SummationByPartsOperators.jl: A Julia library of provably stable semidiscretization techniques with mimetic properties
SummationByPartsOperators.jl is a Julia library of summationbyparts (SBP) operators, which are discrete derivative operators developed to get provably stable semidiscretizations, paying special attention to boundary conditions. Discretizations included in this framework are finite difference, Fourier pseudospectral, continuous Galerkin, and discontinuous Galerkin methods. The main aim of SummationByPartsOperators.jl is to be useful for researchers and students to learn the basic concepts by providing a unified framework of all of these seemingly different discretizations. At the same time, the implementation is optimized to achieve good performance without sacrificing flexibility.
Basic Operators
The following derivative operators are implemented as "lazy"/matrixfree operators, i.e. no large (size of the computational grid) matrix is formed explicitly.
Periodic Domains

periodic_derivative_operator(derivative_order, accuracy_order, xmin, xmax, N)
These are classical central finite difference operators using
N
nodes on the interval[xmin, xmax]
. 
periodic_derivative_operator(Holoborodko2008(), derivative_order, accuracy_order, xmin, xmax, N)
These are central finite difference operators using
N
nodes on the interval[xmin, xmax]
and the coefficients of Pavel Holoborodko. 
fourier_derivative_operator(xmin, xmax, N)
Fourier derivative operators are implemented using the fast Fourier transform of FFTW.jl.
Finite/Nonperiodic Domains

derivative_operator(source_of_coefficients, derivative_order, accuracy_order, xmin, xmax, N)
Finite difference SBP operators for first and second derivatives can be obained by using
MattssonNordström2004()
assource_of_coefficients
. Other sources of coefficients are implemented as well. To obtain a full list for all operators, usesubtypes(SourceOfCoefficients)
. 
legendre_derivative_operator(xmin, xmax, N)
Use Lobatto Legendre polynomial collocation schemes on
N
, i.e. polynomials of degreeN1
, implemented via PolynomialBases.jl.
Dissipation Operators
Additionally, some artificial dissipation/viscosity operators are implemented.
The most basic usage is Di = dissipation_operator(D)
,
where D
can be a (periodic, Fourier, Legendre, SBP FD) derivative
operator. Use ?dissipation_operator
for more details.
Conversion to Other Forms
Sometimes, it can be convenient to obtain an explicit (sparse, banded) matrix form of the operators. Therefore, some conversion functions are supplied, e.g.
julia> using SummationByPartsOperators
julia> D = derivative_operator(MattssonNordström2004(), 1, 2, 0., 1., 5)
SBP 1st derivative operator of order 2 {T=Float64, Parallel=Val{:serial}}
on a grid in [0.0, 1.0] using 5 nodes
and coefficients given in
Mattsson, Nordström (2004)
Summation by parts operators for finite difference approximations of second
derivaties.
Journal of Computational Physics 199, pp.503540.
julia> Matrix(D)
5×5 Array{Float64,2}:
4.0 4.0 0.0 0.0 0.0
2.0 0.0 2.0 0.0 0.0
0.0 2.0 0.0 2.0 0.0
0.0 0.0 2.0 0.0 2.0
0.0 0.0 0.0 4.0 4.0
julia> using SparseArrays
julia> sparse(D)
5×5 SparseMatrixCSC{Float64,Int64} with 10 stored entries:
[1, 1] = 4.0
[2, 1] = 2.0
[1, 2] = 4.0
[3, 2] = 2.0
[2, 3] = 2.0
[4, 3] = 2.0
[3, 4] = 2.0
[5, 4] = 4.0
[4, 5] = 2.0
[5, 5] = 4.0
julia> using BandedMatrices
julia> BandedMatrix(D)
5×5 BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
4.0 4.0 ⋅ ⋅ ⋅
2.0 0.0 2.0 ⋅ ⋅
⋅ 2.0 0.0 2.0 ⋅
⋅ ⋅ 2.0 0.0 2.0
⋅ ⋅ ⋅ 4.0 4.0
Documentation
Examples can be found in the directory notebooks
.
In particular, examples of complete discretisations of
the linear advection equation,
the heat equation,
and the wave equation are supplied.
Further examples are supplied as tests.