Gaussian FSBP operators: Comparison and application to numerical methods for hyperbolic conservation laws
For researchers in numerical methods for hyperbolic PDEs, this work provides a comparative analysis and extension of Gaussian FSBP operators, showing practical advantages in time-dependent problems.
This paper compares open and closed function-space summation-by-parts (FSBP) operators constructed with generalized Gaussian quadratures and applies them to solve hyperbolic conservation laws. It extends the FSBP framework with function-space exact extrapolation operators and demonstrates improved efficiency and accuracy in numerical experiments for 1D linear advection, Burgers, and Euler equations.
Function-space summation-by-parts (FSBP) operators enable conservative and energy-stable numerical methods for hyperbolic conservation laws based on general, non-polynomial approximation spaces. Recent works show that using generalized Gaussian quadrature significantly reduces the number of grid points required compared to existing constructions that have mostly focused on equidistant grids. In this paper, we compare open and closed FSBP operators constructed with generalized Gaussian quadratures and apply them to numerically solve hyperbolic conservation laws. Furthermore, to support open node distributions, we extend the FSBP framework by introducing function-space exact extrapolation operators and operationalize them in numerical schemes for solving hyperbolic conservation laws. Our numerical experiments include the one-dimensional linear advection, non-viscous Burgers, and compressible Euler equations of gas dynamics. We observe that applying FSBP operators in numerical schemes can improve efficiency and accuracy. Notably, we demonstrate these advantages in more challenging time-dependent settings compared to other recent works on Gaussian FSBP operators.