Dual consistency and functional accuracy: a finite-difference perspective

  • Authors:

  • J. E. Hicken;D. W. Zingg

  • Affiliations:

  • Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, United States;Institute for Aerospace Studies, University of Toronto, Toronto, Ontario, M3H 5T6, Canada

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2014

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Abstract

Consider the discretization of a partial differential equation (PDE) and an integral functional that depends on the PDE solution. The discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem. Consequently, a dual-consistent discretization is a synthesis of the so-called discrete-adjoint and continuous-adjoint approaches. We highlight the impact of dual consistency on summation-by-parts (SBP) finite-difference discretizations of steady-state PDEs; specifically, superconvergent functionals and accurate functional error estimates. In the case of functional superconvergence, the discrete-adjoint variables do not need to be computed, since dual consistency on its own is sufficient. Numerical examples demonstrate that dual-consistent schemes significantly outperform dual-inconsistent schemes in terms of functional accuracy and error-estimate effectiveness. The dual-consistent and dual-inconsistent discretizations have similar computational costs, so dual consistency leads to improved efficiency. To illustrate the dual consistency analysis of SBP schemes, we thoroughly examine a discretization of the Euler equations of gas dynamics, including the treatment of the boundary conditions, numerical dissipation, interface penalties, and quadrature by SBP norms.