Many of the basic equations in atmospheric modeling are based on conservation laws. Conservation of mass constitutes the continuity equation, and conservation of momentum establishes the momentum equations. When conservation properties are present in the continuous equations, the numerical (discrete) counterparts should also have conservative properties. Examples for numerical conservation of vorticity or other state variables can be found in [46, 350]. More generally we want a numerical method to adhere to a
structure preservation property
. To achieve conservation is paramount for all kinds of numerical methods that try to discretize conservation laws. However, for adaptive methods this often poses an additional challenge.