Analyse d'un schéma décalé pour l'approximation de systèmes hyperboliques non conservatifs

We consider a model system made of two nonlinear equations which are non conservative. A conservation law can be obtained from these equations through linear operations only, which don't modify the shock waves. A numerical scheme based on a different mesh adapted to each variable is proposed. By choosing a shifted mesh, we have un explicit Riemann solver and we can derive a finite volume scheme. We prove a priori estimates in L^∞ norm and Total Variation for the system, which lead to a strong convergence in L¹ norm towards a solution satisfying the associated conservation law.     

A three‐level linearized finite difference scheme for the camassa–holm equation

The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L_∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

A second‐order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Neumann boundary value conditions

A linearized three‐level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Neumann boundary value problem of a nonlinear parabolic system. It is proved that the difference scheme is uniquely solvable and second‐order convergent in L_∞‐norm. A numerical example is given. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 230–247, 2004     

A second‐order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Dirichlet boundary value conditions

A linearized three‐level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Dirichlet boundary value problem of the nonlinear parabolic systems. It is proved that the difference scheme is uniquely solvable and second order convergent in L_∞‐norm. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 638–652, 2003     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers equation

Numerical solutions of the Benjamin‐Bona‐Mahony‐Burgers equation in one space dimension are considered using Crank‐Nicolson‐type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L^∞‐norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin‐Bona‐Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

In this paper, we consider the initial boundary value problem of the three‐dimensional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well‐posedness of the strong solution is established for any H² initial data. An N‐dimensional logarithmic Sobolev embedding inequality, which bounds the L^∞‐norm in terms of the L^q‐norms up to a logarithm of the L^p‐norm for p > N of the first‐order derivatives, and a system version of the classic Grönwall inequality are exploited to establish the required a~priori H² estimates for global regularity.© 2016 Wiley Periodicals, Inc.     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L^∞(L²) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L^∞(L²) norm.     

Finite element approximations with quadrature for second‐order hyperbolic equations

In this article, the effect of numerical quadrature on the finite element Galerkin approximations to the solution of hyperbolic equations has been studied. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in the L^∞(H¹), L^∞(L²) norms, whereas quasi‐optimal estimate is derived in the L^∞(L^∞) norm using energy methods. The analysis in the present paper improves upon the earlier results of Baker and Dougalis [SIAM J Numer Anal 13 (1976), pp 577–598] under the minimum smoothness assumptions of Rauch [SIAM J Numer Anal 22 (1985), pp 245–249] for a purely second‐order hyperbolic equation with quadrature. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 537–559, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10022     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

An efficient numerical approach to solve Schrödinger equations with space fractional derivative

We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order in L² norm. Numerical examples are illustrated to verify the theoretical results.     

Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations

In this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L² norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l^∞(L²) and l²(H¹) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012     

A second‐order linearized difference scheme for a strongly coupled reaction‐diffusion system

This article deals with the numerical solution to some models described by the system of strongly coupled reaction–diffusion equations with the Neumann boundary value conditions. A linearized three‐level scheme is derived by the method of reduction of order. The uniquely solvability and second‐order convergence in L₂‐norm are proved by the energy method. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Single and multi‐interval Legendre spectral methods in time for parabolic equations

In this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L²‐norm. For the multi‐interval spectral method in time, we obtain the L²‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L^∞(L²) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012