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

What it supports

• Caputo and initialized Riemann-Liouville time derivatives with full-history stepping, direct convolution quadrature or static memory with \(O(m)\) storage.

• Single-exponential memory, including a labelled Caputo-Fabrizio operator.

• Spectral (Dirichlet or Neumann), 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

Each guide covers one capability and links to a runnable example.

• Fractional time derivatives and Time-fractional PDEs: fractional time derivatives.

• Exponential memory and Caputo-Fabrizio: single-exponential and Caputo-Fabrizio memory.

• Fractional space: spectral, Riesz, and periodic fractional Laplacians.

• Refinement workflow, Solvers, field splits, and MPI, Checkpoint and restart, and Classical time stepping with Irksome: workflows.

• Caputo-Wismer imaging: fractionally damped waves and imaging.

Mathematics and methods

• Choosing a method: which method to use, with its primary source.

• Time-memory theory: time derivatives, memory representations, and time stepping.

• Spectral fractional Laplacian, Zero-exterior Riesz fractional Laplacian, and Periodic Fourier fractional Laplacian: the spatial realizations.

• Conventions and normalizations: notation, normalizations, and terminology.

• References: the complete bibliography.