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If you would like to join the seminar and are not currently affiliated with ANU, please contact Kenneth Duru at kenneth.duru@anu.edu.au.

The summation-by-parts framework for the design and analysis of linearly and nonlinearly stable discretizations

Matrix-difference operators having the summation-by-parts (SBP) property were originally developed in the finite-difference community with the purpose of mimicking finite-element energy analysis techniques for proving linear stability. The essential feature of these operators is that they are equipped with a high-order approximation to integration by parts that telescopes (necessary for proving stability). In combination with appropriate procedures for inter-element coupling (for discontinuous approaches) and imposition of initial and boundary conditions, the resulting SBP framework allows for a one-to-one correspondence between discrete and continuous stability proofs and provides a road map for the development of robust algorithms with provable properties.

In recent years, there has been increased interest by the broader numerical methods community in the SBP concept. The SBP framework is discretization agnostic and, besides finite-difference methods, has been applied to nodal/modal discontinuous Galerkin, nodal continuous Galerkin methods, the flux-reconstruction method, and has been shown to have a subcell finite-volume interpretation. The SBP concept has been extended to non-tensor nodal distributions thereby introducing the ability to construct SBP schemes on unstructured tetrahedral meshes. Nonlinearly robust schemes have been constructed by enforcing discrete entropy stability. SBP schemes on non-conforming meshes that remain conservative and stable have been developed. Dual-consistent schemes have been developed that lead to superconvergent functional estimates, etc.

To summarize, the SBP framework is a matrix-based analysis framework that enables the construction and analysis of exible and robust numerical methods that have advantageous properties within a rigorous mathematical framework.

In this talk, I will motivate the SBP framework for the design and analysis of both linearly and nonlinearly stable and conservative methods. I will then demonstrate the matrix analysis technique for both linear and nonlinear problems (linear advection and the Burgers equation, respectively).

Time permitting, I will discuss a number of ongoing projects.