Error Boundedness Of Discontinuous Galerkin Spectral Element Method For Two-dimensional Problems

This article focuses on the error boundedness of discontinuous Galerkin spectral element method(DGSEM)for two-dimensional scalar hyperbolic conservation laws and first-order symmetric positive hyperbolic system with variable coefficients.For scalar hyperbolic conservation laws,it is found that if the divergence of coefficients vector is nonnegative,the error of DGSEM with Legendre-Gauss-Lobatto quadrature is bounded,when there is enough nodal dissipation.DGSEM with Legendre-Gauss quadrature exist a term that cannot be distinguished positive or negative,which might influence the error boundedness.Numerical experiments shows the error of DGSEM with Legendre-Gauss quadrature using central numerical flux can be unbounded.We studied error boundedness of DGSEM for two-dimensional first-order symmetric positive hyperbolic system,and found that the error is bounded providing enough nodal dissipation and the sum of the gradients of coefficients matrices positive semidefinite.Finally,the correctness is verified by numerical experiments.