Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Two-dimensional Linear Sobolev Equation
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In this paper we present an efficient and high-order numerical method to solve two-dimensional linear Sobolev equations, which is based on the local discontinuous Galerkin (LDG) method with the upwind-biased fluxes and generalized alternating fluxes. A weak stability is given for both schemes, and a strong stability is established if the initial solutions exactly satisfy the elemental discontinuous Galerkin discretization. Moreover, the sharp error estimate in \(L^2\)-norm is established, by an elaborate application of the generalized Gauss–Radau projection. A fully-discrete LDG scheme is also considered, where the third-order explicit TVD Runge–Kutta algorithm is adopted. Finally some numerical experiments are given.
KeywordsSobolev equation Local discontinuous Galerkin method Upwind-biased/generalized alternating numerical fluxes Stability and error estimate Generalized Gauss–Radau projection