GN

GN

Structure used to define the moment-based angular discretization method that subdivides the angular domain and uses a polynomial expansion on each subdivision. It generalizes the DPN solver.

Mandatory field(s)

• particle::Particle : particle for which the discretization methods is defined.
• solver_type::String : type of solver for the transport calculations.
• scheme_type::Dict{String,String} : type of schemes for the spatial or energy discretization.
• scheme_order::Dict{String,Int64} : order of the expansion for the discretization schemes.

Optional field(s) - with default values

• legendre_order::Int64 = 16 : global Legendre order.
• legendre_order_local::Int64 = 0 : per-subdivision local Legendre order.
• subdivision::Int64 = 1 : number of angular subdivisions.
• polynomial_basis::String : polynomial basis ("legendre" in 1D, "spherical-harmonics" in 2D or 3D by default).
• angular_fokker_planck::String = "galerkin" : type of discretization for the angular Fokker-Planck operation.
• convergence_criterion::Float64 = 1e-7 : convergence criterion of in-group iterations.
• maximum_iteration::Int64 = 300 : maximum number of in-group iterations.
• acceleration::String = "none" : acceleration method for the in-group iterations.

set_particle(this::GN,particle::Particle)

To set the particle for which the transport discretization method is for.

Input Argument(s)

• this::GN : discretization method.
• particle::Particle : particle.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_particle(electron)

set_solver_type(this::GN,solver_type::String)

To set the solver for the particle transport.

Input Argument(s)

• this::GN : discretization method.
• solver_type::String : solver type, which can take the following values:
  • solver_type = "BTE" : Boltzmann transport equation.
  • solver_type = "BFP" : Boltzmann Fokker-Planck equation.
  • solver_type = "BCSD" : Boltzmann-CSD equation.
  • solver_type = "FP" : Fokker-Planck equation.
  • solver_type = "CSD" : Continuous slowing-down only equation.
  • solver_type = "BFP-EF" : Boltzmann Fokker-Planck without elastic.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_solver_type("BFP")

set_legendre_order(this::GN,legendre_order::Int64,legendre_order_local::Int64)

To set the global Legendre order and the per-subdivision local Legendre order.

Input Argument(s)

• this::GN : discretization method.
• legendre_order::Int64 : global Legendre order.
• legendre_order_local::Int64 : per-subdivision local Legendre order.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_legendre_order(16,0)

set_subdivision(this::GN,subdivision::Int64)

To set the number of angular subdivisions used by the GN solver.

Input Argument(s)

• this::GN : discretization method.
• subdivision::Int64 : number of angular subdivisions (must be at least 1).

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_subdivision(4)

set_polynomial_basis(this::GN,basis::String)

To set the polynomial basis used for the angular discretization.

Input Argument(s)

• this::GN : discretization method.
• basis::String : polynomial basis, which can take the following values:
  • basis = "legendre" : Legendre polynomials (1D only).
  • basis = "spherical-harmonics" : spherical harmonics (1D, 2D or 3D).

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_polynomial_basis("legendre")

set_angular_fokker_planck(this::GN,angular_fokker_planck::String)

To set the discretization method for the angular Fokker-Planck term.

Input Argument(s)

• this::GN : discretization method.
• angular_fokker_planck::String : discretization method for the angular Fokker-Planck term, which can take the following values:
  • angular_fokker_planck = "galerkin" : Galerkin moment-based discretization (diagonal on full-range spherical harmonics, ℳ[p,p] = -ℓ_p(ℓ_p+1)).
  • angular_fokker_planck = "finite-difference" : finite-volume (TPFA) discretization of the Laplace-Beltrami operator on the GN patch graph, using the natural connectivity of the tiling (no Voronoi needed).

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_angular_fokker_planck("finite-difference")

set_scheme(this::GN,axis::String,scheme_type::String,scheme_order::Int64)

To set the type of discretization scheme for derivative along the specified spatial or energy axis.

Input Argument(s)

• this::GN : discretization method.
• axis::String : variable of the derivative for which the scheme is applied, which can take the following values:
  • axis = "x" : spatial x axis (discretization of the streaming term).
  • axis = "y" : spatial y axis (discretization of the streaming term).
  • axis = "z" : spatial z axis (discretization of the streaming term).
  • axis = "E" : energy axis (discretization of the continuous slowing-down term).
• scheme_type::String : type of scheme to be applied, which can take the following values:
  • scheme_type = "DD" : diamond difference scheme (any order).
  • scheme_type = "DG" : discontinuous Galerkin scheme (any order).
  • scheme_type = "AWD" : adaptive weighted scheme (1st and 2nd order only).
• scheme_order::Int64 : scheme order, which takes values greater than 1.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_scheme("x","DD",1)
julia> m.set_scheme("E","DG",2)

set_is_full_coupling(this::GN,isFC::Bool)

Set, for multidimensional high-order schemes, if the high-order moments are fully coupled or not. For example, with two linear schemes, the moments are either fully coupled [00,10,01,11] or not [00,10,01].

Input Argument(s)

• this::GN : discretization method.
• isFC::Bool : boolean indicating if the high-order moments are fully coupled or not.

Output Argument(s)

N/A

set_acceleration(this::GN,acceleration::String,parameter::Int64=0)

To set the acceleration method for the in-group iteration process.

Input Argument(s)

• this::GN : discretization method.
• acceleration::String : acceleration method, which takes the following values:
  • acceleration = "none" : source iteration without acceleration.
  • acceleration = "livolant" : Livolant two-point extrapolation.
  • acceleration = "anderson" : depth-m Anderson acceleration.
  • acceleration = "gmres" : matrix-free restarted GMRES.
  • acceleration = "bicgstab" : matrix-free BiCGStab.
• parameter::Int64 : optional tuning parameter for the chosen method. It is the Krylov subspace size before restart for "gmres" (default 30) and the memory depth for "anderson" (default 3); it is ignored by the other methods. A value of 0 keeps the current default.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_acceleration("gmres")      # GMRES with the default restart of 30
julia> m.set_acceleration("gmres",50)   # GMRES restarted every 50 Krylov vectors
julia> m.set_acceleration("anderson",5) # Anderson acceleration with memory depth 5

set_convergence_criterion(this::GN,convergence_criterion::Float64)

To set the convergence criterion for the in-group iterations.

Input Argument(s)

• this::GN : discretization method.
• convergence_criterion::Float64 : convergence criterion.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_convergence_criterion(1e-5)

set_maximum_iteration(this::GN,maximum_iteration::Int64)

To set the maximum number of in-group iterations.

Input Argument(s)

• this::GN : discretization method.
• maximum_iteration::Int64 : maximum number of in-group iterations.

Output Argument(s)

N/A

Examples

julia> m = GN()
julia> m.set_maximum_iteration(50)

get_particle(this::GN)

Get the particle associated with the discretization methods.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• particle::Particle : particle.

get_solver_type(this::GN)

Get the type of solver for transport calculations.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• solver::Int64 : type of solver for transport calculations.
• isCSD::Bool : indicate if continuous slowing-down term is used or not.

get_legendre_order(this::GN)

Get the global Legendre truncation order.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• legendre_order::Int64 : Legendre truncation order.

get_legendre_order_local(this::GN)

Get the per-subdivision local Legendre truncation order.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• legendre_order_local::Int64 : per-subdivision local Legendre truncation order.

get_subdivision(this::GN)

Get the number of angular subdivisions.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• subdivision::Int64 : number of angular subdivisions.

get_polynomial_basis(this::GN,Ndims::Int64)

Get the polynomial basis used for the angular discretization. If the basis was not set explicitly, the default basis is selected based on the geometry dimension ("legendre" in 1D, "spherical-harmonics" otherwise).

Input Argument(s)

• this::GN : discretization method.
• Ndims::Int64 : geometry dimension.

Output Argument(s)

• polynomial_basis::String : polynomial basis.

get_angular_fokker_planck(this::GN)

Get the type of angular discretization for the Fokker-Planck operator.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• angular_fokker_planck::String : type of angular discretization for the Fokker-Planck operator.

get_schemes(this::GN,geometry::Geometry,isFC::Bool)

Get the space and/or energy schemes informations.

Input Argument(s)

• this::GN : discretization method.
• geometry::Geometry : geometry.
• isFC::Bool : boolean indicating if the high-order moments are fully coupled.

Output Argument(s)

• schemes::Vector{String} : scheme types.
• 𝒪::Vector{Int64} : order of the schemes.
• Nm::Vector{Int64} : numbers of moments.

get_is_full_coupling(this::GN)

Get, for multidimensional high-order schemes, if the high-order moments are fully coupled or not.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• isFC::Bool : boolean indicating if the high-order moments are fully coupled or not.

get_acceleration(this::GN)

Get the acceleration method for in-group iteration convergence.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• acceleration::String : acceleration method.

get_gmres_restart(this::GN)

Get the GMRES restart parameter for in-group iteration convergence.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• gmres_restart::Int64 : Krylov subspace size before restart.

get_anderson_depth(this::GN)

Get the Anderson acceleration memory depth for in-group iteration convergence.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• anderson_depth::Int64 : Anderson memory depth.

get_convergence_criterion(this::GN)

Get the convergence criterion for in-group iteration convergence.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• convergence_criterion::Float64 : convergence criterion.

get_maximum_iteration(this::GN)

Get the maximum number of iterations for in-group iteration convergence.

Input Argument(s)

• this::GN : discretization method.

Output Argument(s)

• maximum_iteration::Int64 : maximum number of iterations.