Stability and error analysis of a third order fully discrete local discontinuous Galerkin method for one-dimensional high order wave equations

Due to the lack of coercivity and the fact that the corresponding local discontinuous Galerkin (LDG) operator is not symmetric, it is challenging to establish an energy analysis for implicit-explicit (IMEX) LDG schemes for high order wave equations, especially for high order schemes. As a first step towards resolving this problem, we study high order IMEX-LDG schemes for high order wave equations, which only contain the highest spatial derivative term (hence the time discretization method becomes a purely implicit one). The unconditional energy stability analysis of the fully discrete schemes, which couple a third order Runge-Kutta type IMEX method with LDG spatial discretization methods, will be presented. The main technique is introducing a series of temporal differences about stage solutions and constructing a semi-negative definite symmetric form about the discretization for the highest order derivative terms. By the aid of a special projection operator and exploiting the stability provided by the temporal differences, we also obtain optimal error estimates for the fully discrete schemes. Numerical experiments for both the third order and the fifth order wave equations are displayed, which show the optimal accuracy of the considered schemes as well as the good performance of the schemes in simulating solitary waves.