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Damped Gardner-Burgers Equation

Overview

This equation is a modified version of the Gardner equation where the diffusion term $u_{xx}$ and damping term $u$ is added for stability. The equation is defined by

\[u_t = -\alpha u_{xxx} + \beta uu_x + \gamma u^2 u_{x} + \delta u_{xx} + \epsilon u\]

Model

The damped Gardner-Burgers equation becomes the exact same as the original Gardner equation:

\[\dot{\mathbf{u}}(t) = \mathbf{Au}(t) + \mathbf{H}(\mathbf{u}(t) \otimes \mathbf{u}(t)) + \mathbf{G}(\mathbf{u}(t) \otimes \mathbf{u}(t) \otimes \mathbf{u}(t)) + \mathbf{Bw}(t)\]

or

\[\dot{\mathbf{u}}(t) = \mathbf{Au}(t) + \mathbf{F}(\mathbf{u}(t) \oslash \mathbf{u}(t)) + \mathbf{E}(\mathbf{u}(t) \oslash \mathbf{u}(t) \oslash \mathbf{u}(t)) + \mathbf{Bw}(t)\]

where

  • $\mathbf{u}\in\mathbb{R}^N$: the state vector
  • $\mathbf{w}\in\mathbb{R}^m$: the input vector
  • $\mathbf{A}\in\mathbb{R}^{N\times N}$: the linear state matrix
  • $\mathbf{H}\in\mathbb{R}^{N\times N^2}$: the quadratic state matrix with redundancy
  • $\mathbf{F}\in\mathbb{R}^{N\times N(N+1)/2}$: the quadratic state matrix without redundancy
  • $\mathbf{G}\in\mathbb{R}^{N\times N^3}$: the cubic state matrix with redundancy
  • $\mathbf{E}\in\mathbb{R}^{N\times N(N+1)(N+2)/6}$: the cubic state matrix without redundancy
  • $\mathbf{B}\in\mathbb{R}^{N\times m}$: the control input matrix

Numerical Integration

For the numerical integration we consider two methods:

  • Semi-Implicit Euler (SIE)
  • Crank-Nicolson Adam-Bashforth (CNAB)

For the exact expressions of the time-stepping check Allen-Cahn equation.

Examples

using CairoMakie
using LinearAlgebra
using PolynomialModelReductionDataset: DampedGardnerBurgersModel

# Setup
Ω = (0.0, 3.0)
Nx = 2^8; dt = 1e-3
dgb = DampedGardnerBurgersModel(
    spatial_domain=Ω, time_domain=(0.0, 3.0), Δx=(Ω[2] + 1/Nx)/Nx, Δt=dt,
    params=Dict(:a => 1, :b => 3, :c => 5, :d => 0.2, :e => 0.5), BC=:dirichlet,
)
DS = 100
dgb.IC = 2 * cos.(2π * dgb.xspan / (Ω[2] - Ω[1])) # + 0.5 * cos.(4π * dgb.xspan / (Ω[2] - Ω[1]))
Ubc1 = 0.5ones(1,dgb.time_dim)
Ubc2 = -0.5ones(1,dgb.time_dim)
Ubc = [Ubc1; Ubc2]

# Operators
A, F, E, B = dgb.finite_diff_model(dgb, dgb.params)

# Integrate
U = dgb.integrate_model(
    dgb.tspan, dgb.IC, Ubc;
    linear_matrix=A, quadratic_matrix=F, cubic_matrix=E, control_matrix=B,
    system_input=true, integrator_type=:CNAB,
)

# Surface plot
fig3, _, sf = CairoMakie.surface(dgb.xspan, dgb.tspan[1:DS:end], U[:, 1:DS:end],
    axis=(type=Axis3, xlabel=L"x", ylabel=L"t", zlabel=L"u(x,t)"))
CairoMakie.Colorbar(fig3[1, 2], sf)
fig3
Example block output
# Flow field
fig4, ax, hm = CairoMakie.heatmap(dgb.xspan, dgb.tspan[1:DS:end], U[:, 1:DS:end])
ax.xlabel = L"x"
ax.ylabel = L"t"
CairoMakie.Colorbar(fig4[1, 2], hm)
fig4
Example block output

API

PolynomialModelReductionDataset.DampedGardnerBurgers.DampedGardnerBurgersModel — Type
mutable struct DampedGardnerBurgersModel <: AbstractModel

Damped Gardner-Burgers equation model

\[\frac{\partial u}{\partial t} = -\alpha\frac{\partial^3 u}{\partial x^3} - \beta u\frac{\partial u}{\partial x} - \gamma u^2\frac{\partial u}{\partial x} + \delta \frac{\partial^2 u}{\partial x^2} + \epsilon u\]

Fields

  • spatial_domain::Tuple{Real,Real}: spatial domain
  • time_domain::Tuple{Real,Real}: temporal domain
  • param_domain::Tuple{Real,Real}: parameter domain
  • Δx::Real: spatial grid size
  • Δt::Real: temporal step size
  • BC::Symbol: boundary condition
  • IC::Array{Float64}: initial condition
  • xspan::Vector{Float64}: spatial grid points
  • tspan::Vector{Float64}: temporal points
  • spatial_dim::Int64: spatial dimension
  • time_dim::Int64: temporal dimension
  • params::Union{Real,AbstractArray{<:Real}}: parameter vector
  • param_dim::Int64: parameter dimension
  • finite_diff_model::Function: model using Finite Difference
  • integrate_model::Function: model integration
source
PolynomialModelReductionDataset.DampedGardnerBurgers.integrate_model — Method
integrate_model(tdata, u0)
integrate_model(tdata, u0, input; kwargs...)

Integrate the Damped Gardner-Burgers model using 2 different methods:

  • Semi-Implicit Euler (SIE)
  • Crank-Nicolson Adam-Bashforth (CNAB)

Arguments

  • tdata::AbstractArray{T}: time data
  • u0::AbstractArray{T}: initial condition
  • input::AbstractArray{T}=[]: input data

Keyword Arguments

  • linear_matrix::AbstractArray{T,2}: linear matrix
  • quadratic_matrix::AbstractArray{T,2}: quadratic matrix
  • cubic_matrix::AbstractArray{T,2}: cubic matrix
  • control_matrix::AbstractArray{T,2}: control matrix
  • system_input::Bool=false: system input
  • const_stepsize::Bool=false: constant step size
  • u2_jm1::Union{Nothing,AbstractArray{T}}=nothing: u2 at j-1
  • u3_jm1::Union{Nothing,AbstractArray{T}}=nothing: u3 at j-1
  • integrator_type::Symbol=:CNAB: integrator type

Returns

  • u::AbstractArray{T}: solution

Notes

  • If system_input=true, then the control matrix must be provided
  • If const_stepsize=true, then the time step size is assumed to be constant
  • If u2_jm1 and u3_jm1 are provided, then the CNAB scheme is used
  • If integrator_type=:SIE, then the Semi-Implicit Euler scheme is used
  • If integrator_type=:CNAB, then the Crank-Nicolson Adam-Bashforth scheme is used
source