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

TL;DR

GP-MOOD is a high-order finite volume method combining Gaussian Process reconstruction with MOOD detection, enabling positivity-preserving solutions for hyperbolic conservation laws without traditional limiters.

Contribution

This paper introduces GP-MOOD, a novel high-order finite volume method that integrates Gaussian Process reconstruction with MOOD detection for stable, accurate solutions of hyperbolic systems.

Findings

Achieves high-order accuracy (up to seventh-order) in multiple dimensions.

Maintains positivity without traditional limiters.

Demonstrates stability and accuracy on smooth and discontinuous flows.

Abstract

We present an a posteriori shock-capturing finite volume method algorithm called GP-MOOD that solves a compressible hyperbolic conservative system at high-order solution accuracy (e.g., third-, fifth-, and seventh-order) in multiple spatial dimensions. The GP-MOOD method combines two methodologies, the polynomial-free spatial reconstruction methods of GP (Gaussian Process) and the a posteriori detection algorithms of MOOD (Multidimensional Optimal Order Detection). The spatial approximation of our GP-MOOD method uses GP's unlimited spatial reconstruction that builds upon our previous studies on GP reported in Reyes et al., Journal of Scientific Computing, 76 (2017) and Journal of Computational Physics, 381 (2019). This paper focuses on extending GP's flexible variability of spatial accuracy to an a posteriori detection formalism based on the MOOD approach. We show that GP's polynomial-free reconstruction provides a seamless pathway to the MOOD's order cascading formalism by utilizing GP's novel property of variable (2R+1)th-order spatial accuracy on a multidimensional GP stencil defined by the GP radius R, whose size is smaller than that of the standard polynomial MOOD methods. The resulting GP-MOOD method is a positivity-preserving method. We examine the numerical stability and accuracy of GP-MOOD on smooth and discontinuous flows in multiple spatial dimensions without resorting to any conventional, computationally expensive a priori nonlinear limiting mechanism to maintain numerical stability.