Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs.

We present a spacetime DG method for 1D spatial domains and three linear hyperbolic, damped hyperbolic, and parabolic PDEs. The latter two correspond to Maxwell-Cattaneo-Vernotte (MCV) and Fourier heat conduction problems. The method is called the tent-pitcher spacetime DG method (tpSDG) due to its resemblance to the causal spacetime DG method (cSDG) wherein the solution advances in time by pitching spacetime patches. The tpSDG method extends the applicability of such methods from hyperbolic to parabolic and hyperbolic PDEs. For problems with a spatially uniform mesh, a transfer matrix approach is derived wherein the inflow, boundary, and source term values are mapped to the solution coefficient and output values. This resembles a finite difference scheme, but with grid points at the Gauss points of the spatial elements and arbitrarily tunable order of accuracy in spacetime. The spectral stability analysis of the method provides stability correction factors for the parabolic case. Numerical examples demonstrate the applicability of the method to problems with heterogeneous material properties. • Extended spacetime tent-pitcher methods from hyperbolic to hyperbolic-parabolic PDEs. • Transfer matrices relate inflow and outflow solutions of a patch at the Gauss points. • Polynomial order-independent stability limit of tent-pitcher method is shown for wave equation. • The stability limit for parabolic heat conduction is obtained using the transfer matrices. • The transfer matrix method advances the solution of 1D uniform grids with tunable order. [ABSTRACT FROM AUTHOR]