Summation by parts for finite difference approximations for d/dx
Journal of Computational Physics
A stable and conservative interface treatment of arbitrary spatial accuracy
Journal of Computational Physics
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates
Journal of Computational Physics
Boundary Procedures for Summation-by-Parts Operators
Journal of Scientific Computing
On Coordinate Transformations for Summation-by-Parts Operators
Journal of Scientific Computing
Stable and Accurate Artificial Dissipation
Journal of Scientific Computing
Summation by parts operators for finite difference approximations of second derivatives
Journal of Computational Physics
On the order of accuracy for difference approximations of initial-boundary value problems
Journal of Computational Physics
Stable and accurate schemes for the compressible Navier-Stokes equations
Journal of Computational Physics
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat)
A stable and conservative high order multi-block method for the compressible Navier-Stokes equations
Journal of Computational Physics
Revisiting and Extending Interface Penalties for Multi-domain Summation-by-Parts Operators
Journal of Scientific Computing
SIAM Journal on Scientific Computing
Output error estimation for summation-by-parts finite-difference schemes
Journal of Computational Physics
Journal of Computational Physics
Summation-by-parts operators and high-order quadrature
Journal of Computational and Applied Mathematics
On the impact of boundary conditions on dual consistent finite difference discretizations
Journal of Computational Physics
Journal of Computational Physics
Dual consistency and functional accuracy: a finite-difference perspective
Journal of Computational Physics
Journal of Computational Physics
High-fidelity numerical solution of the time-dependent Dirac equation
Journal of Computational Physics
PDE-constrained optimization with error estimation and control
Journal of Computational Physics
Optimal diagonal-norm SBP operators
Journal of Computational Physics
A generalized framework for nodal first derivative summation-by-parts operators
Journal of Computational Physics
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Diagonal-norm summation-by-parts (SBP) operators can be used to construct time-stable high-order accurate finite-difference schemes. However, to achieve both stability and accuracy, these operators must use $s$-order accurate boundary closures when the interior scheme is $2s$-order accurate. The boundary closure limits the solution to $(s+1)$-order global accuracy. Despite this bound on solution accuracy, we show that functional estimates can be constructed that are $2s$-order accurate. This superconvergence requires dual-consistency, which depends on the SBP operators, the boundary condition implementation, and the discretized functional. The theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension. In higher dimensions, we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces. The generality of the theoretical results is demonstrated using a two-dimensional Poisson problem and a nonlinear hyperbolic system—the Euler equations of fluid mechanics.