A new convergent and universally bounded interpolation scheme for the treatment of advection: application to viscoelastic simulations

Abstract

Accurate predictions of viscoelastic flows requirediscretisation schemes for convection of at leastsecond-order accuracy. In an important paper,Godunov [1] demonstrated that all monotone(bounded) linear schemes could be at most firstorder accurate. The only way to avoid thislimitation is thus the development of nonlinearcomposite schemes, usually referred to as highresolution schemes (HRS). Several discretisationschemes were proposed in the last years basedon either the total variation diminishing framework(TVD) [2], on the normalised variable formulation(NVF) [3], or others. The NVF approach has beenextended to non-uniform grids yielding thenormalised variable and space formulation(NVSF) [4], to be used here. Although HRSs haveseen widespread usage in the last years, thereare still some convergence difficulties with thesemethods [5]. This work aims at developing a newHRS with good accuracy and enhancedconvergence properties. The new scheme, namedConvergent and Universally BoundedInterpolation Scheme for the Treatment ofAdvection (CUBISTA), has similar accuracy to thewell-known SMART scheme [6], both being thirdorder accurate on uniform meshes.The upper-convected Maxwell (UCM) model wasselected to assess the performance of theCUBISTA scheme due to the hyperbolic natureand the known numerical difficulties of thesestress equations. The benchmark flow of a UCMfluid in a 4:1 planar contraction is used tocompare the performance of different highresolution schemes, and thus serving to illustratethe superiority of the CUBISTA scheme in termsof iterative convergence (robustness).