Incremental unknowns method based on the θ-scheme for time-dependent convection–diffusion equations

A θ-scheme using two-level incremental unknowns is presented for solving time-dependent convection–diffusion equations in two-dimensional case. The IMG algorithm (Inertial Manifold–Multigrid algorithm) including the second-order incremental unknowns is convergent. The incremental unknowns method based on the θ-scheme needs a stability condition as 0≤θ<1/2 and is unconditionally stable as 1/2≤θ≤1. By the GMRES method in the iteration at each time step, numerical results of the convection–diffusion equations are investigated and confirm that oscillations can be controlled by the incremental unknowns method.