zahr2020aiaa.bib

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@conference{zahr2020aiaa,
  abstract = {This work presents a high-order accurate, nonlinearly stable numerical framework for solving steady conservation laws with discontinuous solution features such as shock waves. The method falls into the category of a shock tracking or r-adaptive method and is based on the observation that numerical discretizations such as finite volume or discontinuous Galerkin methods that support discontinuities along element faces can perfectly repre- sent discontinuities and provide appropriate stabilization through approximate Riemann solvers. The difficulty lies in aligning element faces with the unknown discontinuity. The proposed method recasts a discretized conservation law as a PDE-constrained optimization problem whose solution is a (curved) mesh that tracks the discontinuity and the solution of the discrete conservation law on this mesh. The discrete state vector and nodal positions of the high-order mesh are taken as optimization variables. The objective function is a dis- continuity indicator that monotonically approaches a minimum as element faces approach the shock surface in a neighborhood of radius O(h), where h is the mesh size parameter. The discretized conservation law on a parametrized domain defines the equality constraints for the optimization problem. A full space optimization solver is used to simultaneously converge the state vector and mesh to their optimal values. This ensures the solution of the discrete PDE is never required on meshes that are not aligned with discontinuities and it increases the nonlinear stability. The method is demonstrated in one and two dimensions: transonic flow through a nozzle and supersonic flow around a bluff body. In both cases, the framework tracks the discontinuity closely with curved mesh elements and provides accurate solutions on extremely coarse meshes, e.g., O(1000) degrees of freedom to resolve supersonic flow at Mach 4 in two dimensions.},
  address = {Orlando, Florida},
  author = {Zahr, Matthew J. and Persson, Per-Olof},
  booktitle = {Proc. of the AIAA Science and Technology Forum and Exposition (SciTech2020)},
  conftype = {conference},
  date-added = {2020-01-26 21:03:48 -0500},
  date-modified = {2020-01-26 21:04:46 -0500},
  organization = {American Institute of Aeronautics and Astronautics},
  presenter = {Zahr, Matthew J.},
  project = {shktrk},
  status = {contributed},
  title = {An $r$-adaptive, high-order discontinuous {G}alerkin method for flows with attached shocks},
  year = {1/6/2020 -- 1/10/2020}
}

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