API

Problem(ℓ::Real, d::Real, ℐ::Integer, t::AbstractRange{Real}; O = 4, M_s = 0, M_b = 0, static_bottom = true)

Construct an IBV Problem object corresponding to a fluid domain of length ℓ and depth d with ℐ harmonics and N time steps.

Output is a Problem object with fields:

• ℓ is the fluid domain length (m),
• d is the water depth (m),
• ℐ is the number of harmonics,
• t is the time range,
• Δt is the time step (s),
• N is the number of time steps,
• O is the order of the time-stepping scheme,
• κ are wave numbers (rad/m),
• η̂ are free-surface elevation amplitudes (m),
• η̇ are free-surface vertical velocity amplitudes (m/s),
• β̂ are bottom-surface elevation amplitudes (m),
• β̇ are bottom-surface vertical velocity amplitudes (m/s),
• ϕ̂ are flat-bottom velocity potential amplitudes (m²/s),
• ϕ̇ are flat-bottom acceleration potential amplitudes (m²/s²),
• ψ̂ are uneven-bottom velocity potential amplitudes (m²/s),
• ψ̇ are uneven-bottom acceleration potential amplitudes (m²/s²),
• p̂ are surface pressure head amplitudes (m),
• χ is wavemaker paddle displacement (m),
• ξ is wavemaker paddle velocity (m/s),
• ζ is wavemaker paddle acceleration (m/s²),
• 𝒯 are hyperbolic tangent lookup values,
• 𝒮 are hyperbolic secant lookup values,
• static_bottom is a boolean flag to indicate if the bottom is static.

absolute_error(a::Vector{<:Number}, b::Vector{<:Number})

Compute absolute error between vectors a and b.

Examples

julia> a = [1, 2]; b = [1, 1]; absolute_error(a, b)
1.0

bottom_bump!(p::Problem, h, λ, x₀ = 0)

Calculate β̂ coefficients of a problem p for a bottom bump of height h and characteristic length λ at a position x₀.

bottom_slope!(p::Problem, h)

Calculate β̂ coefficients a wave problem p for a bottom slope of height h.

bottom_step!(p::Problem, h)

Calculate β̂ coefficients of a wave problem p for a bottom step of height h.

bottom_surface(p::Problem, x::Real, n=1)

Compute the bottom surface elevation β at position x and for time instant n.

bottom_vector!(p::Problem, x::AbstractRange{<:Real}, β::Vector{<:Real})

Calculate β̂ coefficients of a wave problem p for a bottom profile β(x).

convolution_power(a::Vector{<:Number}, n::Integer)

Compute convolution n-th power of vector a.

Examples

julia> a = [1, 2, 3]; convolution_power(a, 3)
7-element Vector{Int64}:
  1
  6
 21
 44
 63
 54
 27

convolution_range(m::Integral, M::Integral, n::Integer)

Compute range r of indices of a central part of convolution vector corresponding to the m-th order of convolution, where M is the maximum order of convolution, N is the length of convolution vector corresponding to the M-th order of convolution, and ℐ is the number of harmonics.

Examples

julia> m = 1; M = 3; ℐ = 5; c_range = convolution_range(m, M, ℐ)
11:31

convolve(a::Vector{Ta}, b::Vector{Tb}) where {Ta<:Number, Tb<:Number}

Compute direct convolution of vectors a and b.

Examples

julia> a = [1, 2]; b = [1, 1]; convolve(a, b)
3-element Vector{Int64}:
 1
 3
 2

convolve_dsp(a::Vector{Ta}, b::Vector{Tb}) where {Ta<:Number, Tb<:Number}

Compute direct convolution of vectors a and b using DSP package.

Examples

julia> a = [1, 2]; b = [1, 1]; convolve(a, b)
3-element Vector{Int64}:
 1
 3
 2

factorial_lookup(n::Integer)

Compute lookup table for factorials of numbers from 0 to n.

Examples

julia> factorial_lookup(3)
4-element Vector{Float64}:
  1.0
  1.0
  2.0
  6.0

fourier_transform(f::Vector{<:Number}, ω::Real, x::AbstractRange{<:Real})

Compute f̂(ω) using function values f at points x.

Examples

julia> f = [-1, 0, 1, 0, -1]; ω = 1; x = range(-π, π, length = 5);
julia> fourier_transform(f, ω, x)
0.5 + 0.0im

general_error(a::Vector{<:Number}, b::Vector{<:Number})

Compute a general error between vectors a and b as a relative_error(a, b) or an absolute_error(a, b) for norm(a) not equal or equal to 0, respectively.

init_nonlinear_bottom_boundary_condition(κ, 𝒯, 𝒮, ℐ, M)

Initialize expansion coefficients for nonlinear bottom boundary conditions for eigenvalues κ, hyperbolic tangent 𝒯 and secant 𝒮 values, number of harmonics ℐ and order of nonlinear expansion M.

Output is a tuple (Ψ̂′, Ψ̂″, Ψ̃′, Ψ̃″), where:

• Ψ̂′ are surface-potential-amplitude dependent expansion coefficients and
• Ψ̂″ are bottom-potential-amplitude dependent expansion coefficients for computing bottom velocity potential amplitudes and its vertical gradients,
• Ψ̃′ are surface-potential-amplitude dependent expansion coefficients and
• Ψ̃″ are bottom-potential-amplitude dependent expansion coefficients for computing bottom horizontal velocity potential amplitudes and its vertical gradients.

init_nonlinear_surface_boundary_condition(κ, 𝒯, 𝒮, ℐ, M)

Initialize expansion coefficients for nonlinear free-surface boundary conditions for eigenvalues κ, hyperbolic tangent 𝒯 and secant 𝒮 values, number of harmonics ℐ and order of nonlinear expansion M.

Output is a tuple (Φ̇′, Φ̇″, Φ̂′, Φ̂″, Φ̃′, Φ̃″), where:

• Φ̇′ are surface-potential-amplitude dependent expansion coefficients and
• Φ̇″ are bottom-potential-amplitude dependent expansion coefficients for computing

surface acceleration potential amplitudes and its vertical gradients,

• Φ̂′ are surface-potential-amplitude dependent expansion coefficients and
• Φ̂″ are bottom-potential-amplitude dependent expansion coefficients for computing

surface velocity potential amplitudes and its vertical gradients,

• Φ̃′ are surface-potential-amplitude dependent expansion coefficients and
• Φ̃″ are bottom-potential-amplitude dependent expansion coefficients for computing

surface horizontal velocity potential amplitudes and its vertical gradients.

inverse_fourier_transform(f̂::Vector{<:Number}, ω::AbstractRange{<:Real}, x::Real)

Compute f(x) using expansion amplitudes f̂ and eigenvalues ω.

Examples

julia> f̂ = [0.5, 0, 0.5]; ω = -1:1; x = π; inverse_fourier_transform(f̂, ω, x)
-1.0

linear_regular_wave!(p::Problem, H, L, nΔt)

Calculate initial values of η̂, η̇, ϕ̂, ϕ̇ for a regular linear wave of height H and length L.

linear_wavemaker!(p::Problem, H, L, t₀)

Calculate wavemaker paddle displacement χ, velocity ξ, and acceleration ζ for a train of linear waves of height H and period T and a number of ramped periods nT₀.

moving_bottom_bump!(p::Problem, h, λ, u, x₀ = 0)

Calculate β̂ coefficients of a wave problem p for a moving bottom bump of height h, characteristic length λ, and velocity u at initial position x₀.

moving_pressure_bump!(p::Problem, h, λ, u, x₀ = 0)

Calculate p̂ coefficients of a problem p for a moving pressure bump of height h, characteristic length λ, and velocity u at initial position x₀.

nonlinear_dfsbc_correction(η̂, ϕ̂, ϕ̇, ψ̂, ψ̇, Φ̇′, Φ̇″, Φ̂′, Φ̂″, Φ̃′, Φ̃″, κ′, ℐ, F, M, ξ, ζ, ℓ, d)

Calculate nonlinear correction δϕ̇ to dynamic free-surface boundary condition.

nonlinear_kfsbc_correction(η̂, ϕ̂, ψ̂, Φ̂′, Φ̂″, Φ̃′, Φ̃″, κ, κ′, ℐ, F, M)

Calculate nonlinear correction δη̇ to kinematic free-surface boundary condition.

relative_error(a::Vector{<:Number}, b::Vector{<:Number})

Compute relative error between vectors a and b.

Examples

julia> a = [2, 2]; b = [1, 1]; relative_error(a, b)
0.5

solve_problem!(p::Problem; msg_flag = true)

Calculate solution coefficients η̂, η̇, ϕ̂, ϕ̇, ψ̂, ψ̇ of the wave problem.

Modified in-place variables:

• η̂ are free-surface elevation potential amplitudes (m),
• η̇ are free-surface vertical velocity amplitudes (m/s),
• ϕ̂ are flat-bottom velocity potential amplitudes (m²/s),
• ϕ̇ are flat-bottom acceleration potential amplitudes (m²/s²),
• ψ̂ are uneven-bottom velocity potential amplitudes (m²/s),
• ψ̇ are uneven-bottom acceleration potential amplitudes (m²/s²),

Input variables:

• β̂ are bottom-surface elevation amplitudes (m),
• β̇ are bottom-surface vertical velocity amplitudes (m/s),
• p̂ are surface pressure head amplitudes (m),
• κ are wave numbers (rad/m),
• 𝒯 are hyperbolic tangent lookup values,
• 𝒮 are hyperbolic secant lookup values,
• ℐ is the number of harmonics,
• M_s is the order of nonlinear free-surface boundary condition,
• M_b is the order of nonlinear bottom boundary condition,
• Δt is the time step (s),
• O is the order of the time-stepping scheme,
• N is the number of time steps,
• χ is wavemaker paddle displacement (m),
• ξ is wavemaker paddle velocity (m/s),
• ζ is wavemaker paddle acceleration (m/s²),
• ℓ is the fluid domain length (m),
• d is the water depth (m).

Keyword arguments:

• static_bottom is a boolean flag to indicate whether the bottom is static,
• msg_flag is a boolean flag to indicate whether to print progress messages.

surface_bump!(p::Problem, h, λ, x₀ = 0)

Calculate η̂ coefficients of a problem p for a surface bump of height h and characteristic length λ at position x₀.

surface_pressure(p::Problem, x::Real, n=1)

Compute the surface pressure P at position x and for time instant n.

time_integration_coeffs(O::Integer)

Calculate Adams-Bashforth-Moulton time-stepping scheme coefficients for a given order O.

Output is a tuple of two vectors with Adams-Bashforth and Adams-Moulton coefficients, respectively.

toeplitz(a::Vector{T}) where T

Transform vector a to a Toeplitz matrix.

Examples

julia> a = [1, 2, 3]; toeplitz(a)
2×2 Matrix{Int64}:
 2  1
 3  2

update_bbc_sle!(A′, A″, Ψ̂′, Ψ̂″, Ψ̃′, Ψ̃″, w′, β̂, κ, κ′, ℐ, F, M)

Compute coefficients A′, A″, and w′ for the bottom boundary condition system of linear equations (SLE).

Modified in-place variables:

• A′ are constant coefficients of the system of linear equations and
• A″ are coefficients of the system of linear equations,
• w′ are coefficients corresponding to the linear wavemaker term.

water_surface(p::Problem, x::Real, n::Integer)

Compute the water surface elevation η at position x and for time instant n.

water_velocity(p::Problem, x::Real, z::Real, n::Integer, c::Symbol)

Compute the water velocity component u or w using the symbol :x or :z at position (x, z), and for time instant n.