SummationByPartsOperators.jl: A Julia library of provably stable discretization techniques with mimetic properties

Abstract

SummationByPartsOperators.jl is a Julia library of summation-by-parts (SBP) operators, which are discrete derivative operators developed to get provably stable (semi-) discretizations, paying special attention to boundary conditions. Discretizations included in this framework are finite difference, Fourier pseudospectral, continuous Galerkin, and discontinuous Galerkin methods. The main aim of SummationByPartsOperators.jl is to be useful for both students learning the basic concepts and researchers developing new numerical algorithms based on SBP operators. Therefore, SummationByPartsOperators.jl provides a unified framework of all of these seemingly different discretizations. At the same time, the implementation is reasonably optimized to achieve good performance without sacrificing flexibility.

Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing

“…In addition, it allows a user to extend Trixi.jl step-by-step with new capabilities. For example, there are ongoing efforts to incorporate summation-by-parts finite difference methods via Sum-mationByPartsOperators.jl [43] and discontinuous Galerkin methods on simplex elements via StartUpDG.jl 6 in Trixi.jl. This is feasible because the modular design of Trixi.jl gives users the flexibility to pick a subset of the available meshes/methods, selected via multiple dispatch, to test their newly implemented features.…”

On the entropy projection and the robustness of high order entropy stable discontinuous Galerkin schemes for under-resolved flows

“…We note that, because an entropy conservative scheme can be constructed given any summation-by-parts or skew-symmetric operator [12,14,26], we are able to implement an entropy conservative C 0 continuous spectral element method and periodic finite difference method by constructing global difference operators from the tensor product of one-dimensional operators. These one-dimensional operators are provided by the Julia library SummationByPartsOperators.jl [68].…”

Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws

“…Many common numerical schemes can be formulated in terms of SBP operators, including finite difference methods [39,65], finite volume methods [42,43], continuous Galerkin methods [1,28,29], discontinuous Galerkin (DG) methods [5,6,21], and flux reconstruction methods [30,54]. Thus, they can be used to analyze a broad range of numerical methods in a unified fashion [50,53]. For background information on SBP operators and further references, we recommend the review articles [14,66].…”