4 research outputs found
Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
Summation-by-parts (SBP) operators allow us to systematically develop
energy-stable and high-order accurate numerical methods for time-dependent
differential equations. Until recently, the main idea behind existing SBP
operators was that polynomials can accurately approximate the solution, and SBP
operators should thus be exact for them. However, polynomials do not provide
the best approximation for some problems, with other approximation spaces being
more appropriate. We recently addressed this issue and developed a theory for
one-dimensional SBP operators based on general function spaces, coined
function-space SBP (FSBP) operators. In this paper, we extend the theory of
FSBP operators to multiple dimensions. We focus on their existence, connection
to quadratures, construction, and mimetic properties. A more exhaustive
numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their
application will be provided in future works. Similar to the one-dimensional
case, we demonstrate that most of the established results for polynomial-based
multi-dimensional SBP (MSBP) operators carry over to the more general class of
MFSBP operators. Our findings imply that the concept of SBP operators can be
applied to a significantly larger class of methods than is currently done. This
can increase the accuracy of the numerical solutions and/or provide stability
to the methods.Comment: 28 pages, 9 figure
On conservation and stability properties for summation-by-parts schemes
We discuss conservative and stable numerical approximations in summation-by-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.Funding agencies: VINNOVA [2013-01209]</p