Galerkin time stepping along characteristics for Burgers' equation

A numerical method for convection-diffusion problems is presented. Without upwinding, it treats convection by time stepping along the characteristics of the associated pure convection problem and uses finite elements to model diffusion. The method can be applied easily to problems in more than one space dimension. When applied to the nonlinear Burgers equation, it requires only a backsolve (no iteration) at each time step. Numerical results show that the method loses no accuracy with time steps at least an order of magnitude larger than those of standard finite element methods. With a spatial grid fine enough to resolve sharp fronts, the method introduces no appreciable numerical diffusion or spurious oscillations.