Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation (Q2378925)

The authors consider an initial-boundary value problem of Rosenau-Burgers equation \[ u_{t}+u_{xxxxt}-u_{xx}+u_{x}+uu_{x}=0 \] and propose a Crank-Nicolson type finite difference scheme. The existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is stable and convergent in the order of \( O(\Delta t^{2}+\Delta x^{2})\). The method is tested by an example. It will be better if the new scheme is tested by a shock wave problem.