High Order Schemes for Hyperbolic Problems: How to Avoid Mass Matrices for Continuous Finite Elements

When integrating unsteady problems with continuous finite element methods, one typically faces the problem of solving a linear system with a large mass matrix. In some cases, this mass matrix needs to be recomputed at each time step. Moreover, in some methods that are not directly formulated by standard variational principles, it is not clear how to write an invertible mass matrix.
In this paper, we show how to avoid this problem for hyperbolic systems, and we also detail the conditions under which this is possible. Analysis and numerical simulation support our conclusions, namely that it is possible to design the lumping of the mass matrix without sacrificing the accuracy of the scheme.