Results for “RK collocation”

NASA NTRS2013

High-Order Space-Time Methods for Conservation Laws

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 Courant-Friedrichs-Lewy (CFL) 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 p, then the time step sizes are typically proportional to 1/p(exp 2)). 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

Huynh, H. T.↗

NASA NTRSDate not supplied

Explicit Discontinuous Galerkin Methods for Conservation Laws

The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz (2007, 2008) called space–time expansion discontinuous Galerkin or STE-DG scheme, and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction of the data among neighboring cells takes place; the difference, however, is in how the resulting space-time volume integral is estimated. As a consequence of the different estimates, for the case of advection in one spatial dimension (1D), the moment scheme has a CFL (Courant-Friedrichs-Lewy) condition of 1 for all p and is accurate to order 2p+1, i.e., it possesses the super accuracy property, whereas the STE-DG method has a more restrictive CFL condition and is accurate to the expected order of p+1. For 1D advection, compared with the CFL conditions of 1/(2p+1) of standard RK-DG (Runge-Kutta) scheme where space and time discretization are of the same order, the moment scheme allows a significantly larger time step size. It also turns out that the scheme yields a result identical to Van Leer’s scheme III (1977), which amounts to shifting the data a distance of advection corresponding to the time step and projecting the result onto the space of polynomial solutions. Contrary to Van Leer’s approach, however, the space-time ‘predictor-corrector’ formulation facilitates extensions to the case of systems of equations. Concerning 2D extensions, in the case of advection, when the flow is along the diagonal direction, the CFL conditions for the moment schemes become restrictive as will be shown by Fourier (Von Neumann) stability and accuracy analyses. Since the moment scheme employs the right Radau points as collocation points in time, the method is closely related to the implicit Radau IIA scheme, which is stable for any time step size. The role of Radau IIA in relieving stability restriction for these explicit DG schemes remains to be explored

Discontinuous Galerkin↗