Error analysis of a discontinuous Galerkin method for systems of higher-order differential equations

摘要

In this paper, we provide an error analysis of the discontinuous Galerkin (DG) method applied to the system of first-order ordinary differential equations (ODEs) arising from the transformation of an mth-order ODE. We compare this DG method with the DG method introduced in [4], which applies DG directly to the mth-order ODE, and present the advantages and disadvantages of each approach based on certain metrics, such as computational time,~(L2) norm of the approximation error, ~(L2) norm of the derivatives error, and maximum approximation error at the endpoints of each timestep. We generalize the two approaches by introducing a DG method applied to the system of ω-order ODEs arising from an mth-order ODE, where 1 ≤ ω ≤ m. We also consider two DG approaches for solving the second-order wave partial differential equation (PDE). One approach transforms the wave PDE to a system of first-order in time PDEs, then, by the method of lines, to a system of first-order ODEs. It then applies DG to the latter system. We provide an error analysis of this DG method and compare with the one introduced in [5].