Public Documentation

Main module for FourierFlows.jl – an ecosystem for solving partial differential equations on periodic domains using Fourier-based pseudospectral methods."

Exports

• AB3TimeStepper
• AbstractGrid
• AbstractParams
• AbstractTimeStepper
• AbstractVars
• CPU
• Device
• Diagnostic
• ETDRK4TimeStepper
• FilteredAB3TimeStepper
• FilteredETDRK4TimeStepper
• FilteredForwardEulerTimeStepper
• FilteredLSRK54TimeStepper
• FilteredRK4TimeStepper
• ForwardEulerTimeStepper
• GPU
• LSRK54TimeStepper
• OneDGrid
• Output
• RK4TimeStepper
• ThreeDGrid
• TimeStepper
• TwoDGrid
• @createarrays
• cxtype
• dealias!
• device_array
• @devzeros
• fft
• fltype
• gridpoints
• groupsize
• ifft
• increment!
• innereltype
• irfft
• resize!
• rfft
• savediagnostic
• saveoutput
• saveproblem
• step_until!
• stepforward!
• supersize
• @superzeros
• superzeros
• update!
• @zeros

Abstract supertype for grids.

Abstract supertype for parameters.

Abstract supertype for timesteppers.

Abstract supertype for variables.

CPU device.

Abstract supertype for supported devices.

GPU device.

OneDGrid(dev::Device = CPU();
         nx, Lx,
         x0 = -Lx/2, nthreads = Sys.CPU_THREADS, effort = FFTW.MEASURE, 
         T = Float64, aliased_fraction = 1/3)

Construct a OneDGrid on device; by default on CPU(). Grid size is Lx, resolution is nx, and leftmost position is x0. FFT plans are generated for nthreads CPUs using FFTW flag effort. The float type is T. The aliased_fraction keyword determines the highest wavenubers that are being zero-ed out by dealias! function; 1/3 is the nominal value for quadratic nonlinearities.

struct OneDGrid{T<:AbstractFloat, A, Tx, Tfft, Trfft, Talias} <: AbstractGrid{T, A, Talias, D}

A one-dimensional grid.

• device::Any: device which the grid lives on

• nx::Int64: number of points in $x$

• nk::Int64: number of wavenumbers in $x$

• nkr::Int64: number of positive wavenumbers in $x$ (real Fourier transforms)

• dx::AbstractFloat: grid spacing in $x$

• Lx::AbstractFloat: domain extent in $x$

• x::Any: range with $x$-grid-points

• k::Any: array with $x$-wavenumbers

• kr::Any: array with positive $x$-wavenumbers (real Fourier transforms)

• invksq::Any: array with inverse squared $k$-wavenumbers, $1 / k²$

• invkrsq::Any: array with inverse squared $kᵣ$-wavenumbers, $1 / kᵣ²$

• fftplan::Any: the FFT plan for complex-valued fields

• rfftplan::Any: the FFT plan for real-valued fields

• aliased_fraction::AbstractFloat: the fraction of wavenumbers that are aliased (e.g., 1/3 for quadradic nonlinearities)

• kalias::Any: range of the indices of aliased $x$-wavenumbers

• kralias::Any: range of the indices of aliased positive $x$-wavenumbers (real Fourier transforms)

ThreeDGrid(dev::Device=CPU(); nx, Lx, ny=nx, Ly=Lx, nz=nx, Lz=Lx,
           x0=-Lx/2, y0=-Ly/2, z0=-Lz/2,
           nthreads=Sys.CPU_THREADS, effort=FFTW.MEASURE, T=Float64,
           aliased_fraction=1/3)

Construct a ThreeDGrid on device; by default on CPU(). Grid size is Lx, Ly, Lz, resolution is nx, ny, nz and leftmost positions are x0, y0, z0. FFT plans are generated for nthreads CPUs using FFTW flag effort. The float type is T. The aliased_fraction keyword determines the highest wavenubers that are being zero-ed out by dealias! function; 1/3 is the nominal value for quadratic nonlinearities.

struct ThreeDGrid{T<:AbstractFloat, Tk, Tx, Tfft, Trfft, Talias} <: AbstractGrid{T, Tk, Talias}

A three-dimensional grid.

• device::Any: device which the grid lives on

• nx::Int64: number of points in $x$

• ny::Int64: number of points in $y$

• nz::Int64: number of points in $z$

• nk::Int64: number of wavenumbers in $x$

• nl::Int64: number of wavenumbers in $y$

• nm::Int64: number of wavenumbers in $z$

• nkr::Int64: number of positive wavenumers in $x$ (real Fourier transforms)

• dx::AbstractFloat: grid spacing in $x$

• dy::AbstractFloat: grid spacing in $y$

• dz::AbstractFloat: grid spacing in $z$

• Lx::AbstractFloat: domain extent in $x$

• Ly::AbstractFloat: domain extent in $y$

• Lz::AbstractFloat: domain extent in $z$

• x::Any: range with $x$-grid-points

• y::Any: range with $y$-grid-points

• z::Any: range with $z$-grid-points

• k::Any: array with $x$-wavenumbers

• l::Any: array with $y$-wavenumbers

• m::Any: array with $z$-wavenumbers

• kr::Any: array with positive $x$-wavenumbers (real Fourier transforms)

• Ksq::Any: array with squared total wavenumbers, $k² + l² + m²$

• invKsq::Any: array with inverse squared total wavenumbers, $1 / (k² + l² + m²)$

• Krsq::Any: array with squared total wavenumbers for real Fourier transforms, $kᵣ² + l² + m²$

• invKrsq::Any: array with inverse squared total wavenumbers for real Fourier transforms, $1 / (kᵣ² + l² + m²)$

• fftplan::Any: the FFT plan for complex-valued fields

• rfftplan::Any: the FFT plan for real-valued fields

• aliased_fraction::AbstractFloat: the fraction of wavenumbers that are aliased (e.g., 1/3 for quadradic nonlinearities)

• kalias::Any: range of the indices of aliased $x$-wavenumbers

• kralias::Any: range of the indices of aliased positive $x$-wavenumbers (real Fourier transforms)

• lalias::Any: range of the indices of aliased $y$-wavenumbers

• malias::Any: range of the indices of aliased $m$-wavenumbers

TwoDGrid(dev::Device=CPU(); nx, Lx, ny=nx, Ly=Lx,
         x0=-Lx/2, y0=-Ly/2, nthreads=Sys.CPU_THREADS, effort=FFTW.MEASURE,
         T=Float64, aliased_fraction=1/3)

Construct a TwoDGrid on device; by default on CPU(). Grid size is Lx, Ly, resolution is nx, ny, and leftmost positions are x0, y0. FFT plans are generated for nthreads CPUs using FFTW flag effort. The float type is T. The aliased_fraction keyword determines the highest wavenubers that are being zero-ed out by dealias! function; 1/3 is the nominal value for quadratic nonlinearities.

struct TwoDGrid{T<:AbstractFloat, A, Tx, Tfft, Trfft, Talias, D} <: AbstractGrid{T, Tk, Talias, D}

A two-dimensional grid.

• device::Any: device which the grid lives on

• nx::Int64: number of points in $x$

• ny::Int64: number of points in $y$

• nk::Int64: number of wavenumbers in $x$

• nl::Int64: number of wavenumbers in $y$

• nkr::Int64: number of positive wavenumers in $x$ (real Fourier transforms)

• dx::AbstractFloat: grid spacing in $x$

• dy::AbstractFloat: grid spacing in $y$

• Lx::AbstractFloat: domain extent in $x$

• Ly::AbstractFloat: domain extent in $y$

• x::Any: range with $x$-grid-points

• y::Any: range with $y$-grid-points

• k::Any: array with $x$-wavenumbers

• l::Any: array with $y$-wavenumbers

• kr::Any: array with positive $x$-wavenumbers (real Fourier transforms)

• Ksq::Any: array with squared total wavenumbers, $k² + l²$

• invKsq::Any: array with inverse squared total wavenumbers, $1 / (k² + l²)$

• Krsq::Any: array with squared total wavenumbers for real Fourier transforms, $kᵣ² + l²$

• invKrsq::Any: array with inverse squared total wavenumbers for real Fourier transforms, $1 / (kᵣ² + l²)$

• fftplan::Any: the FFT plan for complex-valued fields

• rfftplan::Any: the FFT plan for real-valued fields

• aliased_fraction::AbstractFloat: the fraction of wavenumbers that are aliased (e.g., 1/3 for quadradic nonlinearities)

• kalias::Any: range of the indices of aliased $x$-wavenumbers

• kralias::Any: range of the indices of aliased positive $x$-wavenumbers (real Fourier transforms)

• lalias::Any: range of the indices of aliased $y$-wavenumbers

dealias!(fh, grid)

Dealias array fh on the grid based on the grids's aliased_fraction.

gridpoints(grid::OneDDGrid)
gridpoints(grid::TwoDGrid)
gridpoints(grid::ThreeDGrid)

Return the collocation points of the grid in 1D (X), 2D (X, Y) or 3D arrays (X, Y, Z).

cxtype(T)

Returns T when T is Complex, or Complex{T} when T is Real.

device_array(grid::AbstractGrid)

Return the proper array type according to the grid's device, i.e., Array for CPU and CuArray for GPU.

device_array(device::Device)
device_array(device::Device, T, dim)

Return the proper array type according to the device, i.e., Array for CPU and CuArray for GPU.

fltype(T)

Returns T when T<:AbstractFloat or Tf when T<:Complex{Tf}.

innereltype(x)

Recursively determine the 'innermost' type in by the collection x (which may be, for example, a collection of a collection).

superzeros(T, A)

Returns an array like A, but full of zeros. If innereltype(A) can be promoted to T, then the innermost elements of the array will have type T.

@devzeros dev T dims a b c...

Create arrays of all zeros with element type T, size dims, and global names a, b, c (for example) on device dev.

@superzeros T a b c d...
@superzeros T dims b c d...

Generate arrays b, c, d... with the super-dimensions of a and innereltype T.

@zeros T dims a b c...

Create arrays of all zeros with element type T, size dims, and global names a, b, c (for example). An arbitrary number of arrays may be created.

mutable struct Diagnostic{T, N} <: AbstractDiagnostic

A diagnostic that includes N elements of type T.

• calc::Function: function that returns the diagnostic via calc(prob)

• prob::FourierFlows.Problem: the relevant problem for this diagnostic

• data::Vector: vector where the diagnostic time-series is saved

• t::Vector{Float64}: vector with the times for which the diagnostic was saved

• steps::Vector{Int64}: vector with the problem's step for which the diagnostic was saved

• freq::Int64: integer denoting how often (every how many problem.steps) to save the diagnostic

• i::Int64: integer denoting how many times the diagnostic.data was updated

Diagnostic(calc, prob; freq=1, nsteps=100, ndata=ceil(Int, (nsteps+1)/freq))

Construct a diagnostic that stores the result of calc(prob) with frequency freq.

Keywords

• freq: Diagnostic is saved every freq steps.
• nsteps: The total number of steps in problem.
• ndata: The number of diagnostics to be saved.

increment!(diag)

Increment the Diagnostic diag, or an array of Diagnostics diags.

update!(diag)

Update diag.

struct Output

The composite type for output.

• prob::FourierFlows.Problem: the relevant problem for the output

• path::String: the path for the output file

• fields::Dict{Symbol, Function}: the fields to be saved; the relevant problem for this diagnostic

Output(prob, filename, fieldtuples...)

Define output for prob with fields and functions that calculate the output in the list of tuples fieldtuples = (fldname, func)....

savediagnostic(diagnostic, diagname, filename)

Save diagnostic to filename under name diagname. Only the computed diagnostic is saved, that is, everything up to diagnostic's iteration diagnostic.i.

saveoutput(out)

Save the fields in out.fields to out.path.

saveproblem(prob, filename)

Save certain aspects of a problem prob to filename.

AB3TimeStepper(equation::Equation, dev::Device=CPU())

Construct a 3rd order Adams-Bashforth timestepper for equation on device dev.

struct AB3TimeStepper{T} <: AbstractTimeStepper{T}

A 3rd-order Adams-Bashforth timestepper for time-stepping ∂u/∂t = RHS(u, t) via:

uⁿ⁺¹ = uⁿ + dt/12 * (23 * RHS(uⁿ, tⁿ) - 16 * RHS(uⁿ⁻¹, tⁿ⁻¹) + 5 * RHS(uⁿ⁻², tⁿ⁻²))

Adams-Bashforth is a multistep method, i.e., it not only requires information from the n-th time-step (uⁿ) but also from the previous two timesteps (uⁿ⁻¹ and uⁿ⁻²). For the first two timesteps, it falls back to a forward Euler timestepping scheme:

uⁿ⁺¹ = uⁿ + dt * RHS(uⁿ, tⁿ)

ETDRK4TimeStepper(equation::Equation, dt, dev::Device=CPU())

Construct a 4th-order exponential-time-differencing Runge-Kutta timestepper with timestep dt for equation on device dev.

struct ETDRK4TimeStepper{T,TL} <: AbstractTimeStepper{T}

A 4th-order exponential-time-differencing Runge-Kutta timestepper for time-stepping ∂u/∂t = L * u + N(u). The scheme treats the linear term L exactly while for the nonlinear terms N(u) it uses a 4th-order Runge-Kutta scheme (RK4TimeStepper). That is,

uⁿ⁺¹ = exp(L * dt) * uⁿ + RK4(N(uⁿ))

For more info refer to Kassam and Trefethen (2005).

FilteredAB3TimeStepper(equation::Equation, dev::Device=CPU(); filterkwargs...)

Construct a 3rd order Adams-Bashforth timestepper with spectral filtering for equation on device dev.

struct FilteredAB3TimeStepper{T} <: AbstractTimeStepper{T}

A 3rd order Adams-Bashforth timestepper with spectral filtering. See AB3TimeStepper.

FilteredETDRK4TimeStepper(equation, dt; filterkwargs...)

Construct a 4th-order exponential-time-differencing Runge-Kutta timestepper with timestep dt and spectral filtering for equation on device dev.

FilteredETDRK4TimeStepper{T,TL,Tf} <: AbstractTimeStepper{T}

A 4th-order exponential-time-differencing Runge-Kutta timestepper with spectral filtering. See ETDRK4TimeStepper.

FilteredForwardEulerTimeStepper(equation, dev; filterkwargs...)

Construct a Forward Euler timestepper with spectral filtering for equation on device dev.

struct FilteredForwardEulerTimeStepper{T,Tf} <: AbstractTimeStepper{T}

A forward Euler timestepper with spectral filtering. See ForwardEulerTimeStepper.

FilteredRK4TimeStepper(equation::Equation, dev::Device=CPU(); filterkwargs...)

Construct a 4th-order 5-stages 2-storage Runge-Kutta timestepper with spectral filtering for equation on device dev.

struct FilteredLSRK54TimeStepper{T,V,Tf} <: AbstractTimeStepper{T}

A 4th-order 5-stages low-storage Runge-Kutta timestepper with spectral filtering. See LSRK54TimeStepper.

FilteredRK4TimeStepper(equation::Equation, dev::Device=CPU(); filterkwargs...)

Construct a 4th-order Runge-Kutta timestepper with spectral filtering for equation on device dev.

struct FilteredRK4TimeStepper{T,Tf} <: AbstractTimeStepper{T}

A 4th-order Runge-Kutta timestepper with spectral filtering. See RK4TimeStepper.

ForwardEulerTimeStepper(equation::Equation, dev::Device=CPU())

Construct a forward Euler timestepper for equation on device dev.

struct ForwardEulerTimeStepper{T} <: AbstractTimeStepper{T}

A forward Euler timestepper for time-stepping ∂u/∂t = RHS(u, t) via:

uⁿ⁺¹ = uⁿ + dt * RHS(uⁿ, tⁿ)

LSRK54TimeStepper(equation::Equation, dev::Device=CPU())

Construct a 4th-order 5-stages low-storage Runge-Kutta timestepper for equation on device dev.

struct LSRK54TimeStepper{T} <: AbstractTimeStepper{T}

A 4th-order 5-stages 2-storage Runge-Kutta timestepper for time-stepping ∂u/∂t = RHS(u, t) via:

S² = 0

for i = 1:5
  S² = Aᵢ * S² + dt * RHS(uⁿ, t₀ + Cᵢ * dt)
  uⁿ += Bᵢ * S²
end

uⁿ⁺¹ = uⁿ

where Aᵢ, Bᵢ, and Cᵢ are the $A$, $B$, and $C$ coefficients from the LSRK table at the $i$-th stage. For details, refer to Carpenter and Kennedy (1994).

RK4TimeStepper(equation::Equation, dev::Device=CPU())

Construct a 4th-order Runge-Kutta timestepper for equation on device dev.

struct RK4TimeStepper{T} <: AbstractTimeStepper{T}

A 4th-order Runge-Kutta timestepper for time-stepping ∂u/∂t = RHS(u, t) via:

uⁿ⁺¹ = uⁿ + dt/6 * (k₁ + 2 * k₂ + 2 * k₃ + k₄)

where

k₁ = RHS(uⁿ, tⁿ)
k₂ = RHS(uⁿ + k₁ * dt/2, tⁿ + dt/2)
k₃ = RHS(uⁿ + k₂ * dt/2, tⁿ + dt/2)
k₄ = RHS(uⁿ + k₃ * dt, tⁿ + dt)

TimeStepper(stepper, equation, dt=nothing, dev=CPU(); kw...)

Instantiate the Timestepper for equation with timestep dt and on the device. The keyword arguments, kw, are passed to the timestepper constructor.

step_until!(prob, stop_time)

Step forward prob until stop_time.

See also: stepforward!

stepforward!(prob::Problem, diags, nsteps::Int)

Step forward prob for nsteps, incrementing diags along the way. diags may be a single Diagnostic or a Vector of Diagnostics.

stepforward!(prob::Problem, nsteps::Int)

Step forward prob for nsteps.

stepforward!(prob::Problem)

Step forward prob one time step.

A test-bed module that solves the 1D diffusion equation.

Exports

• Problem
• set_c!
• updatevars!

Problem(dev::Device = CPU();
                 nx = 128,
                 Lx = 2π,
                  κ = 0,
                 dt = 0.01,
            stepper = "RK4",           
   aliased_fraction = 0,
                  T = Float64)

Construct a constant diffusivity problem.

Keyword arguments

• dev: (required) CPU() or GPU(); computer architecture used to time-step problem.
• nx: Number of grid points in $x$-domain.
• Lx: Extent of the $x$-domain.
• κ: Diffusivity coefficient.
• dt: Time-step.
• stepper: The extent of the $y$-domain.
• aliased_fraction: the fraction of high-wavenubers that are zero-ed out by dealias!().
• T: Float64 or Float32; floating point type used for problem data.

set_c!(prob, c)

Set the solution sol as the transform of c and update vars.

updatevars!(vars, grid, sol)

Update the variables in vars on the grid with the solution in sol.