TY - JOUR T1 - Asymptotically Exact Local Discontinuous Galerkin Error Estimates for the Linearized Korteweg-de Vries Equation in One Space Dimension JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 162 EP - 195 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/483.html KW - Local discontinuous Galerkin method, KdV, superconvergence, Radau points, a posteriori error estimates. AB -

We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (2014), pp. 441-455] to prove new superconvergence results towards particular projections of the exact solutions for the two auxiliary variables in the LDG method that approximate the first and second derivatives of the solution. The order of convergence is proved to be $k+3/2$, when polynomials of total degree not exceeding $k$ are used. These results allow us to prove that the significant parts of the spatial discretization errors for the LDG solution and its spatial derivatives (up to second order) are proportional to $(k+1)$-degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates and prove that, for smooth solutions, these a $posteriori$ LDG error estimates for the solution and its spatial derivatives, at a fixed time $t$, converge to the true errors at $\mathcal{O}(h^{k+3/2})$ rate in the $L^2$-norm. Finally, we prove that the global effectivity indices, for the solution and its spatial derivatives, converge to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Numerical results are presented to validate the theory.