Abstract
In this paper we will present the stability in L2-norm and the optimal a priori error estimate for the Runge-Kutta discontinuous Galerkin method to solve linear conservation law with inflow boundary condition. Semi-discrete version and fully-discrete version of this method are considered respectively, where time is advanced by the explicit third order total variation diminishing Runge-Kutta algorithm. To avoid the reduction of accuracy, two correction techniques are given for the intermediate boundary condition. Numerical experiments are also given to verify the above results.
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Carpenter, M.H., Gottlieb, D., Abarbanel, S., Don, W.S.: The theoretical accuracy of Runge-Kutta discretization for the initial-boundary value problem: a study of the boundary error. SIAM J. Sci. Comput. 16, 1241–1252 (1995)
Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for the convection-diffusion problems. Math. Comput. 71, 455–478 (2001)
Ciarlet, P.G.: Finite Element Method for Elliptic Problems. North–Holland, Amsterdam (1978)
Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)
Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous Galerkin Methods. Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)
Johnson, C., Pitkäranta, J.: Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM J. Numer. Anal. 20, 951–966 (1983)
Merlet, B.: L ∞- and L 2-error estimates for a finite volume approximation of linear advection. SIAM J. Numer. Anal. 46(1), 124–150 (2007)
Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory report LA-UR-73-479, Los Alamos, NM, 1973
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Zhang, Q., Shu, C.-W.: Error estimates to smooth solution of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM. J. Numer. Anal. 42, 641–666 (2004)
Zhang, Q., Shu, C.-W.: Error estimates to smooth solution of Runge-Kutta discontinuous Galerkin methods for symmetrizable system of conservation laws. SIAM. J. Numer. Anal. 44, 1702–1720 (2006)
Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimates to the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM. J. Numer. Anal. doi:10.1137/090771363
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The research of this author is supported by NSFC grant 10871093.
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Zhang, Q. Third Order Explicit Runge-Kutta Discontinuous Galerkin Method for Linear Conservation Law with Inflow Boundary Condition. J Sci Comput 46, 294–313 (2011). https://doi.org/10.1007/s10915-010-9403-0
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DOI: https://doi.org/10.1007/s10915-010-9403-0