High-Order Space-Time Methods for Conservation Laws

Abstract

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a CFL (Courant-Friedrichs-Lewy) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree , then the time step sizes are typically proportional to .) Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown.

On High-Order Upwind Methods for Convection

“…Resolving these waves for the multi-dimensional cases appears to be an impossible task. Extension of scheme III in a manner that avoids tracking these waves was carried out using a space-time Taylor series expansion by this author in (Huynh 2006(Huynh , 2013 and Marcus Lo in his PhD dissertation under Van Leer (2011).…”

“…Such an extension was independently obtained by Lo and was studied in combination with Van Leer's recovery scheme for diffusion in his PhD dissertation (2011). Extensions of the moment scheme to arbitrary order in multiple dimensions were carried out in (Huynh 2013); it was shown that extensions to high-order in two spatial dimensions encounter the drawback of a restrictive CFL condition, as opposed to the case of systems of equations in one spatial dimension where the CFL condition is 1 for all polynomial degrees. For scheme V, the difficulty of extension is being tackled by Roe and collaborators in the "active flux scheme" (Eymann and Roe 2013, Fan and Roe 2015).…”

Nodal Space-Time Flux Reconstruction Methods for Conservation Laws

“…method is developed. An implicit space-time method with right Radau points in time (equivalent to the Radau IIA method) has been developed by Huynh [19] for conservation laws. The FR approach is used in the derivation to reduce the weighted residual form to the differentiation one.…”