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We present a new polynomial-free prolongation scheme for Adaptive Mesh Re- finement (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 two previous studies on the GP methods in- troduced by A. Reyes et al., Journal of Scientific Computing, 76 (2017), and Journal of Computational Physics, 381 (2019). In this paper, we extend the previous GP interpolations and reconstructions to a new GP-based AMR pro- longation 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 man- ner. 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 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. 

Heat conduction is strongly suppressed in turbulent, weakly collisional, and magnetized plasmas.