  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 reconstruction. A kernel-based approach inspired by radial
basis functions (RBF) and Gaussian process (GP) modeling, which we call
KFVM-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 (MPI)
is also provided.
