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 multidimensional kernel-based Gaussian process (GP)
prolongation model. The formulation for this scheme was inspired by the two previous
studies on the GP methods introduced by A. Reyes et al. (Journal of Scientific Computing,
76 (2017), and Journal of Computational Physics, 381 (2019)). We extend the previous
GP interpolations and reconstructions to a new GP-based AMR prolongation method
that delivers a third-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. For this, we utilize the shock handling strategy
using the GP-based smoothness indicators developed in the previous GP work by Reyes
et al. We compare our GP-AMR results with the test results using the second-order linear
AMR method to demonstrate the efficacy of the GP-AMR method in a series of test suite
problems using the AMReX library, in which the GP-AMR method has been implemented.
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