Adaptive hyperviscosity stabilisation for the RBF-FD method in solving advection-dominated transport equations

TL;DR

This paper introduces an adaptive hyperviscosity stabilization method for the RBF-FD technique, effectively solving advection-dominated equations with improved efficiency and stability across various node layouts.

Contribution

It develops a PDE-independent, adaptive hyperviscosity procedure for RBF-FD that reduces computational costs and enhances stability without empirical tuning.

Findings

Stable performance on linear advection and Burgers' equation

Supports general node layouts and reduces computational cost

Employs hybrid spline strategies for improved stability

Abstract

This paper presents an adaptive hyperviscosity stabilisation procedure for the Radial Basis Function-generated Finite Difference (RBF-FD) method, aimed at solving linear and non-linear advection-dominated transport equations on domains without a boundary. The approach employs a PDE-independent algorithm that adaptively determines the hyperviscosity constant based on the spectral radius of the RBF-FD evolution matrix. The proposed procedure supports general node layouts and is not tailored for specific equations, avoiding the limitations of empirical tuning and von Neumann-based estimates. To reduce computational cost, it is shown that lower monomial augmentation in the approximation of the hyperviscosity operator can still ensure consistent stabilisation, enabling the use of smaller stencils and improving overall efficiency. A hybrid strategy employing different spline orders for the advection and hyperviscosity operators is also implemented to enhance stability. The method is evaluated on pure linear advection and non-linear Burgers' equation, demonstrating stable performance with limited numerical dissipation. The two main contributions are: (1) a general hyperviscosity RBF-FD solution procedure demonstrated on both linear and non-linear advection-dominated problems, and (2) an in-depth analysis of the behaviour of hyperviscosity within the RBF-FD framework, addressing the interplay between key free parameters and their influence on numerical results.