Nonlinear Diffusion

This function handles expressions of the form ðₙ(D(ðₘu)) where n,m > 0 and D is a function of u i.e. they vary as u(x,t) and D(u). The expansion can be carried out via general Leibniz rule.

A boundary condition operator bc is first operated on u resulting in a boundary padded vector bc*u. Since D is a function of u, its discrete values can be obtained at grid points once u has been padded. After producing these two functions in the grid range, we can expand the given expression via binomial expansion through the nonlinear_diffusion and nonlinear_diffusion! functions and produce the final discretized derivatives.

The functions implicitly put the CenteredDifference operator to use for computing derivates of various orders, e.g. uᵏ = CenteredDifference(k,approx_order,dx,nknots)*u, helping us generate a symmetric discretization. The two functions differ in terms of memory allocation, since the non-! one will allocate memory to the output whereas the ! one can be used for non-allocating applications.

Functions

The two functions are as follows :

nonlinear_diffusion(second_differential_order::Int, first_differential_order::Int, approx_order::Int,
                    p::AbstractVector{T}, q::AbstractVector{T}, dx::Union{T , AbstractVector{T} , Real},
                    nknots::Int) where {T<:Real, N}

nonlinear_diffusion!(du::AbstractVector{T}, second_differential_order::Int, first_differential_order::Int,
                     approx_order::Int,p::AbstractVector{T}, q::AbstractVector{T},
                     dx::Union{T , AbstractVector{T} , Real}, nknots::Int) where {T<:Real, N}

Arguments :

• du : an input AbstractVector similar to u, to store the final discretized expression.
• second_differential_order : the overall order of derivative on the expression.(n)
• first_differential_order : the inner order of derivative to discretize for u.(m)
• approx_order : the order of the discretization in terms of O(dx^order).
• p : boundary padded D.
• q : boundary padded u obtained by bc*u.
• dx: spacing of the discretization. If dx is a Number, the discretization is uniform. If dx is an array, then the discretization is non-uniform.
• nknots : the length of discretization in the direction of operator.