Capturing Near-Equilibrium Solutions: A Comparison Between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes (Communications in Computational Physics, 2019)
Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutionsare affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order toovercome this problem, we compare two commonly adopted strategies: going to veryhigh order and reduce drastically the truncation errors on the equilibrium solution,or design a specific scheme that preserves by construction the equilibrium exactly, theso-called well-balanced approach. We present a modern numerical implementation ofthese two strategies and compare them in details, using hydrostatic but also dynamicalequilibrium solutions of several simple test cases. Finally, we apply our methodologyto the simulation of a protoplanetary disc in centrifugal equilibrium around its starand model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.
Keywords: Numerical methods; benchmark; well-balanced methods; discontinuous Galerkin methods.