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