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 stencils and reconstruction. A kernel-based approach inspired by Gaussian process (GP) modeling is presented here. This approach allows the creation of a scheme of arbitrary order with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows 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, as well as an easy-to-implement effective limiter for positivity preservation, both of 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.

翻译：本文提出了一种用于双曲守恒律系统（尤其关注可压缩欧拉方程）有限体积方法的全多维核重构方案。通过将自适应阶数加权本质无振荡（WENO-AO）方法推广至适用于多维模板与重构的形式，实现了无振荡重构。文中引入了一种受高斯过程（GP）建模启发的核方法，该方法仅需定义简单的多维模板和子模板即可构建任意阶格式。此外，重构的全多维特性使得向高维空间的扩展更加直接，并消除了维度交替方法中对中间量施加复杂边界条件的需要。本文还引入了一组简单有效的新型重构变量，以及一种易于实现的有效保正限制器，两者均可通过微小修改应用于现有格式。通过一系列严苛且具有参考价值的基准算例验证了所提格式的有效性与实用性。