Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy properties are verified using numerical simulations of the Euler equations in two spatial dimensions.Key words: high-order finite difference methods, block interface, numerical stability, interpolation, adaptive grids challenge and that has received considerable past attention. (For examples, see references [15,32,37,29,1,2,10,30,9]).A robust and well-proven HOFD methodology that ensures the strict stability of time-dependent partial differential equations (PDEs) is the summationby-parts simultaneous approximation term (SBP-SAT) method. The SBP-SAT method simply combines finite difference operators that satisfy a summationby-parts (SBP) formula [12], with physical boundary conditions implemented using either the Simultaneous Approximation Term (SAT) method [3], or the projection method [27,28,19]. Examples of the SBP-SAT approach can be found in references [24,23,25,18,20,22,26,17,33,34,14,5].An added benefit of the SBP-SAT method is that it naturally extends to multi-block geometries while retaining the essential single-block properties: strict stability, accuracy and conservation [4]. Thus, problems involving complex domains or non-smooth geometries are easily amenable to the approach. References [17,20,35] report applications of the SBP-SAT HOFD methods to problems involving non-trivial geometries.Current multi-block SBP-SAT methods suffer from two significant impediments: 1) the collocation points in adjoining blocks must match along block interfaces (i.e., "conforming" grids must be used), and 2) identical SBP schemes must be used tangential to the block interfaces. 1 These impediments prevent SBP-SAT operators from use on adaptively refined nonconforming multi-block grids and from use where hybrid-approaches involving two or more discretization techniques is desirable.Although neither impediment (i.e., conforming grids or identical elements) significantly limits Finite-and Spectral-Elements methods, 2 to date, mitigating their influence in the context of HOFDM has rarely been successful. A noteworthy exception appears in reference [25], where a hybrid method is developed (based on the SBP-SAT technique) to merge a HOFDM to an unstructured finite-volume method. This was accomplished by alternating the finite volume scheme at the interface boundary.The goals of the present study are two-fold. The first is to develop a systematic methodology for coupling arbitrary SBP methods between adjoining blocks. This task essentially requires identification of the conditions that adjoining interpolation operators must satisfy to guarantee interface ...