Discontinuous Galerkin Methods with Optimal \(L^2\) Accuracy for One Dimensional Linear PDEs with High Order Spatial Derivatives

In this paper, we formulate and analyze discontinuous Galerkin (DG) methods to solve several partial differential equations (PDEs) with high order spatial derivatives, including the heat equation, a third order wave equation, a fourth order equation and the linear Schrödinger equation in one dimension. Following the idea of local DG methods, we first rewrite each PDE into its first order form and then apply a general DG formulation. The numerical fluxes are introduced as linear combinations of average values of fluxes, and jumps of the solution as well as the auxiliary variables at cell interfaces. The main focus of the present work is to identify a sub-family of the numerical fluxes by choosing the coefficients in the linear combinations, so the solution and some auxiliary variables of the proposed DG methods are optimally accurate in the \(L^2\) norm. In our analysis, one key component is to design some special projection operator(s), tailored for each choice of numerical fluxes in the sub-family, to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates. Our theoretical findings are validated by a set of numerical examples.