Second-order characteristic methods for advection-diffusion equations and comparison to other schemes

We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order Runge-Kutta approximation of the characteristics within the framework of the Eulerian-Lagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two Runge-Kutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from fluid dynamics.