ranocha / summationbypartsoperators.jl Goto Github PK

• Gegenbauer reconstruction for Fourier methods, cf. Script 3.18 of Hesthaven (2018) Numerical methods for conservation laws
• Some Padé reconstructions?
• Total variation denoising, cf. Condat (2013) A Direct Algorithm for 1-D Total Variation Denoising
• Total variation denoising with overlapping group sparsity, cf. Selesnick and Chen (2013)

• Order 2, first derivative operator
• Order 3, first derivative operator
• Order 4, first derivative operator
• Order 5, first derivative operator
• Order 6, first derivative operator
• Order 7, first derivative operator
• Order 8, first derivative operator
• Order 9, first derivative operator
• Order 2, second derivative operators
• Order 3, second derivative operators
• Order 4, second derivative operators
• Order 5, second derivative operators
• Order 6, second derivative operators
• Order 7, second derivative operators
• Order 8, second derivative operators
• Order 9, second derivative operators
• Difference operator as dissipation operator
• Something collecting them all (#156)

Should both boundary nodes be included? Possibly not, since the value at one boundary should be exactly the same as the one at the other. Otherwise some errors may occur...

This could be useful, e.g. to discretize something like (I - D^2) \ D * u, where D is a Fourier first derivative operator, more efficiently.

See Mattsson (http://link.springer.com/article/10.1007/s10915-011-9525-z).

How should the variable coefficients be implemented? Possibly as an additional argument to mul! in order to simplify time-dependent coefficients and multidimensional problems?

These could be used for some varying coefficient derivatives.

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It would be nice to have mul!(du, transpose(D), u) for D::AbstractDerivativeOperator.

• Rational functions of a single operator
• Quotients of polynomials of two different operators? It might be worth to use // for this operation and allow also Fourier polynomials...

The first derivative operators seem to be the same as for MattssonSvärdShoeybi2008 for orders 2, 4, and 6.

https://doi.org/10.2514/6.2016-1329

Conversion to BandedMatrix of BandedMatrices.jl, possibly using Requires.jl. Possibly also to the types of BlockBandedMatrices.jl when they are implemented?

https://blog.travis-ci.com/2020-11-02-travis-ci-new-billing:

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Any ideas, suggestions?

See LoopVectorization.jl.

See Jameson (1993) On the wavelet based differentiation matrix. These schemes are identical with classical centered FD schemes for orders of accuracy 2 and 4 but differ slightly for higher orders of accuracy.

See https://doi.org/10.1016/j.jcp.2018.06.010 and https://bitbucket.org/martinalmquist/optimized_sbp_operators/src/master

• MattssonAlmquistVanDerWeide2018Minimal
  • Interior order 4
  • Interior order 6
  • Interior order 8
  • Interior order 10
  • Interior order 12
• MattssonAlmquistVanDerWeide2018Accurate
  • Interior order 4
  • Interior order 6
  • Interior order 8
  • Interior order 10
  • Interior order 12

Derivative and filter operators are implemented

ERROR: LoadError: LoadError: invalid subtyping in definition of ODEIntegrator
Stacktrace:
 [1] top-level scope at none:0
 [2] include at ./boot.jl:317 [inlined]
 [3] include_relative(::Module, ::String) at ./loading.jl:1044
 [4] include at ./sysimg.jl:29 [inlined]
 [5] include(::String) at /home/sck/.julia/packages/OrdinaryDiffEq/RqsYx/src/OrdinaryDiffEq.jl:1
 [6] top-level scope at none:0
 [7] top-level scope at none:2
in expression starting at /home/sck/.julia/packages/OrdinaryDiffEq/RqsYx/src/integrators/type.jl:49
in expression starting at /home/sck/.julia/packages/OrdinaryDiffEq/RqsYx/src/OrdinaryDiffEq.jl:91
ERROR: LoadError: Failed to precompile OrdinaryDiffEq [1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] to /home/sck/.julia/compiled/v1.0/OrdinaryDiffEq/DlSvy.ji.
Stacktrace:
 [1] top-level scope at none:2
in expression starting at /home/sck/.julia/packages/DiffEqCallbacks/rV4BA/src/DiffEqCallbacks.jl:9
ERROR: LoadError: Failed to precompile DiffEqCallbacks [459566f4-90b8-5000-8ac3-15dfb0a30def] to /home/sck/.julia/compiled/v1.0/DiffEqCallbacks/TKs5l.ji.
Stacktrace:
 [1] top-level scope at none:2
in expression starting at /home/sck/.julia/packages/SummationByPartsOperators/Acxc0/src/SummationByPartsOperators.jl:15
ERROR: LoadError: Failed to precompile SummationByPartsOperators [9f78cca6-572e-554e-b819-917d2f1cf240] to /home/sck/.julia/compiled/v1.0/SummationByPartsOp
Stacktrace:
 [1] compilecache(::Base.PkgId, ::String) at ./loading.jl:1203
 [2] _require(::Base.PkgId) at ./loading.jl:960
 [3] require(::Base.PkgId) at ./loading.jl:858
 [4] require(::Module, ::Symbol) at ./loading.jl:853
 [5] include(::String) at ./client.jl:392
 [6] top-level scope at none:0
in expression starting at /home/sck/plasma/alfven_currents/plot_curr.jl:3

I got this error when i tried to use the SBP-Operator package. Im using Julia 1.0.3 from the rpm repository of Fedora 29 Amd64 KDE.

There is some inconsistency between the number of grid points requested:

julia> using SummationByPartsOperators

julia> D_FD = periodic_derivative_operator(1, 2, 0., 1., 11)
Periodic 1st derivative operator of order 2 {T=Float64, Parallel=Val{:serial}} 
on a grid in [0.0, 1.0] using 11 nodes, 
stencils with 1 nodes to the left, 1 nodes to the right, and coefficients from 
  Fornberg (1998) 
  Calculation of Weights in Finite Difference Formulas. 
  SIAM Rev. 40.3, pp. 685-691. 


julia> grid(D_FD)
0.0:0.1:0.9

julia> D_Fourier = fourier_derivative_operator(0., 1., 11)
Periodic 1st derivative Fourier operator {T=Float64} 
on a grid in [0.0, 1.0] using 11 nodes and 6 modes. 


julia> grid(D_Fourier)
0.0:0.09090909090909091:0.9090909090909091

• Fourier
• Legendre

-> Implement several filter functions.

• continuosly on periodic grids
• discontinuosly on periodic grids
• continuosly on bounded grids
• discontinuosly on bounded grids

Both with and without additional interface dissipation.

Cf. Mattsson, Svärd, Nordström (2004).

Add an implementation of mul!(dest, ..., u, α) in order to avoid problems with dest containing NaNs etc.

Big picture: SBP operators on possibly multiple domains correspond to Hilbert spaces, so the coefficients (on multiple domains) should reflect this behaviour and define scalar products and norms using the corresponding mass/norm matrices.

See also JuliaLang/julia#25565

Up to now, continuous and discontinuous coupling is only implemented for first derivative operators.

Cf. Lundquist, Nordström "Stable and accurate filtering procedures" and https://ntrs.nasa.gov/search.jsp?R=19980004143.

For Dirichlet BCs, the transposed boundary derivative operators are needed additionally.

It would be nice to have mul!(du, D, u) for u::AbstractMatrix.