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:本文介绍了一种针对双曲守恒律系统的有限体积法的全面多维核重构方案,特别是压缩欧拉方程。非振荡重构通过适应性阶加权本质非振荡(WENO-AO)方法实现,该方法适合于多维模板和重构。这里介绍了一种受高斯过程 (GP) 建模启发的基于核的方法。该方法允许创建任意阶数的方案,并具有简单定义的多维模板和子模板。此外,重构的全面多维性质使其更容易扩展到更高的空间维度,并消除了修改逐维度方法中中间量的复杂边界条件的需求。此外,还引入了一种新的简单而有效的重构变量和一种易于实现的有效正性限制器,两者都可以在现有的方案中进行少量修改而有用。提出的方案被应用于一套严格和有信息量的基准测试问题,以证明其功效和实用性。