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

Abstract

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].