Development of arbitrary-order multioperators-based schemes for parallel calculations. 1: Higher-than-fifth-order approximations to convection terms

Further results concerning arbitrary-order approximations to grid functionals via linear combinations of basis operators obtained by fixing sets of free parameters (multioperators) are presented. A parallel algorithm for their calculations is described. As basis operators, a version of one-parametric families of the fifth-order compact upwind differencing operators (CUD) as well as the fourth-order non-centered approximations to first derivatives are considered. The resulting conservative schemes for fluid dynamics type of equations (or other equations with convection terms) are outlined. The existence and uniqueness of the corresponding multioperators are discussed. It is shown that for properly chosen parameters, multioperators preserve the upwind (downwind) properties of the basis operators, that is their positivity (negativity) in appropriate Hilbert spaces of grid functions. As examples, the seventh- and ninth-order multioperators-based schemes with very good dispersion and dissipation properties are described, their possible optimization being discussed. Numerical examples illustrating their extremely high accuracy are presented.