  We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for
numerically solving convection and convection-diffusion equations.
Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly
due to their ability to allow large time steps. Our proposed scheme is
formulated by integrating the PDE on a space-time region partitioned by
approximations of the characteristics determined from the Rankine-Hugoniot jump
condition; and then rewriting the time-integral form into a time differential
form to allow application of Runge-Kutta (RK) methods via the method-of-lines
approach. The scheme can be viewed as a generalization of the standard
Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is
partitioned by approximate characteristics with zero velocity. The high-order
spatial reconstruction is achieved using the recently developed weighted
essentially non-oscillatory schemes with adaptive order (WENO-AO); and the
high-order temporal accuracy is achieved by explicit RK methods for convection
equations and implicit-explicit (IMEX) RK methods for convection-diffusion
equations. Our algorithm extends to higher dimensions via dimensional
splitting. Numerical experiments demonstrate our algorithm's robustness,
high-order accuracy, and ability to handle extra large time steps.
