Summation by parts for finite difference approximations for d/dx
Journal of Computational Physics
Summation by parts, projections, and stability. I
Mathematics of Computation
Summation by parts, projections, and stability. II
Mathematics of Computation
A stable and conservative interface treatment of arbitrary spatial accuracy
Journal of Computational Physics
High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates
Journal of Computational Physics
Boundary Procedures for Summation-by-Parts Operators
Journal of Scientific Computing
On Coordinate Transformations for Summation-by-Parts Operators
Journal of Scientific Computing
Stable and Accurate Artificial Dissipation
Journal of Scientific Computing
Steady-State Computations Using Summation-by-Parts Operators
Journal of Scientific Computing
The cactus framework and toolkit: design and applications
VECPAR'02 Proceedings of the 5th international conference on High performance computing for computational science
A spectral solver for evolution problems with spatial S3-topology
Journal of Computational Physics
Stable Boundary Treatment for the Wave Equation on Second-Order Form
Journal of Scientific Computing
Revisiting and Extending Interface Penalties for Multi-domain Summation-by-Parts Operators
Journal of Scientific Computing
Stable and Accurate Interpolation Operators for High-Order Multiblock Finite Difference Methods
SIAM Journal on Scientific Computing
Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations
SIAM Journal on Scientific Computing
Output error estimation for summation-by-parts finite-difference schemes
Journal of Computational Physics
Summation-by-parts operators and high-order quadrature
Journal of Computational and Applied Mathematics
Journal of Computational Physics
High-fidelity numerical solution of the time-dependent Dirac equation
Journal of Computational Physics
Optimal diagonal-norm SBP operators
Journal of Computational Physics
A generalized framework for nodal first derivative summation-by-parts operators
Journal of Computational Physics
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We construct optimized high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators' spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions. In particular, we find that the constructed dissipation operators are effective in suppressing instabilities that are sometimes otherwise present in the restricted full norm case.