points become dense. The density measure, which is typically used to express these estimates, is the mesh norm or Hausdorff distance of the discrete points to the bounded domain. Such a density measure intrinsically suffers from the curse of dimension. The curse of dimension can be circumvented, at least to a certain extend, by considering additional structures. Here, we will focus on bounded mixed regularity. In this situation sparse grid constructions have been proven to overcome the curse of dimension to a certain extend. In this paper, we will concentrate on a special construction for such sparse grids, namely Smolyak's method and provide sampling inequalities for mixed regularity functions on such sparse grids in terms of the number of points in the sparse grid. Finally, we will give some applications of these sampling inequalities.

English

Quaternion algebras ((-1,b)/Q) are investigated and isomorphisms between them are described. Furthermore, the orders of these algebras are presented and the uniqueness of the discrete norm for such orders is proved.

Space–time isogeometric analysis of parabolic evolution problems

We present and analyze a new stable space–time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces.

The usual and discrete norms of some polynomials with coefficients on the unit circle

The relation between the usual and discrete norms of polynomials was first introduced by Zygmund. In this paper, we present some behaviors of both norms on different subclasses of the polynomials with coefficients on the unit circle.

Summation-by-parts operators for correction procedure via reconstruction

The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

Abstract In this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.

A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem

We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter e). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.

Framing Human Rights and the Production of Translation Legal Consciousness

This paper presents a case study on a complaint document filed by the British NGO “Survival International” against the multinational mining conglomerate Vedanta Resources for alleged human rights violations. By examining the internal and external dynamics of the text of the complaint document, this paper seeks to delineate the strategic framing of rights violations by Survival International within the context of international legal discourse. The document operates beyond the contours of international legal formalism. The complaint document functions as a polycentric legal document, as it seeks to bridge the gap between “hard” and “soft” international legal structures through the invocation of multiple discrete norm systems in a unified fashion. Most importantly, by bringing together civil-political and socioeconomic rights into a singular, indivisible human rights narrative frame, the documents serves as an example of the ways which non-state actors utilize the discursive dynamics of the polycentric global norm structures for disciplining corporate social responsibility.

A class of stable and conservative finite difference schemes for the Cahn-Hilliard equation

In this paper, we propose a class of stable finite difference schemes for the initial-boundary value problem of the Cahn-Hilliard equation. These schemes are proved to inherit the total mass conservation and energy dissipation in the discrete level. The dissipation of the total energy implies boundness of the numerical solutions in the discrete H 1 norm. This in turn implies boundedness of the numerical solutions in the maximum norm and hence the stability of the difference schemes. Unique existence of the numerical solutions is proved by the fixed-point theorem. Convergence rate of the class of finite difference schemes is proved to be O(h 2 + τ2) with time step τ and mesh size h. An efficient iterative algorithm for solving these nonlinear schemes is proposed and discussed in detail.

Numerical treatment of a quasilinear initial value problem with boundary layer

The paper deals with the singularly perturbed quasilinear initial value problem exhibiting initial layer. First the nature of solution of differential problem before presenting method for its numerical solution is discussed. The numerical solution of the problem is performed with the use of a finite-fitted difference scheme on an appropriate piecewise uniform mesh (Shishkin-type mesh). An error analysis shows that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Finally, numerical results supporting the theory are presented.