A New Class of High-Order Methods for Fluid Dynamics Simulations Using Gaussian Process Modeling: One-Dimensional Case

Abstract

We introduce an entirely new class of high-order methods for computational fluid dynamics based on the Gaussian process (GP) family of stochastic functions. Our approach is to use kernel-based GP prediction methods to interpolate/reconstruct high-order approximations for solving hyperbolic PDEs. We present a new high-order formulation to solve (magneto)hydrodynamic equations using the GP approach that furnishes an alternative to conventional polynomial-based approaches.

A variable high-order shock-capturing finite difference method with GP-WENO

“…However, for the square wave the GP-R2 solution introduces some Gibbs phenomena at the discontinuities. Similar oscillations associated with flat discontinuities like those shown here have been also observed in shock-tube problems in our previous finite volume version of GP-WENO [95]. We find that this issue stems from the use of the zero-mean prior in the calculation of the smoothness indicators, which biases the stencil choice towards those that are more compatible with the zero-mean prior (i.e., data that are closer to zero).…”

“…In this section, we briefly outline the statistical theory underlying the construction of GP-based Bayesian prior and posterior distributions (see Section 3.1). Interested readers are encouraged to refer to our previous paper [95] for a more detailed discussion in the context of applying GP for the purpose of achieving high-order algorithms for FVM schemes. For a more general discussion of GP theory see [93].…”

“…For other choices of kernel functions and the related discussion in the context of designing high-order approximations for numerical PDEs, readers are referred to [95].…”

“…However, is a length scale that controls the characteristic scale on which the GP sample functions vary. As was demonstrated in [95], plays a critical role in the solution accuracy of a GP interpolation/reconstruction scheme and ideally should match the length scales of the problem that are to be resolved.…”

GP-MOOD: A positive-preserving high-order finite volume method for hyperbolic conservation laws

“…A more popular and traditional path has been the a priori approach, in which all cells are spatially discretized by utilizing one or more nonlinear limiting procedures, integrated with a stable temporal update to the next time step. Well-known examples of a priori limiting methods include second-order piecewise linear TVD (Total Variation Diminishing) methods (e.g., [1,2,3,4]), higher-order polynomial approximations such as piecewise parabolic method (PPM) [5,6], essentially non-oscillatory methods (ENO) (e.g., [7,8]), weighted ENO (WENO) methods (e.g., [9,10,11,12]) and other variants such as central WENO (CWENO) [13,14,15,16], Hermite WENO (HWENO) [17], adaptive-order WENO (AO-WENO) [18,19], polynomial-free Gaussian Process WENO (GP-WENO) [20,21,22], to name a few.…”