BSplineKit.jl

Tools for B-spline based Galerkin and collocation methods in Julia.

Features

This package provides:

• B-spline bases of arbitrary order on uniform and non-uniform grids;

• evaluation of splines and their derivatives and integrals;

• spline interpolations and function approximation;

• basis recombination, for generating bases satisfying homogeneous boundary conditions using linear combinations of B-splines. Supported boundary conditions include Dirichlet, Neumann, Robin, and generalisations of these;

• banded Galerkin and collocation matrices for solving differential equations, using B-spline and recombined bases;

• efficient "banded" 3D arrays as an extension of banded matrices. These can store 3D tensors associated to quadratic terms in Galerkin methods.

Example usage

The following is a very brief overview of some of the functionality provided by this package.

• Interpolate discrete data using cubic splines (B-spline order k = 4):

  xdata = (0:10).^2  # points don't need to be uniformly distributed
  ydata = rand(length(xdata))
  itp = interpolate(xdata, ydata, BSplineOrder(4))
  itp(12.3)  # interpolation can be evaluated at any intermediate point

• Create B-spline basis of order k = 6 (polynomial degree 5) from a given set of breakpoints:

  breaks = log2.(1:16)  # breakpoints don't need to be uniformly distributed either
  B = BSplineBasis(BSplineOrder(6), breaks)

• Approximate known function by a spline in a previously constructed basis:

  f(x) = exp(-x) * sin(x)
  fapprox = approximate(f, B)
  f(2.3), fapprox(2.3)  # (0.07476354233090601, 0.0747642348243861)

• Create derived basis satisfying homogeneous Robin boundary conditions on the two boundaries:

  bc = Derivative(0) + 3Derivative(1)
  R = RecombinedBSplineBasis(B, bc)  # satisfies u ∓ 3u' = 0 on the left/right boundary

• Construct mass matrix and stiffness matrix for the Galerkin method in the recombined basis:

  # By default, M and L are Hermitian banded matrices
  M = galerkin_matrix(R)
  L = galerkin_matrix(R, (Derivative(1), Derivative(1)))

• Construct banded 3D tensor associated to non-linear term of the Burgers equation:

  T = galerkin_tensor(R, (Derivative(0), Derivative(1), Derivative(0)))

See the heat equation example in the docs for the use of these tools to solve partial differential equations.

References

• C. de Boor, A Practical Guide to Splines. New York: Springer-Verlag, 1978.

• J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition. Mineola, N.Y: Dover Publications, 2001.

• O. Botella and K. Shariff, B-spline Methods in Fluid Dynamics, Int. J. Comput. Fluid Dyn. 17, 133 (2003).