  We introduce Spline Moment Equations (SME) for kinetic equations using a new
weighted spline ansatz of the distribution function and investigate the ansatz,
the model, and its performance by simulating the one-dimensional Boltzmann-BGK
equation. The new basis is composed of weighted constrained splines for the
approximation of distribution functions that preserves mass, momentum, and
energy. This basis is then used to derive moment equations using a Galerkin
approach for a shifted and scaled Boltzmann-BGK equation, to allow for an
accurate and efficient discretization in velocity space with an adaptive grid.
The equations are given in compact analytical form and we show that the
hyperbolicity properties are similar to the well-known Grad moment model. The
model is investigated numerically using the shock tube, the symmetric two-beam
test and a stationary shock structure test case. All tests reveal the good
approximation properties of the new SME model when the parameters of the spline
basis functions are chosen properly. The new SME model outperforms existing
moment models and results in a smaller error while using a small number of
variables for efficient computations.
