This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs). To avoid solving the nonlinear system, a predictor-corrector scheme is proposed. Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

A meshless method using the radial basis functions for numerical solution of the regularized long wave equation

On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type

https://hal.archives-ouvertes.fr/hal-00783827/document

An adaptation of Nitscheʼs method to the Tresca friction problem

Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method

A finite element approximation of the Stokes equations under a certain nonlinear boundary condition, namely, the slip boundary condition of friction type, is considered. We propose an approximate problem formulated by a variational inequality, prove an existence and uniqueness result, present an error estimate, and discuss a numerical realization using an iterative Uzawa-type method. Several numerical examples are provided to support our theoretical results.

On a finite element approximation of the Stokes equations under a slip boundary condition of the friction type

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

https://www.onlinelibrary.wiley.com/doi/pdf/10.1002/num.20256

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

In this paper, we present and study a mixed variational method in order to approximate, with the finite element method, a Stokes problem with Tresca friction boundary conditions. These non-linear boundary conditions arise in the modeling of mold filling process by polymer melt, which can slip on a solid wall. The mixed formulation is based on a dualization of the non-differentiable term which define the slip conditions. Existence and uniqueness of both continuous and discrete solutions of these problems is guaranteed by means of continuous and discrete inf-sup conditions that are proved. Velocity and pressure are approximated by P1 bubble-P1 finite element and piecewise linear elements are used to discretize the Lagrange multiplier associated to the shear stress on the friction boundary. Optimal a priori error estimates are derived using classical tools of finite element analysis and two uncoupled discrete inf-sup conditions for the pressure and the Lagrange multiplier associated to the fluid shear stress.

/pdf/error-estimates-for-stokes-problem-with-tresca-friction-2hh5bl21bd.pdf

Error estimates for stokes problem with tresca friction conditions

Mixed formulation for Stokes problem with Tresca friction

In this article, a fully discrete Galerkin scheme based on a nonlinear Crank–Nicolson method to approximate the solution of the DGRLW equation is constructed. Some a priori bounds are proved as well as error estimates. Then, a linearized modification scheme by an extrapolation method is discussed. The two schemes are time second order convergence. The last part is devoted to some numerical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A fully Galerkin method for the damped generalized regularized long‐wave (DGRLW) equation

A finite difference scheme is derived for the initial-boundary problem for the nonlinear equation system 

$$\frac{\partial u}{\partial t}=A\frac{\partial^{2}u}{\partial x^{2}}+f(u),$$

where A is a complex diagonal matrix, f is a complex vector function. The stability and convergence in discrete L∞-norm of proposed Crank-Nicolson type finite difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed. Some numerical experiments have been conducted in order to validate the theoretical results.

/pdf/stability-and-convergence-of-difference-scheme-for-nonlinear-3n49uxeysg.pdf

Mekki Ayadi

Papers

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

Error estimates for stokes problem with tresca friction conditions

Mixed formulation for Stokes problem with Tresca friction

A fully Galerkin method for the damped generalized regularized long‐wave (DGRLW) equation

Stability and convergence of difference scheme for nonlinear evolutionary type equations