Separating error sources

A fractional simulation has several independent error sources. Reporting their sum as one “fractional error” hides which control actually needs refining. Vary one at a time.

Each operator has its own numerical controls. Mesh and solver refinement come after the representation-specific error is small enough not to dominate.

Time-memory operators

Full history

Refine the timestep. There is no quadrature spectrum or mode count to tune.

Diffusive memory

Fix a sufficiently small timestep, then refine the mode count. Check the representation’s rate scaling separately rather than compensating for a coarse time grid with more modes.

Spatial operators

Spectral fractional Laplacian

Fix the mesh, tighten the sinc truncation target, and compare against generalized eigenvalues. Only then refine the finite-element mesh.

Riesz/restricted fractional Laplacian

Fix the mesh, refine the target quadrature rule and degree, and compare against an over-resolved direct energy calculation. Refine H-matrix compression tolerance separately from quadrature.

Periodic Fourier fractional Laplacian

There is no representation quadrature to refine. Increase the uniform grid resolution while keeping the represented Fourier modes below the Nyquist limit. For nonsmooth periodic data, check aliasing and the decay of the highest retained modes separately.

Shared controls

Refine the mesh only after representation and quadrature controls are tight enough not to dominate. Keep algebraic solver tolerances below the measured discretization error.

Two habits are worth enforcing in your own studies: never report a single unexplained error figure, and never compare spectral and Riesz solutions as though they shared a boundary realization (they do not).

The Benchmarks and comparisons give concrete numerical comparisons and vary one control at a time by construction.