Development of arbitrary-order multioperators-based schemes for parallel calculations.: Part 2: Families of compact approximations with two-diagonal inversions and related multioperators

One-parameter families of compact approximations to grid functionals with inverses of two-point operators and their properties are described. As particular examples, interpolation/extrapolations operators, quadratures formulas and approximations to derivatives are presented. Using operators from the families with fixed parameters values as basis operators, their linear combinations providing formally arbitrary-order approximations (multioperators) are constructed. Numerical illustrations are presented. Special emphasis is placed on first derivatives discretizations in the context of conservation laws. As an example, a highly accurate tenth-order scheme is outlined and tested against the Burgers’ equation. It is shown how extrapolation multioperators can be used to create boundary closures.