7 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
Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations
High-order Hadamard-form entropy stable multidimensional summation-by-parts
discretizations of the Euler and compressible Navier-Stokes equations are
considerably more expensive than the standard divergence-form discretization.
In search of a more efficient entropy stable scheme, we extend the
entropy-split method for implementation on unstructured grids and investigate
its properties. The main ingredients of the scheme are Harten's entropy
functions, diagonal- summation-by-parts operators with diagonal
norm matrix, and entropy conservative simultaneous approximation terms (SATs).
We show that the scheme is high-order accurate and entropy conservative on
periodic curvilinear unstructured grids for the Euler equations. An entropy
stable matrix-type interface dissipation operator is constructed, which can be
added to the SATs to obtain an entropy stable semi-discretization.
Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta
method. Entropy stable viscous SATs, applicable to both the Hadamard-form and
entropy-split schemes, are developed for the compressible Navier-Stokes
equations. In the absence of heat fluxes, the entropy-split scheme is entropy
stable for the compressible Navier-Stokes equations. Local conservation in the
vicinity of discontinuities is enforced using an entropy stable hybrid scheme.
Several numerical problems involving both smooth and discontinuous solutions
are investigated to support the theoretical results. Computational cost
comparison studies suggest that the entropy-split scheme offers substantial
efficiency benefits relative to Hadamard-form multidimensional SBP-SAT
discretizations.Comment: 34 pages, 8 figure