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. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with 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 is also provided.

中文翻译：

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KFVM-WENO：一种基于高阶精确核的可压缩流体动力学有限体积方法


本文提出了一种应用于双曲守恒定律系统的有限体积方法的完全多维基于核的重构方案，特别强调可压缩欧拉方程。非振荡重建是通过自适应阶加权本质非振荡（WENO）方法实现的，该方法被转换成适合多维重建的形式。这里提出了一种受径向基函数和高斯过程建模启发的基于核的方法，我们将其称为带有 WENO 的基于核的有限体积方法。这种方法允许使用简单定义的多维模板和子模板创建任意精度顺序的方案。此外，重建的完全多维性质允许更直接地扩展到更高的空间维度，并且消除了修改的逐维方法中中间量的复杂边界条件的需要。此外，还引入了一组新的简单而有效的重建变量，只需稍作修改即可在现有方案中发挥作用。所提出的方案应用于一系列严格且信息丰富的基准问题，以证明其有效性和实用性。还提供了使用 Kokkos 和消息传递接口的高度并行多 GPU 实现。