Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws

In this paper we apply implicit two-derivative multistage time integrators to viscous conservation laws in one and two dimensions. The one dimensional solver discretizes space with the classical discontinuous Galerkin (DG) method, and the two dimensional solver uses a hybridized discontinuous Galerkin (HDG) spatial discretization for efficiency. We propose methods that permit us to construct implicit solvers using each of these spatial discretizations, wherein a chief difficulty is how to handle the higher derivatives in time. The end result is that the multiderivative time integrator allows us to obtain high-order accuracy in time while keeping the number of implicit stages at a minimum. We show numerical results validating and comparing methods.