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In this paper, we develop a well-balanced oscillation-free discontinuousGalerkin (OFDG) method for solving the shallow water equations with a non-flatbottom topography. One notable feature of the constructed scheme is thewell-balanced property, which preserves exactly the hydrostatic equilibriumsolutions up to machine error. Another feature is the non-oscillatory property,which is very important in the numerical simulation when there exist some shockdiscontinuities. To control the spurious oscillations, we construct an OFDGmethod with an extra damping term to the existing well-balanced DG schemesproposed in [Y. Xing and C.-W. Shu, CICP, 1(2006), 100-134.]. With a carefulconstruction of the damping term, the proposed method achieves both thewell-balanced property and non-oscillatory property simultaneously withoutcompromising any order of accuracy. We also present a detailed procedure forthe construction and a theoretical analysis for the preservation of thewell-balancedness property. Extensive numerical experiments including one- andtwo-dimensional space demonstrate that the proposed methods possess the desiredproperties without sacrificing any order of accuracy.

A well-balanced oscillation-free discontinuous Galerkin method for shallow water equations