Inverse Lax-Wendroff boundary treatment for solving conservation laws with finite difference HWENO methods
Guangyao Zhu, Yan Jiang, Zhuang Zhao, Mengping Zhang
TL;DR
The paper tackles high-order boundary treatment for conservation-law solvers on irregular geometries by integrating an inverse Lax-Wendroff (ILW) boundary method with a fifth-order finite-difference HWENO-R scheme. A novel least-squares extrapolation augments the ILW procedure to reconstruct ghost-point values and derivatives while reducing the number of low-order terms needed, improving efficiency and stability. A linear stability analysis yields a CFL bound and parameter ranges that preserve the inner scheme's stability, with recommendations such as $(k_d,k)=(3,2)$ in 1D and $(2,4)$ in 2D. Numerical tests in 1D and 2D, including curved boundaries and complex flows, confirm fifth-order accuracy, robustness, and practical efficiency gains for challenging geometries.
Abstract
This paper presents a novel inverse Lax-Wendroff (ILW) boundary treatment for finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes to solve hyperbolic conservation laws on arbitrary geometries. The complex geometric domain is divided by a uniform Cartesian grid, resulting in challenge in boundary treatment. The proposed ILW boundary treatment could provide high order approximations of both solution values and spatial derivatives at ghost points outside the computational domain. Distinct from existing ILW approaches, our boundary treatment constructs the extrapolation via optimized through a least squares formulation, coupled with the spatial derivatives at the boundary obtained via the ILW procedure. Theoretical analysis indicates that compared with other ILW methods, our proposed one would require fewer terms by using the relatively complicated ILW procedure and thus improve computational efficiency while preserving accuracy and stability. The effectiveness and robustness of the method are validated through numerical experiments.