Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations

We deal with the implicit spatial discretization method as applied to the advection-diffusion-reaction equations in 1-Dimensional situation. We present the Order conditions, stability criteria and the monotonicity properties of the method.  In particular, we apply the variants of the method, namely, implicit Adaptive Upwinding, Central Difference Schemes and Compact schemes for the test problem. We also present the results of the comparison of this method with the explicit spatial discretization methods. We show the advantages and the limitations of the method.