Optimal Superconvergence of Energy Conserving Local Discontinuous Galerkin Methods for Wave Equations (Communications in Computational Physics, 2018)
This paper is concerned with numerical solutions of the LDG method for1D wave equations. Superconvergence and energy conserving properties have beenstudied. We first study the superconvergence phenomenon for linear problems whenalternating fluxes are used. We prove that, under some proper initial discretization, thenumerical trace of the LDG approximation at nodes, as well as the cell average, convergewith an order 2k+1. In addition, we establish k+2-th order and k+1-th ordersuperconvergence rates for the function value error and the derivative error at Radaupoints, respectively. As a byproduct, we prove that the LDG solution is superconvergentwith an order k+2 towards the Radau projection of the exact solution. Numericalexperiments demonstrate that in most cases, our error estimates are optimal, i.e., theerror bounds are sharp. In the second part, we propose a fully discrete numericalscheme that conserves the discrete energy. Due to the energy conserving property, afterlong time integration, our method still stays accurate when applied to nonlinearKlein-Gordon and Sine-Gordon equations.