A monotonicity preserving, nonlinear, finite element upwind method for the transport equation

We propose a simple upwind finite element method that is monotonicity preserving and weakly consistent of order O(h32). The scheme is nonlinear, but since an explicit time integration method is used the added cost due to the nonlinearity is not prohibitive. We prove the monotonicity preserving property for the forward Euler method and for a second order Runge–Kutta method. The convergence properties of the Runge–Kutta finite element method are verified on a numerical example.