Dispersion and dissipation error in high-order Runge-Kutta ...

of m-stage linear SSP-RK schemes, given recursively by
u (0) = u n ,
u (i) = u (i−1) + ∆tLu (i−1) , i = 1, . . .,m − 1 (21)
u (m) =
m−2
∑
k=0
u n+1 = u (m)
where α 1,0 = 1 and
α m,k u (k) + α m,m−1
(
u (m−1) + ∆tLu (m−1)) ,
α m,k = 1 k α m−1,k−1, k = 1, . . ., m − 2
α m,m−1 = 1 m! ,
α m,0 = 1 −
m−1
∑
α m,k ,
k=1
are mth-order accurate. This was extended to linear non-autonomous systems by Chen
et al. [6]. In that work the authors demonstrated that when applied together with the
classical discontinuous Galerkin method [9, 11], the SSP-RK scheme gives (p + 1)st-order
convergence with the stability bound
CFL(p) = C 1
2p + 1
(22)
with C = 1, as long as for a given spatial discretisation of polynomial order p, the
corresponding SSP-RK method has order p + 1. The stability regions of several SSP-RK
methods and the low-storage five-stage fourth-order Runge-Kutta method are displayed
in Figure 2.
5 Analysis of the dispersion and dissipation error
A critical factor in the numerical simulation of wave-propagation is the artificial dissipation
and/or dispersion inflicted on the waves due to numerical discretisation errors. In
order to analyse these properties of the different schemes, we resort to the one-dimensional
and two-dimensional forms of the Maxwell equations with periodic boundary conditions.
First, these reduced models are formulated and then we perform a numerical Fourier
analysis of the fully discrete schemes to investigate the dispersion and dissipation errors
as a function of mesh size per wave length and time step. This analysis provides important
information on the accuracy of the schemes regarding wave motion and the relation
between time step, mesh size and polynomial order.
5.1 Wave equation in one and two dimensions
The Maxwell equations in one dimension read
ε r
∂E
∂t = −∂H ∂x ,
µ ∂H
r
∂t = −∂E ∂x , (23)
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