DG Balance Law Method

BalanceLaw

An abstract type representing a PDE balance law of the form

elements for balance laws of the form

Subtypes L should define the methods below

vars_state_conservative(::L, FT)

a tuple of symbols containing the state variables given a float type FT.

vars_state_auxiliary(::L, FT)

a tuple of symbols containing the auxiliary variables given a float type FT.

vars_state_gradient(::L, FT)

a tuple of symbols containing the transformed variables of which gradients are computed given a float type FT.

vars_integrals(::L, FT)

a tuple of symbols containing variables to be integrated along a vertical stack, given a float type FT.

vars_integrals(m::AtmosModel, FT)

vars_integral(::HBModel)

location to store integrands for bottom up integrals ∇hu = the horizontal divegence of u, e.g. dw/dz

vars_reverse_integrals(::L, FT)

a tuple of symbols containing variables to be integrated along a vertical stack, in reverse, given a float type FT.

vars_reverse_integrals(m::AtmosModel, FT)

vars_reverse_integral(::HBModel)

location to store integrands for top down integrals αᵀθ = density perturbation

vars_state_gradient_flux(::L, FT)

a tuple of symbols containing the diffusive variables given a float type FT.

init_state_conservative!(
  ::L,
  state_conservative::Vars,
  state_auxiliary::Vars,
  coords,
  args...)

Initialize the conservative state variables at $t = 0$

init_state_conservative!(
    m::AtmosModel,
    state::Vars,
    aux::Vars,
    coords,
    t,
    args...)

Initialise state variables. args... provides an option to include configuration data (current use cases include problem constants, spline-interpolants)

init_state_auxiliary!(
  ::L,
  state_auxiliary::Vars,
  coords,
  args...)

Initialize the auxiliary state, at $t = 0$

init_state_auxiliary!(
    m::AtmosModel,
    aux::Vars,
    geom::LocalGeometry
    )

Initialise auxiliary variables for each AtmosModel subcomponent. Store Cartesian coordinate information in aux.coord.

No need to init, initialize by full model

flux_first_order!(
    ::L,
    flux::Grad,
    state_conservative::Vars,
    state_auxiliary::Vars,
    t::Real
)

Compute first-order flux terms in balance law equation

flux_first_order!(
    m::AtmosModel,
    flux::Grad,
    state::Vars,
    aux::Vars,
    t::Real
)

Computes and assembles non-diffusive fluxes in the model equations.

flux_first_order!(::HBModel)

calculates the hyperbolic flux contribution to state variables this computation is done pointwise at each nodal point

arguments:

m -> model in this case HBModel F -> array of fluxes for each state variable Q -> array of state variables A -> array of aux variables t -> time, not used

computations

∂ᵗu = ∇⋅(g*η + g∫αᵀθdz + v⋅u) ∂ᵗθ = ∇⋅(vθ) where v = (u,v,w)

flux_second_order!(
    ::L,
    flux::Grad,
    state_conservative::Vars,
    state_gradient_flux::Vars,
    hyperdiffusive::Vars,
    state_auxiliary::Vars,
    t::Real
)

Compute second-order flux terms in balance law equation

flux_second_order!(
    atmos::AtmosModel,
    flux::Grad,
    state::Vars,
    diffusive::Vars,
    hyperdiffusive::Vars,
    aux::Vars,
    t::Real
)

Diffusive fluxes in AtmosModel. Viscosity, diffusivity are calculated in the turbulence subcomponent and accessed within the diffusive flux function. Contributions from subcomponents are then assembled (pointwise).

flux_second_order!(::HBModel)

calculates the parabolic flux contribution to state variables this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• F: array of fluxes for each state variable
• Q: array of state variables
• D: array of diff variables
• A: array of aux variables
• t: time, not used

computations

∂ᵗu = -∇⋅(ν∇u) ∂ᵗθ = -∇⋅(κ∇θ)

flux_second_order!(::HBModel)

calculates the parabolic flux contribution to state variables this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• F: array of fluxes for each state variable
• Q: array of state variables
• D: array of diff variables
• A: array of aux variables
• t: time, not used

computations

∂ᵗu = -∇⋅(ν∇u) ∂ᵗθ = -∇⋅(κ∇θ)

source!(
    ::L,
    source::Vars,
    state_conservative::Vars,
    diffusive::Vars,
    state_auxiliary::Vars,
    t::Real
)

Compute non-conservative source terms in balance law equation

source!(
    m::AtmosModel,
    source::Vars,
    state::Vars,
    diffusive::Vars,
    aux::Vars,
    t::Real,
    direction::Direction
)

Computes (and assembles) source terms S(Y) in:

∂Y
-- = - ∇ • F + S(Y)
∂t

source!(::HBModel)
calculates the source term contribution to state variables
this computation is done pointwise at each nodal point

arguments:
m -> model in this case HBModel
F -> array of fluxes for each state variable
Q -> array of state variables
A -> array of aux variables
t -> time, not used

computations
∂ᵗu = -f×u
∂ᵗη = w|(z=0)

integral_load_auxiliary_state!

Specify how to compute integrands. Can be functions of the conservative state and auxiliary variables.

integral_set_auxiliary_state!

Specify which auxiliary variables are used to store the output of the integrals.

reverse_integral_load_auxiliary_state!

Specify auxiliary variables need their integrals reversed.

reverse_integral_set_auxiliary_state!

Specify which auxiliary variables are used to store the output of the reversed integrals.

compute_gradient_flux!(
    ::L,
    state_gradient_flux::Vars,
    ∇transformstate::Grad,
    state_auxiliary::Vars,
    t::Real
)

transformation of gradients to the diffusive variables

compute_gradient_flux!(::TurbulenceClosure, _...)

Post-gradient-transformed variables specific to turbulence models.

compute_gradient_flux!(::HBModel)

copy ∇u and ∇θ to var_diffusive this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• D: array of diffusive variables
• G: array of gradient variables
• Q: array of state variables
• A: array of aux variables
• t: time, not used

compute_gradient_flux!(::LinearHBModel)

copy ν∇u and κ∇θ to var_diffusive this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• D: array of diffusive variables
• G: array of gradient variables
• Q: array of state variables
• A: array of aux variables
• t: time, not used

compute_gradient_argument!(
    ::L,
    transformstate::Vars,
    state_conservative::Vars,
    state_auxiliary::Vars,
    t::Real
)

transformation of state variables to variables of which gradients are computed

compute_gradient_argument!

Assign pre-gradient-transform variables specific to turbulence models.

compute_gradient_argument!(::HBModel)

copy u and θ to var_gradient this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• G: array of gradient variables
• Q: array of state variables
• A: array of aux variables
• t: time, not used

compute_gradient_argument!(::LinearHBModel)

copy u and θ to var_gradient this computation is done pointwise at each nodal point

arguments:

• m: model in this case HBModel
• G: array of gradient variables
• Q: array of state variables
• A: array of aux variables
• t: time, not used

transform_post_gradient_laplacian!(
    ::L,
    Qhypervisc_div::Vars,
    ∇Δtransformstate::Grad,
    state_auxiliary::Vars,
    t::Real
)

transformation of laplacian gradients to the hyperdiffusive variables

wavespeed(
    ::L,
    n⁻,
    state_conservative::Vars,
    state_auxiliary::Vars,
    t::Real
)

wavespeed

boundary_state!(
    ::NumericalFluxGradient,
    ::L,
    state_conservative⁺::Vars,
    state_auxiliary⁺::Vars,
    normal⁻,
    state_conservative⁻::Vars,
    state_auxiliary⁻::Vars,
    bctype,
    t
)
boundary_state!(
    ::NumericalFluxFirstOrder,
    ::L,
    state_conservative⁺::Vars,
    state_auxiliary⁺::Vars,
    normal⁻,
    state_conservative⁻::Vars,
    state_auxiliary⁻::Vars,
    bctype,
    t
)
boundary_state!(
    ::NumericalFluxSecondOrder,
    ::L,
    state_conservative⁺::Vars,
    state_gradient_flux⁺::Vars,
    state_auxiliary⁺:
    Vars, normal⁻,
    state_conservative⁻::Vars,
    state_gradient_flux⁻::Vars,
    state_auxiliary⁻::Vars,
    bctype,
    t
)

Apply boundary conditions for

• NumericalFluxGradient numerical flux (internal method)
• NumericalFluxFirstOrder first-order unknowns
• NumericalFluxSecondOrder second-order unknowns

update_auxiliary_state!(
    dg::DGModel,
    m::HBModel,
    Q::MPIStateArray,
    t::Real,
    elems::UnitRange,
)

Update the auxiliary state variables

update_auxiliary_state_gradient!

Update the auxiliary state gradient variables