Association of the time discretisation with discontinuous Galerkin spatial discretization has been subject to several work, like the contribution of [1], [2] and [3]. In the linear case they have performed a von-Neumann stability analysis on the CFL in function of the space and time order of discretization.
Those results are recall for the L_{2}
norm in table 1. We can summarize the main results by retaining that for a discontinuous Galerkin spatial discretization with polynomial approximation of degree k
and a Runge-Kutta time scheme of order k+1
we have the stability condition [3] :
CFL_{L_{2}} = \dfrac{1}{2k+1}
k | 0 | 1 | 2 | 3 |
---|---|---|---|---|
v = 1 | 1.000 | \star |
\star |
\star |
v = 2 | 1.000 | 0.333 | \star |
\star |
v = 3 | 1.256 | 0.409 | 0.209 | 0.130 |
v = 4 | 1.392 | 0.464 | 0.235 | 0.145 |
Tab 1: CFL pour des polynômes de degré k
et des méthodes de Runge Kutta d’ordre v
dans le cas linéaire. \star
indique un schéma L_{2}
-instable
Remark : when using artificial viscosity to stabilize the scheme on discontinuities, the system to solve is :
\partial_{t} W = L(W^{n}, \nu^{n})
In this case, the artificial viscosity \nu^{n}(W^{n}, z)
is computed only once at each time step and is constant over Runge-Kutta stages.
References
[1] G. Chavent and B. Cockburn. The local projection P0P1 Discontinuous-Galerkin Finite Element Method For Scalar Conservation Laws. 1987 (cit.on p. 19).
[2] B. Cockburn and C-W Shu. “TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II.General framework”. Mathematics of Computation 52.186 (1989), pp. 411–411 (cit. on p. 19).
[3] B. Cockburn and C-W Shu. “Runge – Kutta Discontinuous Galerkin Methods for Convection Dominated Problems”. Journal of Scientific Computing 16.3 (2001), p. 89 (cit. on pp. 6, 19).