Dual consistency and functional accuracy: a finite-difference perspective

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. (C) 2013 Elsevier Inc. All rights reserved.