  We present a new polynomial-free prolongation scheme for Adaptive Mesh
Refinement (AMR) simulations of compressible and incompressible computational
fluid dynamics. The new method is constructed using a multi-dimensional
kernel-based Gaussian Process (GP) prolongation model. The formulation for this
scheme was inspired by the GP methods introduced by A. Reyes et al. (A New
Class of High-Order Methods for Fluid Dynamics Simulation using Gaussian
Process Modeling, Journal of Scientific Computing, 76 (2017), 443-480; A
variable high-order shock-capturing finite difference method with GP-WENO,
Journal of Computational Physics, 381 (2019), 189-217). In this paper, we
extend the previous GP interpolations and reconstructions to a new GP-based AMR
prolongation method that delivers a high-order accurate prolongation of data
from coarse to fine grids on AMR grid hierarchies. In compressible flow
simulations special care is necessary to handle shocks and discontinuities in a
stable manner. To meet this, we utilize the shock handling strategy using the
GP-based smoothness indicators developed in the previous GP work by A. Reyes et
al. We demonstrate the efficacy of the GP-AMR method in a series of testsuite
problems using the AMReX library, in which the GP-AMR method has been
implemented.
