Monday, November 17, 11:30 (HH312)
Summation-by-parts (SBP) methods provide a powerful algebraic framework for constructing high-order discretizations of partial differential equations that rigorously mimic various properties of the continuous problem. By enforcing a discrete form of integration by parts, SBP schemes enable discrete analogues of conservation, stability, and other structure-preserving proofs—without relying on assumptions such as exact integration often required in finite-element and discontinuous Galerkin formulations. SBP methods can be expressed in finite-difference, finite-volume, or finite-element form, allowing them to inherit the geometric flexibility, h-p adaptivity, and analytical tools developed within each of these frameworks.
The first part of this talk will introduce SBP methods through the linear convection equation, illustrating the key concepts of stability and conservation. We then extend these ideas to nonlinear conservation laws, such as the compressible Euler equations, by incorporating the framework of entropy stability. Finally, I will discuss recent numerical experiments addressing a key open question in the field—local linear stability—which examines how linear perturbations evolve in nonlinear entropy-stable schemes: if a scheme is provably stable for the nonlinear problem, can its linear perturbations still be unstable?