Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Abstract

A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative term plays a more important role than in dissipation.

On the use of spectral discretizations with time strong stability preserving properties to Dirichlet pseudo-parabolic problems

“…Spectral convergence for Chebyshev collocation approximation. The Chebyshev collocation scheme is now used to approximate the BBM-Burgers problem (2.14), (2.15) in Ω = (−20, 30) with homogeneous boundary conditions, - −1/2 and, [56] v(x, 0) = sech(x),…”

“…They include the BBM-Burgers equation, a modification of the Benjamin-Bona-Mahony equation, [12], which includes a dissipative term. On the other hand, most of the work presented in the literature about the numerical approximation of pseudo-parabolic equations seems to be focused on the use of finite differences, [5,6,80,7,26,32,8], as well as finite elements, [9,56], and finite volumes of different type for the discretization in space, sometimes combined with a domain decomposition method for a more accurate approximation of convective and diffusive effects, [4,82]. This different numerical treatment of the advective and diffusive fluxes, mentioned above, also makes influence in some choices of time integrators for pseudo-parabolic problems.…”

Error estimates for semidiscrete Galerkin and collocation approximations to pseudo-parabolic problems with Dirichlet conditions

“…Concerning the numerical approximation of equations of the form (1.1), the literature contains many references involving finite differences, [4,5,54,6,22,28], as well as finite elements and finite volumes, [38,45,2,56]. We also mention some convergence results.…”

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

“…Many works on nonlinear evolution equations have been studied, such as the Hamiltonian structure [1,2], the infinite conservation laws [3,4], the Bäcklund transformation [5,6] and so on [7][8][9]. Besides, the exact solution of these equations, which can be expressed in various forms by different methods, is also a significant subject of soliton research [10][11][12][13][14][15][16][17][18][19][20][21][22]. In recent years, with the development of soliton theory, more and more researchers pay attention to the Riemann-Hilbert approach.…”