FourierFlows.jl Documentation

Overview

FourierFlows provides solvers for partial differential equations on doubly-periodic domains using Fourier-based pseudospectral methods. A central intent of the software's design is also to provide a framework for writing new, fast solvers for new physical problems. The code is written in Julia.

Installation

FourierFlows is a registered package and so can be installed via Pkg.add.

Pkg.add("FourierFlows")

For now, this package supports Julia 0.6. Support for version 0.7 is on its way.

Basic Notation

The code solves partial differential equations of the general form:

\[\partial_t u = \mathcal{L}u + \mathcal{N}(u)\ .\]

We decompose the right hand side of the above in a linear part ($\mathcal{L}u$) and a nonlinear part ($\mathcal{N}(u)$). The time steppers treat the linear and nonlinear parts differently.

The coefficients for the linear operator $\mathcal{L}$ are stored in array LC. The term $\mathcal{N}(u)$ is computed for by calling the function calcN!.

Source code organization

The code is divided along conceptual lines into problem-agnostic and problem-specific components. Files that contain problem-agnostic parts of the code are stored in /src. Files in /src define the domain, 'AbstractTypes' that supertype problem-specific types, and time-stepper types and routines. Problem-specific modules are stores in /src/physics.

Here's an overview of the code structure:

/src/
- FourierFlows.jl
  - Defines supertyping AbstractParams, AbstractGrid, etc.
  - Defines a Problem type to organize the grid, vars, params, equation, and timestepper into a single structure.
  - Includes all sources files and physics files.
- timesteppers.jl: defines modules and stepforward! routines for various time-steppers. Current implemented time-steppers are:
  - Forward Euler
  - 3rd-order Adams-Bashforth (AB3)
  - 4th-order Runge-Kutta (RK4)
  - 4th-order Runge-Kutta Exponential Time Differencing (ETDRK4)
  - 4th-order Dual Runge-Kutta (DualRK4)
  - 4th-order Dual Runge-Kutta Exponential Time Differencing (DualETDRK4)
  For each time-stepper exists also a "filtered" version that filters out high-wavenumber spectral components of the solution. The Dual time-steppers evolve a state variable that comprises both of real valued and complex valued fields.
- physics/
  - twodturb.jl: Defines a TwoDTurb module that provides a solver for the two-dimensional vorticity equation.
  - barotropicqg.jl: Defines a BarotropicQG module that provides several solvers for the barotropic QG model that permit beta, topography, beta + topography, and forcing.
  - kuramotosivashinsky.jl: Defines a KuramotoSivashinsky module that solves the Kuramoto-Sivashinsky.
  - verticallyfourierboussinesq.jl: Defines a VerticallyFourierBoussinesq module that solves the two-mode truncation of the Fourier series thin-layer approximation to the hydrostatic Boussinesq equations.
  - verticallycosinerboussinesq.jl: Defines a VerticallyCosineBoussinesq module that solves the two-mode truncation of the Sin/Cos series thin-layer approximation to the hydrostatic Boussinesq equations.
  - traceradvdiff.jl: Defines a TracerAdvDiff module that provides a solver for a two-dimensional and periodic tracer field in a given 2D flow (u, w), which can be an arbitrary function of x, z, and t.

Writing fast solvers

The performance-intensive part of the code involves just two functions: the time-stepping scheme stepforward!, and the function calcN! that calculates the nonlinear part of the given equation's right-hand side. Optimization of these two functions for a given problem will produce the fastest possible code.

Examples

examples/twodturb/McWilliams.jl: A script that simulates decaying two-dimensional turbulence reproducing the results of the paper by
McWilliams, J. C. (1984). The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech., 146, 21-43
examples/barotropicqg/decayingbetaturb.jl: An example script that simulates decaying quasi-geostrophic flow on a beta-plane demonstrating zonation.
examples/barotropicqg/ACConelayer.jl: A script that simulates barotropic quasi-geostrophic flow above topography reproducing the results of the paper by
Constantinou, N. C. (2018). A barotropic model of eddy saturation. J. Phys. Oceanogr., 48 (2), 397-411

Tutorials

BarotropicQG Module

Future work

The code is in the chaotic stage of development. A main goal for the future is to permit the use of shared memory parallelism in the compute-intensive routines (shared-memory parallelism provided already by FFTW/MKLFFT, but is not yet native to Julia for things like element-wise matrix multiplication, addition, and assignment). This feature may possibly be enabled by Intel Lab's ParallelAccelerator package.