  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.
