In this dissertation, we consider high order accurate, implicit, finite volume, weighted essentially non-oscillatoy (WENO) schemes for solving advection-diffusion equations. Our schemes are locally mass conservative and suppress oscillations in the solution.

WENO reconstruction is used for the space discretization. We analyze standard WENO reconstructions and WENO reconstructions with adaptive order (WENO-AO). We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth and discontinuious cases. The new WENO-AO reconstruction maintains the accuracy of the largest stencil over which the solution is smooth.

For the evolution of time, we use an implicit Runge-Kutta time integrator. The strong stability preserving (SSP) methods are only guaranteed to be stable under the SSP timestep limits. We compare them with L-stable Runge-Kutta methods. We also develop a new Runge-Kutta method for solving our advection-diffusion problems, which is high order accurate in time but drops order near discontinuities in the solution. We show the new method is unconditionally L-stable for the linear problem with smooth solutions on the uniform grid. Numerical results show the new method improves the stability of the solutions over other Runge-Kutta methods.

Applications to two-phase flow in porous media in two-dimensional space on quadrilateral computational meshes are discussed. Computational results show the scheme can handle various test problems well.