Refinement workflow

Vary one control at a time and follow the sequence for the operator in use.

Operator	Likely refinement sequence	Compare against
Full history	Reduce \(\Delta t\)	Exact solution or the preceding timestep
Diffusive memory	Increase mode count \(L\) → grade the time levels if the solution has an initial singularity → reduce \(\Delta t\) → compare linear and quadratic interpolation if needed	Analytic or high-accuracy reference
Sum of exponentials	Lower target_error → reduce \(\Delta t\) and rebuild with the new min_step	Analytic or full-history reference
Sine diffusive memory	Increase mode count \(L\) → reduce \(\Delta t\) → extend the comparison interval	Full history on the same time grid
Exponential memory	Reduce \(\Delta t\)	The preceding timestep
Spectral fractional Laplacian	Tighten sinc_truncation_target → refine the mesh	Generalized eigenvalues
Riesz, dense or matrix-free	Increase quadrature_degree → refine the mesh	Over-resolved direct energy
Riesz, H-matrix	Increase quadrature_degree → lower compression_tolerance → refine the mesh	Over-resolved direct energy
Periodic Fourier fractional Laplacian	Increase the uniform grid resolution	Resolved Fourier modes

For a time-dependent PDE, refine the time and space controls separately. Keep iterative solver tolerances below the measured discretization error. See Accuracy and performance for numerical examples.

For smooth solved problems, linear and quadratic recurrence have orders \(2-\alpha\) and \(3-\alpha\). Both are approximately first order on uniform steps for a \(t^\alpha\) initial singularity, where grading the steps gains far more than the interpolant does.