Direct convolution methods

The diffusive representations replace the power-law kernel with a fixed set of memory modes. Yonderdrake also provides three uniform-grid methods that act on the convolution itself. They are useful when a published discretization is part of the model, when second-order time accuracy is important, or when a long run makes direct history too expensive.

Lubich convolution quadrature

LubichCQ generates convolution weights from a backward difference symbol. For the Caputo derivative, it applies the weights to \(u-u(0)\):

\[ D_C^\alpha u(t_n) \approx h^{-\alpha}\sum_{j=0}^n \omega_{n-j}\left(u_j-u_0\right), \qquad \sum_{j=0}^\infty \omega_j\zeta^j=\delta(\zeta)^\alpha. \]

The available symbols are

\[ \delta_{\mathrm{BDF1}}(\zeta)=1-\zeta, \qquad \delta_{\mathrm{BDF2}}(\zeta)=\frac32-2\zeta+\frac12\zeta^2. \]

BDF2 is the default. num_corrections adds Lubich starting weights that are exact for the first powers \(t^{\alpha},t^{2\alpha},\ldots\). The default is one correction for BDF1 and two for BDF2.

It uses \(O(n)\) distributed-field storage and \(O(n)\) work at step \(n\), giving \(O(N^2)\) total history work. It requires a uniform timestep. BDF2 gives second-order convergence for sufficiently regular data. Initial singularities can reduce the observed order, which is why the starting corrections are part of the default.

Alikhanov L2-1-sigma

AlikhanovL21Sigma uses Alikhanov’s quadratic L2-1\(\sigma\) approximation with

\[ \sigma=1-\frac{\alpha}{2}. \]

The complete residual is evaluated at \(t_{n+\sigma}\). Yonderdrake therefore replaces an ordinary occurrence of the solution by \(\sigma u_{n+1}+(1-\sigma)u_n\) and replaces the symbolic time by \(t_n+\sigma h\). The fractional marker and every other term consequently use the same offset equation.

The method is second order on smooth solutions and usually retains better accuracy than uncorrected high-order formulas near a weak initial singularity. It requires a uniform timestep, Caputo markers with one shared \(\alpha\), and stores the full increment history. Its storage and total history work are \(O(N)\) fields and \(O(N^2)\) respectively.

Fast oblivious convolution quadrature

FastObliviousCQ accelerates BDF1 convolution quadrature. Recent increments are applied exactly. Older increments are grouped into dyadic blocks and integrated on conjugate Talbot contours. Complex contour states are represented internally by coupled real arrays, so the public Firedrake problem still uses real scalar fields.

For \(N\) accepted steps, the history has \(O(\log N)\) work per step and \(O(\log N)\) distributed-field storage, giving \(O(N\log N)\) total history work. The constants include the contour nodes at every active level, so direct history can remain smaller and faster for short runs. The time discretization is BDF1 and is therefore first order.

The principal controls are:

Control	Meaning
target_error	Selects the default number of contour nodes
nodes_per_level	Explicit contour quadrature size per dyadic level
direct_steps	Number of recent increments kept exact
num_levels	Dyadic depth, supporting at most \(2^{\mathtt{num\_levels}}-1\) steps
contour	Currently the published "talbot" contour

The method requires a uniform timestep. Increase nodes_per_level to check contour convergence independently of timestep convergence. Increase num_levels before a run if its configured maximum step count is too small.

To choose between these and the diffusive representations, see Choosing a method.