This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Non-oscillatory reconstruction is achieved through an adaptive order weighted essentially non-oscillatory (WENO-AO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions (RBF) and Gaussian process (GP) modeling, which we call KFVM-WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple-yet-effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message passing interface (MPI) is also provided.

翻译：本文提出了一种完全多维的核基重构方案，用于求解双曲守恒律系统的有限体积方法，特别关注可压缩欧拉方程。通过将自适应阶加权本质无振荡（WENO-AO）方法转化为适用于多维重构的形式，实现了无振荡重构。本文提出了一种受径向基函数（RBF）和高斯过程（GP）建模启发的核基方法，称为KFVM-WENO。该方法能够通过简单定义的多维模板和子模板构建任意精度阶的格式。此外，完全多维重构的特性使其能够更直接地扩展到更高空间维度，并消除了对修正分维方法中中间量的复杂边界条件的需求。本文还引入了一组简单而高效的重构变量，可几乎不作修改地应用于现有格式。通过一系列严格且具有代表性的基准问题验证了所提方案的有效性和实用性，并提供了基于Kokkos和消息传递接口（MPI）的高度并行多GPU实现。