In this thesis we carry out a further study of the back and forth error compensation and correction (BFECC) method. The first part discusses the time reversibility of numerical schemes. Motivated by the BFECC method, a variety of new numeri- cal schemes that aim at improving the time reversibility are developed and studied. We also introduce an interpolation algorithm based on BFECC in this part. In the second part we introduce a new limiting strategy which requires another backward advection in time so that overshoots/undershoots at the new time level get exposed when they are transformed back to compare with the solution at the old time level. This new technique is very simple to implement even for unstructured meshes and is able to eliminate artifacts induced by jump discontinuities in the solution itself or in its derivatives.