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Equilibrium or stationary solutions usually proceed through the exact balancebetween hyperbolic transport terms and source terms. Such equilibrium solutionsare affected by truncation errors that prevent any classical numerical schemefrom capturing the evolution of small amplitude waves of physical significance.In order to overcome this problem, we compare two commonly adopted strategies:going to very high order and reduce drastically the truncation errors on theequilibrium solution, or design a specific scheme that preserves byconstruction the equilibrium exactly, the so-called well-balanced approach. Wepresent a modern numerical implementation of these two strategies and comparethem in details, using hydrostatic but also dynamical equilibrium solutions ofseveral simple test cases. Finally, we apply our methodology to the simulationof a protoplanetary disc in centrifugal equilibrium around its star and modelits interaction with an embedded planet, illustrating in a realisticapplication the strength of both methods.

Capturing near-equilibrium solutions: a comparison between high-order discontinuous Galerkin methods and well-balanced schemes