Discontinuous Galerkin methods Lecture 1 - Brown University

Why
14
high-order
From local
accuracy
to global approximation
?
350
300
250
200
150
100
50
e p = 0.1
0
0 0.25 0.5 0.75 1
n
W 1
W 2
W 3
3500
3000
2500
2000
1500
1000
500
0
e p = 0.01
W 1
W 2
W 3
0 0.25 0.5 0.75 1
n
High-order Figure 1.3 is The the growthright of the work solution function, Wm, forif various : finite difference
schemes is given as a function of time, ν, in terms of periods. On the left we show
the growth for a required phase error of εp = 0.1, while the right shows the result
• High accuracy is required - and it increasingly is !
of a similar computation with εp = 0.01, i.e., a maximum phase error of less than
1%. • Long time integration is needed
• High-dimensional problems (3D) are considered
• Memory restrictions become a bottleneck
• .. apart from that, these methods are superior
for hardware with deep memory hierarchies
where C F Lm = c t
refers to the C F L bound for stability. We assume that the
x
fourth-order Runge–Kutta method will be used for time discretization. For this
method it can be shown that C F L1 = 2.8, C F L2 = 2.1, and C F L3 = 1.75.
Thus, the estimated work for second, fourth, and sixth-order schemes is
W1 30ν ν
ν
, W2 35ν , W3 48ν 3
ν
. (1.15)
εp
εp
εp