General Linear Methods for Time-Dependent PDEs

The hybridized discontinuous Galerkin method has been successfully applied to time-dependent problems using implicit time integrators. These integrators stem from the 'classical' class of backward differentiation formulae (BDF) and diagonally implicit Runge-Kutta (DIRK) methods. We extend this to the class of general linear methods (GLMs) that unify multistep and multistage methods into one framework. We focus on diagonally implicit multistage integration methods (DIMSIMs) that can have the same desirable stability properties like DIRK methods while also having high stage order. The presented numerical results confirm that the applied DIMSIMs achieve expected approximation properties for linear and nonlinear problems.