A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms (Communications in Computational Physics, 2018)
Hyperbolic balance laws have steady state solutions in which the flux gradients arenonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designedhigh order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper, we present a different approach to the same purpose: designing high orderwell-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKuttadiscontinuous Galerkin (RKDG) finite element methods. We make the observation thatthe traditional RKDG methods are capable of maintaining certain steady states exactly, if asmall modification on either the initial condition or the flux is provided. The computationalcost to obtain such a well balanced RKDG method is basically the same as the traditionalRKDG method. The same idea can be applied to the finite volume WENO schemes. Wewill first describe the algorithms and prove the well balanced property for the shallow waterequations, and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of theseschemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities, and the genuine highorder accuracy in smooth regions.