2 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
Encapsulated high order difference operators on curvilinear non-conforming grids
Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development