  Latitude on the choice of initialisation is a shared feature between one-step
extended state-space and multi-step methods. The paper focuses on lattice
Boltzmann schemes, which can be interpreted as examples of both previous
categories of numerical schemes. We propose a modified equation analysis of the
initialisation schemes for lattice Boltzmann methods, determined by the choice
of initial data. These modified equations provide guidelines to devise and
analyze the initialisation in terms of order of consistency with respect to the
target Cauchy problem and time smoothness of the numerical solution. In detail,
the larger the number of matched terms between modified equations for
initialisation and bulk methods, the smoother the obtained numerical solution.
This is particularly manifest for numerical dissipation. Starting from the
constraints to achieve time smoothness, which can quickly become prohibitive
for they have to take the parasitic modes into consideration, we explain how
the distinct lack of observability for certain lattice Boltzmann schemes --
seen as dynamical systems on a commutative ring -- can yield rather simple
conditions and be easily studied as far as their initialisation is concerned.
This comes from the reduced number of initialisation schemes at the fully
discrete level. These theoretical results are successfully assessed on several
lattice Boltzmann methods.
