Operator Bounds and Time Step Conditions for the DG and Central DG Methods

Abstract

Discontinuous Galerkin (DG) and central DG methods are two related families of finite element methods. They can provide high order spatial discretizations that are often combined with explicit high order time discretizations to solve initial boundary value problems. In this context, it has been observed that the central DG method allows larger time steps, especially for schemes with high accuracy. In this paper, we estimate bounds for the DG and central DG spatial operators for the linear advection equation. Based on these estimates and Kreiss-Wu theory, we obtain time step conditions to ensure the numerical stability of the DG and central DG methods when the methods are combined with locally stable time discretizations. In particular, for a fixed time discretization, the time step allowed for the DG method is proportional to $$h/k^2$$ h / k 2 , while the time step allowed for the central DG method is proportional to $$h/k$$ h / k , where $$h$$ h is the spatial mesh size and $$k>0$$ k > 0 is the polynomial degree of the discrete space of the spatial discretization. In addition, the analysis provides new insight into the role of a parameter in the central DG formulation. We verify our results numerically, and we also discuss extensions of our analysis to some related discretizations.