The paper proposes an equation-independent adaptive hyperviscosity stabilization for radial-basis-function-generated finite differences on periodic, boundaryless domains. It assembles the RBF-FD evolution matrix, uses its largest-magnitude eigenvalue to determine whether the selected time integrator is stable, and applies bisection to find the smallest hyperviscosity constant that places the spectrum inside the stability region. An error estimate shows that the high-order hyperviscosity operator can use lower-degree monomial augmentation consistently, reducing stencil size and sparsity costs. The authors also recommend different polyharmonic-spline orders for advection and hyperviscosity, using order three for advection and
- Gave a spectral-radius-based algorithm that adaptively selects a stable hyperviscosity constant without equation-specific von Neumann tuning.
- Proved consistency for hyperviscosity RBF-FD approximations using monomial degree below the derivative order, enabling smaller stencils.
- Introduced a hybrid spline-order prescription using order three for advection and
$2\alpha+1$ for hyperviscosity. - Mapped interactions among hyperviscosity, spline order, stencil size, time step, stability, and dissipation in linear advection.
- Demonstrated time-dependent stabilizer recomputation for nonlinear Burgers flow and quantified sensitivity to recomputation frequency.