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

TL;DR

This work introduces an adaptive hyperviscosity stabilisation scheme for the PDE-independent RBF-FD method to solve advection-dominated transport equations on boundaryless domains. Stability is achieved by dynamically selecting the hyperviscosity constant from the spectral radius of the evolution matrix, with a practical procedure that recomputes the parameter for nonlinear problems. The authors demonstrate consistency and reduced computational cost by employing lower monomial augmentation for hyperviscosity and by using different spline orders for the advection and hyperviscosity operators. Numerical experiments on linear advection and Burgers' equation show stable, low-dissipation solutions and provide guidance on parameter choices and re-computation frequency, highlighting the method's potential for robust meshless simulations of advection-dominated transport problems.

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.

Figures (16)

• Figure 1: Eigenvalue spectra of the stabilised advection operator $\hat{\bm{D}}_h$\ref{['eq:advection_op']} using $N \approx 10^3$ with respect to different values of $c$ for $\alpha = 2$. The shaded region in the figure is the implicit Euler stability region. The black dots represent stable eigenvalues, whereas the red dots represent eigenvalues that lie outside of the stability region. Eigenvalues are scaled with $h$ for ease of visualisation.
• Figure 2: Spectral radius $\rho$ of the evolution matrix $\bm{G}_h$\ref{['eq:dynamical_system_hv']} at $\Delta t = 10^{-4}$ with respect to $c \in [10^{-4}, 10^2]$ for internodal distances $h \in \{0.01, 0.02, 0.03, 0.04\}$ and orders of hyperviscosity $\alpha \in \{2,3,4\}$. The solid line shows $\rho$ calculated with the full order monomial augmentation as suggested for the RBF-FD approximation while the dotted line displays effects of using a reduced monomial order of $m=2$ with stencil size $n = 30$.
• Figure 3: Eigenvalue spectra of the stabilised advection operator $\hat{\bm{D}}_h$\ref{['eq:advection_op']} using $N \approx 10^3$ with respect to different hyperviscosity orders $\alpha$ using $c_{opt}$. The shaded region in the figure is the implicit Euler stability region. Eigenvalues are scaled with $h$ for ease of visualisation.
• Figure 4: Eigenvalue spectra of the advection operator $\bm{D}_h$ for different orders of polyharmonic splines $k$ at $h=0.03$. The eigenvalues are scaled by $\lambda = h\tilde{\lambda}$. The shaded region represents the implicit Euler stability region.
• Figure 5: Solution of the linear advection equation at two different times $t \in \{0.24, 0.32\}$ with $h = 0.01$. The stabilised solution (left) admits no visible artefacts, compared to the non-stabilised solution (right) where major instabilities can be observed.

Theorems & Definitions (2)

• Proposition 1
• proof