Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
arXiv:2301.12996 · doi:10.1016/j.jcp.2023.112370
Abstract
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.
arXiv comment28 pages, 9 figures
References in corpus (15)
- Review of Summation-by-parts schemes for initial-boundary-value problems
- The AAA algorithm for rational approximation
- Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations
- On discretely entropy conservative and entropy stable discontinuous Galerkin methods
- Summation-by-parts operators for correction procedure via reconstruction
- Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
- Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians
- Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
- Stable high-order cubature formulas for experimental data
- Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
- Summation-by-parts operators for general function spaces
- Construction and application of provable positive and exact cubature formulas
- Stable High Order Quadrature Rules for Scattered Data and General Weight Functions
- Towards Stable Radial Basis Function Methods for Linear Advection Problems
- Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions