# Discretization

MethodOfLines.MOLFiniteDifferenceType
MOLFiniteDifference(dxs, time=nothing;
approx_order = 2, advection_scheme = UpwindScheme(),
grid_align = CenterAlignedGrid(), kwargs...)

A discretization algorithm.

Arguments

• dxs: A vector of pairs of parameters to the grid step in this dimension, i.e. [x=>0.2, y=>0.1]. For a non uniform rectilinear grid, replace any or all of the step sizes with the grid you'd like to use with that variable, must be an AbstractVector but not a StepRangeLen.
• time: Your choice of continuous variable, usually time. If time = nothing, then discretization yeilds a NonlinearProblem. Defaults to nothing.

Keyword Arguments

• approx_order: The order of the derivative approximation.
• advection_scheme: The scheme to be used to discretize advection terms, i.e. first order spatial derivatives and associated coefficients. Defaults to UpwindScheme(). This is the only relevant scheme at present.
• grid_align: The grid alignment types. See CenterAlignedGrid() and EdgeAlignedGrid().
• kwargs: Any other keyword arguments you want to pass to the ODEProblem.
MethodOfLines.DiscreteSpaceType
DiscreteSpace(domain, depvars, indepvars, discretization::MOLFiniteDifference)

A type that stores informations about the discretized space. It takes each independent variable defined on the space to be discretized and create a corresponding range. It then takes each dependant variable and create an array of symbolic variables to represent it in its discretized form.

Arguments

• domain: The domain of the space.
• depvars: The independent variables to be discretizated.
• indepvars: The independent variables.
• discretization: The discretization algorithm.

Fields

• ū: The vector of dependant variables.
• args: The dictionary of the operations of dependant variables and the corresponding arguments, which include the time variable if given.
• discvars: The dictionary of dependant variables and the discrete symbolic representation of them. Note that this includes the boundaries. See the example below.
• time: The time variable. nothing for steady state problems.
• x̄: The vector of symbolic spatial variables.
• axies: The dictionary of symbolic spatial variables and their numerical discretizations.
• grid: Same as axies if CenterAlignedGrid is used. For EdgeAlignedGrid, interpolation will need to be defined ±dx/2 above and below the edges of the simulation domain where dx is the step size in the direction of that edge.
• dxs: The discretization symbolic spatial variables and their step sizes.
• Iaxies: The dictionary of the dependant variables and their CartesianIndices of the discretization.
• Igrid: Same as axies if CenterAlignedGrid is used. For EdgeAlignedGrid, one more index will be needed for extrapolation.
• x2i: The dictionary of symbolic spatial variables their ordering.

Examples

julia> using MethodOfLines, DomainSets, ModelingToolkit
julia> using MethodOfLines:DiscreteSpace

julia> @parameters t x
julia> @variables u(..)
julia> Dt = Differential(t)
julia> Dxx = Differential(x)^2

julia> eq  = [Dt(u(t, x)) ~ Dxx(u(t, x))]
julia> bcs = [u(0, x) ~ cos(x),
u(t, 0) ~ exp(-t),
u(t, 1) ~ exp(-t) * cos(1)]

julia> domain = [t ∈ Interval(0.0, 1.0),
x ∈ Interval(0.0, 1.0)]

julia> dx = 0.1
julia> discretization = MOLFiniteDifference([x => dx], t)
julia> ds = DiscreteSpace(domain, [u(t,x).val], [x.val], discretization)

julia> ds.discvars[u(t,x)]
11-element Vector{Num}:
u[1](t)
u[2](t)
u[3](t)
u[4](t)
u[5](t)
u[6](t)
u[7](t)
u[8](t)
u[9](t)
u[10](t)
u[11](t)

julia> ds.axies
Dict{Sym{Real, Base.ImmutableDict{DataType, Any}}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}} with 1 entry:
x => 0.0:0.1:1.0