Rosenau Burgers Equation Research Articles

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

THE FRACTAL MODIFICATION OF THE ROSENAU–BURGERS EQUATION AND ITS FRACTAL VARIATIONAL PRINCIPLE

A new fractal Rosenau–Burgers equation with He’s fractal derivative is proposed in this work. Exerting the semi-inverse method, two different kinds of the generalized fractal variational principles (GFVPs) of the fractal Rosenau–Burgers equation are constructed. The GFVPs extracted in this paper are expected to bring some new inspiration for the study and application of the variational method in the fractal space.

A two-grid spectral method to study of dynamics of dense discrete systems governed by Rosenau-Burgers' equation

A numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. We show how the spectral element method and the two-grid idea can be used together. The two-grid idea is combined with the spectral element method for simulating the nonlinear high-dimensional Rosenau-Burgers' equation. The proposed technique is based on the three-level algorithm. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated and also the error analysis is discussed.

Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity

This paper studies the asymptotic behavior of a solution of dispersive shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation. The energy decay rates of the solution to the model are examined through the Fourier transform method. Moreover, we develop a pseudo-compact finite difference scheme for solving the model, study solution behavior, and confirm our theoretical results. The fundamental energy-decreasing property, which is obtained from the model of coupling the Rosenau-RLW equation with the Rosenau-Burgers equation, is derived and preserved by the present numerical scheme. The existence, uniqueness, convergence, and stability of the numerical solution are theoretically analyzed. Some numerical experiments are also conducted to demonstrate the accuracy and robustness of the present method. Finally, to verify the optimal decay rates, the numerical results are carried out at variant time scales by applying the moving boundary technique. The simulations are successfully constructed to support the theoretical results.

On the numerical solution of two-dimensional Rosenau–Burgers (RB) equation

Our purpose is to solve numerically the nonlinear evolutionary problem called Rosenau–Burgers (RB) equation. First, we have derived error estimates of semidiscrete finite element method for the approximation of the Rosenau–Burgers problem. For a second-order accuracy in time, we propose the Galerkin–Crank–Nicolson fully discrete method. Second, a linearized difference scheme for the Rosenau–Burgers equation is considered. It is proved that the proposed difference scheme is uniquely solvable, and the method is shown to be second-order convergent both in time and space in maximum norm. Finally, some numerical experiments are given to demonstrate the validity and accuracy of our linearized difference scheme.

A new implicit nonlinear discrete scheme for Rosenau–Burgers equation based on multiple integral finite volume method

In this paper, we study the initial-boundary value problem of the Rosenau–Burgers equation by the multiple integral finite volume method (MIFVM). The MIFVM can keep the original equation property very well. We propose a two-level implicit nonlinear discrete scheme, which preserves the energy decline property of the original equation. Existence and uniqueness of the numerical solution are derived. The convergence with the order of O(τ2 + h3) and unconditional stability of the numerical scheme are verified. Numerical examples demonstrate that the scheme is reliable and effective.

Numerical Methods for a Shallow Water Rosenau-Burgers Equation

We turn to study the numerical solution of a shallow water of Rosenau-Burgers equation. We propose a scheme to numerically solve this equation. A detailed analysis is carried out for the scheme, and analysis shows that the scheme is unconditionally stable; At last, we use the proposed methods to investigate the asymptotical behavior of the solutions to the shallow water wave equation.

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.

The stability and convergence of time-stepping/spectral methods with asymptotic behaviour for the Rosenau–Burgers equation

ABSTRACT This study is concerned with numerical methods for the periodic initial boundary value problem of Rosenau–Burgers equation. The main objective is to construct and analyze stable time stepping scheme for the time discretization and use a spectral method for the spatial discretization. It is proved that the combined method is unconditionally stable, and convergent in order of , where , N and m are respectively time step size, polynomial degree, and regularity in the space variable. The numerical tests carried out in the paper show that the proposed method is efficient. In addition, the asymptotic analysis of the solution is discussed by using some existing theorem.

Numerical scheme for a model of shallow water waves in [formula omitted]-dimensions

A nonlinear difference scheme is considered for the two-dimensional Rosenau–Burgers equation. Some priori estimates, existence and uniqueness of the difference solution have been shown. A second-order convergence in the uniform norm and stability are proved. Also a convergent iterative algorithm is presented. All results are obtained without any restrictions on the meshsizes. At last numerical experiments are carried out to support the theoretical claims.

CUBIC B-SPLINE FINITE ELEMENT METHOD FOR THE ROSENAU-BURGERS EQUATION

Numerical solutions of the Rosenau-Burgers equation based on the cubic B-spline finite element method are introduced. The backward Euler method is used for discretization in time, and the obtained nonlinear algebraic system is changed to a linear system by the Newton's method. We show that those methods are unconditionally stable. Two test problems are studied to demonstrate the accuracy of the proposed method. The computational results indicate that numerical solutions are in good agreement with exact solutions.

Analysis and computational method based on quadratic B-spline FEM for the Rosenau-Burgers equation

L∞-error estimates for B-spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B-spline Galerkin scheme is studied using appropriate projections. For fully discrete B-spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016

A Modified Three-Level Average Linear-Implicit Finite Difference Method for the Rosenau-Burgers Equation

We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers equation is obtained using a five-point stencil. We prove the existence and uniqueness of the numerical solution. Moreover, the convergence and stability of the numerical solution are also shown. The numerical results show that our method improves the accuracy of the solution significantly.

Local solvability and blowup of the solution of the Rosenau-Bürgers equation with different boundary conditions

We consider several models of initial boundary-value problems for the Rosenau-Burgers equation with different boundary conditions. For each of the problems, we prove the unique local solvability in the classical sense, obtain a sufficient condition for the blowup regime, and estimate the time of the solution decay. The proof is based on the well-known test-function method.

A new finite difference scheme for generalized Rosenau–Burgers equation

In this paper, the numerical solution of the initial-boundary value problem of generalized Rosenau–Burgers equation is considered. A new linear implicit finite difference scheme of two-level is proposed. And the prior estimate of the finite difference solution is obtained. The unique solvability of numerical solutions has been shown. It is proved that the finite difference scheme is convergent and stable. Numerical experiments indicate the method is efficient.

A new finite difference scheme for the Rosenau–Burgers equation

In this paper, a linear-implicit finite difference scheme is given for the initial-boundary problem of Rosenau–Burgers equation, which is convergent and unconditionally stable. The unique solvability of numerical solutions has been shown. A priori estimate and second-order convergence of the finite difference approximate solution are discussed using energy method. Numerical results demonstrate that the scheme is efficient and accurate.

Blowup of solutions of the three-dimensional Rosenau-Burgers equation

We consider the initial boundary value problem for the well-known three-dimensional Rosenau-Burgers equation in the cylinder \((0,L) \otimes \mathbb{S}\) (where \(\mathbb{S} \subset \mathbb{R}^2\)) for some boundary conditions. Using the test-function method, we obtain the result on the blowup of solutions of this initial boundary value problem during a finite time. This is one of the first results in the “blowup” direction for this equation.

Average implicit linear difference scheme for generalized Rosenau–Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau–Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O( τ 2 + h 2) and stable. Numerical simulations show that the method is efficient.

Editorial Notice on “Decay of solutions of some nonlinear equations” by Mohammed Aassila [Portugaliae Mathematica, Vol. 60, No. 4 (2003), 389–409

Editorial Notice: We have found recently that this paper was plagiarized from work by Liping Liu and Ming Mei, presented in their paper entitled “A better asymptotic profile of Rosenau–Burgers equation”, published in Applied Mathematics and Computation 131 (2002), pp. 147–170. We very much regret this error and offer our apologies to the authors. The Editors

Crank–Nicolson finite difference scheme for the Rosenau–Burgers equation

In this paper, a Crank–Nicolson finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau–Burgers equation is proposed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is convergent in the order of o ( τ 2 + h 2 ) and stable. Numerical simulations show that the method is efficient.