A convergent and universally bounded interpolation scheme for the treatment of advection

A high resolution scheme with improved iterative convergence properties was devised by incorporating total-variation diminishing constraints, appropriate for unsteady problems, into an implicit time-marching method used for steady flow problems. The new scheme, referred to as Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA), has similar accuracy to the well-known SMART scheme, both being formally third-order accurate on uniform meshes for smooth flows. Three demonstration problems are considered: (1) advection of three scalar profiles, a step, a sine-squared, and a semi-ellipse; (2) Newtonian flow over a backward-facing step; and (3) viscoelastic flow through a planar contraction and around a cylinder. For the case of the viscoelastic flows, in which the high resolution schemes are also used to represent the advective terms in the constitutive equation, it is shown that only the new scheme is able to provide a converged solution to the prescribed tolerance.