Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations

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.

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