Yonderdrake¶

Alpha software: Yonderdrake has been broadly tested, but bugs and missing features may remain. Please report them as issues.

Yonderdrake adds fractional time derivatives, fading-memory operators, and nonlocal spatial operators to UFL forms solved with Firedrake. You write an ordinary residual with a marker in it. Yonderdrake does the rest. Equations containing both a nonlocal time operator and a nonlocal space operator are supported as well.

F = (inner(CaputoDerivative(u, 0.6), v) + inner(u, v)) * dx
stepper = FractionalTimeStepper(F, BirkSong(48), t, dt, u)
stepper.advance()

What it supports¶

Caputo and initialized Riemann-Liouville time derivatives.
Full-history stepping and static memory with \(O(m)\) storage.
Single-exponential memory, including a labelled Caputo-Fabrizio operator.
Dirichlet spectral, zero-exterior Riesz, and periodic Fourier fractional Laplacians, each with its corresponding boundary or exterior realization. The Riesz operator supports CG1 and CG2 on triangular 2D and tetrahedral 3D meshes.
Direct 2D and 3D Caputo-Wismer wave support for power-law attenuation, with heterogeneous density, volume and boundary sources, impedance boundaries, PML, user-defined sensor arrays, exact adjoints, time reversal, and regularized inverse reconstruction.
Variable steps, MPI, PETSc options, and checkpoint/restart.
Spatial fractional operators combined with Irksome for classical time integration.

Start here¶

Installation: install into a Firedrake environment.
Quickstart: the core workflow in one page.
Guides and examples: one capability at a time.
Gallery: larger applications and their animations.
Mathematics and methods: definitions, methods, and the papers behind them.
API and supported scope: signatures and exact limitations.

Guides and examples¶

After the Quickstart, choose the time-memory workflow or the space-fractional workflow. Each guide links to a runnable example. The Gallery collects larger applications and animations. Refinement, parallel execution, checkpointing, and performance guidance are grouped under Applications and workflow in the sidebar.

Mathematics and methods¶

The method map is the short overview of the available realizations and their sources. Detailed definitions and numerical guidance live with each operator family. The complete bibliography is in References.

Method map¶

Operator	Method	What it is	Use	Primary source
Caputo and Riemann-Liouville	`Cayley`	Diffusive modes, Gauss-Jacobi after a Cayley map of selectable exponent	Default time memory	Generalizes Diethelm (2008) and Birk and Song (2010)
	`Jacobi`	Diffusive modes with independent low-rate and high-rate map exponents	Expert option for measured asymmetric rate ranges	Diethelm (2023)
	`SumOfExponentials`	Tolerance-driven positive exponential sum on a declared time interval	Supported alternative	Jiang et al. (2017)
	`Diethelm2022`	Truncated log-rate quadrature	Comparison only	Diethelm (2022), (2023)
	`YuanAgrawal`	Original Gauss-Laguerre rule	Comparison only	Yuan and Agrawal (2002)
	`SineDiffusive`	Generalized Gauss-Laguerre quadrature of undamped sine modes	Comparison only	Khosravian-Arab and Dehghan (2024)
	`LubichCQ`	Uniform-grid BDF1 or BDF2 convolution quadrature with starting corrections	Direct first- or second-order method	Lubich (1986), (1988, Part I), (1988, Part II)
	`FastObliviousCQ`	BDF1 CQ with dyadic Talbot-contour history	Long uniform-grid histories	Schädle, López-Fernández, and Lubich (2006)
	`AlikhanovL21Sigma`	Uniform-grid quadratic formula at \(t_{n+\sigma}\)	Direct second-order method	Alikhanov (2015)
	`FullHistory`	Variable-step L1 history integral	Reference method	Lin and Xu (2007)
Exponential memory	`ExponentialMemory`	One-timescale exponential convolution	Direct fading-memory operator	Standard one-state realization
Spectral fractional Laplacian	`SpectralFractionalLaplacian`	Power of the homogeneous-Dirichlet or natural-Neumann Laplacian	Spectral realization	Bonito and Pasciak (2015)
Riesz fractional Laplacian	`RieszFractionalLaplacian`	Whole-space integral on the zero extension	Integral realization	Acosta and Borthagaray (2017)
Periodic fractional Laplacian	`PeriodicFractionalLaplacian`	Fourier multiplier on a uniform periodic cell	Periodic realization	Standard Fourier-series multiplier

Convenience constructors¶

Published members of the families above. Each is the general class at a fixed setting and behaves identically to it.

Constructor	Equivalent to	Use	Primary source
`BirkSong(n)`	`Cayley(n, power=4)`	Default representation	Birk and Song (2010)
`Diethelm2008(n)`	`Cayley(n, power=2)`	Narrow rate spans	Diethelm (2008)
`CaputoFabrizioOperator`	Rescaled `ExponentialMemory`	Labelled Caputo-Fabrizio interface	Caputo and Fabrizio (2015), with the classification in Ortigueira and Machado (2018)