# 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;

• 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. See the examples in the sidebar for more details.

• 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)))

## Comparison with similar packages

This project presents several similarities with the excellent BSplines package. This includes various types and functions which have the same names (e.g. BSplineBasis, Spline, knots, order). In most cases this is pure coincidence, even though some inspiration was later taken from that package (for instance, the idea of a Derivative type).

Some design differences with the BSplines package include:

• in BSplineKit, the B-spline order k is considered a compile-time constant, as it is encoded in the BSplineBasis type. This leads to important performance gains when evaluating splines. It also enables the construction of efficient 3D banded structures based on stack-allocated StaticArrays;

• we do not assume that knots are repeated k times at the boundaries, even though this is still the default when creating a B-spline basis. This is to allow the possibility of imposing periodic boundary conditions on the basis.

In addition to the common functionality, BSplineKit provides easy to use tools for solving boundary-value problems using B-splines. This includes the generation of bases satisfying a chosen set of boundary conditions (basis recombination), as well as the construction of arrays for solving such problems using collocation and Galerkin methods.

## 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).