Finite element galerkin solutions for the rosenau equation

Abstract

Finite element Galerkin approximate solutions for a KdV-like Rosenau equation which models the dynamics of dense discrete systems are cdnsidered. Existence and uniqueness of exact solutions are shown and the error estimates of the continuous time Galerkin solutions are discussed. For the fully discrete time Galerkin solutions, we consider the backward Euler method which results the first order convergence in the temporal direction. For the second order convergence in time, we consider a three-level backward method and the Crank-Nicolson method which give optimal convergence in the spatial direction.

A high-accuracy conservative difference approximation for Rosenau-KdV equation

“…In the study of the dynamics of dense discrete systems, Rosenau [22,23] proposed the so-called Rosenau equation u t + u xxxxt + u x + uu x = 0, x ∈ R, t > 0. (1.4) From then on, there are many studies about the existence, the uniqueness and numerical methods for the equation (1.4) (see [5,6,7,14,16,18,21]). As the further consideration of nonlinear wave, Zuo added viscous term to Rosenau equation (1.4) and obtained Rosenau-KdV equation (1.1).…”

Cubic B-Spline Finite Element Method for the Rosenau-Burgers Equation

“…We use a variational formulation to help to define a finite element method to approximate (1). A variational formulation of the problem (1) is as following:…”

“…The existence and uniqueness of the solution to (1) were proved in [12], but it is still difficult to find the analytical solution to (1). For this reason, many works have been done on the numerical methods for solving (1)(see e.g., [1,2,5,8] and also the 54 G.X.…”

Explicit multistep methods for nonstiff partial differential equations

“…Park, in [13], proved existence, uniqueness and reqularity results for the solution u of (5a.1). Also Chung and Ha, in [6], discretize (5a.1) using finite elements in space and the backward Euler or the Crank-Nicolson method in time. For the space discretization, consider a family {V h } 0<h≤1 of finite dimensional subspaces of H 2 0 .…”