  We propose an Exponential DG approach for numerically solving partial
differential equations (PDEs). The idea is to decompose the governing PDE
operators into linear (fast dynamics extracted by linearization) and nonlinear
(the remaining after removing the former) parts, on which we apply the
discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete
system is then integrated using exponential time-integrators: exact for the
former and approximate for the latter. By construction, our approach i) is
stable with a large Courant number (Cr > 1); ii) supports high-order solutions
both in time and space; iii) is computationally favorable compared to IMEX DG
methods with no preconditioner; iv) requires comparable computational time
compared to explicit RKDG methods, while having time stepsizes orders magnitude
larger than maximal stable time stepsizes for explicit RKDG methods; v) is
scalable in a modern massively parallel computing architecture by exploiting
Krylov-subspace matrix-free exponential time integrators and compact
communication stencil of DG methods. Various numerical results for both Burgers
and Euler equations are presented to showcase these expected properties. For
Burgers equation, we present detailed stability and convergence analyses for
the exponential Euler DG scheme.
