Optimal Convergence for Time-Dependent Stokes Equation: A New Approach

In our book ”Navier-Stokes Equations in Planar Domains”, Imperial College Press, 2013, we have suggested a fourth-order compact scheme for the Navier-Stokes equations in streamfunction formulation ∂t(∆ψ)+ (∇⊥ψ) · ∇(∆ψ) = ν∆ψ. Here we present a new approach for the analysis of a high-order compact scheme for the Navier-Stokes equations in cases where the convective term vanishes, or in cases where the viscous term dominates the convective term. In these cases the Navier-Stokes equations is replaced by the time-dependent Stokes equation ∂t(∆ψ) = ν∆ ψ. The same type of fourth-order compact schemes that were proposed for the Navier-Stokes equations, may be adopted to the time-dependent Stokes problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence is actually four, thus the error tends to zero as O(h), where h is the size of the mesh.