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This paper designs and analyzes positivity-preserving well-balanced (WB)central discontinuous Galerkin (CDG) schemes for the Euler equations withgravity. A distinctive feature of these schemes is that they not only are WBfor a general known stationary hydrostatic solution, but also can preserve thepositivity of the fluid density and pressure. The standard CDG method does notpossess this feature, while directly applying some existing WB techniques tothe CDG framework may not accommodate the positivity and keep other importantproperties at the same time. In order to obtain the WB andpositivity-preserving properties simultaneously while also maintaining theconservativeness and stability of the schemes, a novel spatial discretizationis devised in the CDG framework based on suitable modifications to thenumerical dissipation term and the source term approximation. The modificationsare based on a crucial projection operator for the stationary hydrostaticsolution, which is proposed for the first time in this work. This novelprojection has the same order of accuracy as the standard $L^2$-projection, canbe explicitly calculated, and is easy to implement without solving anyoptimization problems. More importantly, it ensures that the projectedstationary solution has the same cell averages on both the primal and dualmeshes, which is a key to achieve the desired properties of our schemes. Basedon some convex decomposition techniques, rigorous positivity-preservinganalyses for the resulting WB CDG schemes are carried out. Several one- andtwo-dimensional numerical examples are performed to illustrate the desiredproperties of these schemes, including the high-order accuracy, the WBproperty, the robustness for simulations involving the low pressure or density,high resolution for the discontinuous solutions and the small perturbationsaround the equilibrium state.

Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Schemes for the Euler Equations under Gravitational Fields