Discontinuous Galerkin in time

Abstract

In the previous two chapters, we have used finite differences to approximate the time derivative in the space semi-discrete parabolic problem. We now adopt a different viewpoint directly relying on a space-time weak formulation. The time approximation is realized by using piecewise polynomial functions over the time mesh. The test functions are discontinuous at the time nodes, thereby allowing for a time-stepping process, i.e., the discrete formulation decouples into local problems over each time step. This leads to two new families of schemes. In the present chapter, we study the discontinuous Galerkin method in time, where the trial functions are also discontinuous at the time nodes. In the next chapter, we study the continuous Petrov–Galerkin methods where the trial functions are continuous. The lowest-order version of the discontinuous Galerkin technique is the implicit Euler scheme, and the lowest-order version of the Petrov–Galerkin technique is the Crank–Nicolson scheme. All these schemes are implicit Runge–Kutta methods.