10:35 AM
Classical finite-difference operators having the summation-by-parts (SBP) property were originally constructed with the goal of mimicking finite-element linear stability proofs. The crucial feature of such operators is that their constituent matrices are high-order approximations to integral bilinear forms and therefore lead to high-order approximations to integration by parts. The combination of SBP operators with appropriate weak imposition of boundary and inter-element coupling results in an SBP framework that allows for a one-to-one correspondence between discrete and continuous stability proofs and thereby naturally guides the construction of robust algorithms. Since their introduction in the early 70s, much of the concentration in the SBP community has been on developing rigorous means of imposing inter-element coupling and boundary conditions for a variety of PDEs including the Navier-Stokes equations. Recently, there have been a number of exciting developments in the SBP community. The SBP framework has been extended to include nodal discontinuous and continuous Galerkin methods, the flux-reconstruction method, and has been shown to have a subcell finite-volume interpretation. SBP operators can now be constructed on non-tensor product nodal distributions consequently intruding the ability to construct SBP schemes on unstructured meshes. Nonlinearly robust schemes can be constructed by enforcing discrete entropy stability and dual consistent schemes have been developed that lead to superconvergent functional estimates, etc. The appealing feature of the SBP concept is that it provides a rigorous mathematical framework within which to construct flexible and robust numerical methods with advantageous properties. In this talk I will give a brief introduction to the SBP concept and highlight a number of recent developments.