General Formulation of Second-Order Semi-Lagrangian Methods for Convection-Diffusion Problems

and Applied Analysis 3 andX E = x−uΔt (similarly we see thatX E = x−2uΔt for the two-step methods). In (7) with X being replaced by X E , we have φ n+1 − φ n (x − uΔt) Δt − νΔφ n+1 = f n+1 . (9) On the other hand, (9) can be derived by MMOC [5]. In fact, with u = (u 1 , u 2 ), let s denote the direction vector (1, u 1 , u 2 ), and define the operator d ds := 1 θ ( ∂ ∂t + u ⋅ ∇) , (10) with θ(x, t) := [1 + |u(x, t)|]1/2 = [1 + |u 1 (x, t)| + |u 2 (x, t)|]1/2. So (1a) can be written to the form θ dφ ds − νΔφ = f. (11) Using the backward difference quotient, we have dφ ds = φ n+1 − φ n (x − Δtu/θ)