We present an a posteriori shock-capturing finite volume method algorithm called GP-MOOD that solves a compressible hyperbolic conservative system at high-order solution accuracy (e.g., third-, fifth-, and seventh-order) in multiple spatial dimensions. The GP-MOOD method combines two methodologies, the polynomial-free spatial reconstruction methods of GP (Gaussian Process) and the a posteriori detection algorithms of MOOD (Multidimensional Optimal Order Detection). The spatial approximation of our GP-MOOD method uses GP's unlimited spatial reconstruction that builds upon our previous studies on GP reported in Reyes et al., Journal of Scientific Computing, 76 (2017) and Journal of Computational Physics, 381 (2019). This paper focuses on extending GP's flexible variability of spatial accuracy to an a posteriori detection formalism based on the MOOD approach. We show that GP's polynomial-free reconstruction provides a seamless pathway to the MOOD's order cascading formalism by utilizing GP's novel property of variable (2R+1)th-order spatial accuracy on a multidimensional GP stencil defined by the GP radius R, whose size is smaller than that of the standard polynomial MOOD methods. The resulting GP-MOOD method is a positivity-preserving method. We examine the numerical stability and accuracy of GP-MOOD on smooth and discontinuous flows in multiple spatial dimensions without resorting to any conventional, computationally expensive a priori nonlinear limiting mechanism to maintain numerical stability.
翻译:我们提出了一种事后冲击采集有限体积法算法,称为GP-MOOD,它以多种空间层面的高度溶解精确度(例如第三、第五和第七级)解决了压缩的超偏保守系统,其基础是多个空间层面的高度溶解度(例如第三、第五和第七级)。GP-MOOD方法结合了两种方法,即GP(Gausian Process)的无多边空间重建方法,以及MOOD的事后检测算法(Mlobal Obtimal Actimal) 。我们GP-MOD方法的空间近似近似利用GP在雷耶斯等人(Reyes et al.),《科学计算学期刊》,第76(2017年)和第381(2019年)《比较物理物理杂志》,它侧重于将GPA的灵活空间精确度变异性变异性,以MOD法的形式延伸。我们表明,GPO-P-F-F的重建为MO的平稳秩序提供了一种无缝路径,通过使用GPGP’nal-deal-dealal-deal-deal-deal-deal rode rodeal deal rodeal laismal dealmal rol rol rol rol 和Syal-stal-stal-stal-altimeal-stal-stal-stal-stal-al-stal-stal-stal-al-stal-al-stalizal-stal-stald-stal-stald-stal-stal-stal-stal-stal-stal-stal-stal-st-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stald-stal-stald-stald-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-stal-al-al-al-al-al-al-al-sal-stal-stal