Optimal boundary closures for diagonal-norm upwind SBP operators

TL;DR

The paper addresses the challenge of achieving high-order accuracy near boundaries for diagonal-norm SBP operators on nonuniform grids. It introduces boundary-optimized upwind SBP operators constructed with non-equispaced boundary points, parameterized $H$ and $Q_+$, and optimized by minimizing leading boundary errors while enforcing $H>0$ and $S\le0$, with dissipation provided by $D_+$ and $D_-$ in conjunction with SAT or projection. The authors demonstrate stability and accuracy on 1D hyperbolic systems and extend to 2D compressible Euler equations, showing improved convergence rates, enhanced robustness, and explicit SBP-SAT/projection discretizations on multiblock grids. These results suggest substantial benefits for long-time simulations and boundary-interaction-dominated flows, with practical implementation available in a Julia package for SBP operators.

Abstract

By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP) framework, ensuring linear stability on piecewise curvilinear multiblock grids. Boundary and interface conditions are imposed using either weak enforcement through simultaneous approximation terms (SAT) or strong enforcement via the projection method. The proposed operators yield significantly improved accuracy and computational efficiency compared with SBP operators constructed on equidistant grids. The resulting SBP--SAT and SBP--projection discretizations produce fully explicit systems of ordinary differential equations. The accuracy and stability properties of the proposed operators are demonstrated through numerical experiments for linear hyperbolic problems in one spatial dimension and for the compressible Euler equations in two spatial dimensions.

Figures (5)

• Figure 1: Comparing central difference SBP (both traditional and boundary-optimized) against the boundary-optimized $9^\text{th}$-order upwind SBP operator for two grid resolutions. The simulations are run to $t=1.8$ with CFL=0.05 and $\alpha=3$.
• Figure 2: Domain for the isentropic vortex convergence test and an example discretization of the two block domain with a non-uniform grid.
• Figure 3: Isentropic vortex on a chevron domain with two blocks each containing $61\times 61$ nodes. The vortex strength is $\varepsilon=5$. All results use $8^{\text{th}}$ order diagonal norm SBP operators with $4^{\text{th}}$ order boundary stencils. The boundary optimized upwind (a) and traditional upwind SBP operators run to the final time $t=75$, although the tranditional upwind operator exhibits small discrepancies. The boundary optimized traditional SBP operator crashes at $t\approx 41.438$ and the traditional SBP operator crashes early in the simulation at $t\approx 0.2919$.
• Figure 4: Isentropic vortex on a chevron domain with two blocks each containing $61\times 61$ nodes. The vortex strength is $\varepsilon=10$. The $8^{\text{th}}$ order boundary optimized upwind SBP operator (left) runs to the final time $t=75$ whereas the traditional $8^{\text{th}}$ order upwind SBP operator (right) crashes at $t\approx 72.255$.
• Figure 5: Visualization of numerical solutions from Kelvin--Helmholtz instability simulations at time $t = 3.63$. We use interior order $p=6$ upwind SBP operators on a mesh of $16^2$ Cartesian elements with $17^2$ nodes per element. The boundary-optimized simulation reaches the final time, whereas the traditional one crashes; green regions in (b) mark points with negative density.

Theorems & Definitions (7)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Remark 3.1