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

DoF / λ
DoF / λ
1
0.9
−3e−05
14
−0.0001
12
10
−0.0003
8
−0.001
6
−0.01
4
1
0.9
1
0.9
−3e−07
−1e−06
15
−3e−06
−1e−05
10
5
1
0.9
0.8
0.8
0.8
0.8
∆ t / ∆ t max
0.7
0.6
0.5
−1e−05
−3e−05
−0.0001
−0.0003
−0.001
−0.01
0.7
0.6
0.5
∆ t / ∆ t max
∆ t / ∆ t max
0.7
0.6
0.5
−3e−07
−1e−06
−3e−06
−1e−05
−0.0001
0.7
0.6
0.5
∆ t / ∆ t max
0.4
0.3
0.2
25
−3e−05
20
−0.0001
15
λ / h
10
−0.001
−0.01
5
0.4
0.3
0.2
0.4
0.3
0.2
−1e−07
25
−3e−07
20
−1e−06
15
λ / h
10
−0.0001
5
0.4
0.3
0.2
(a) p = 2
(b) p = 3
DoF / λ
DoF / λ
1
0.9
20
−1e−08
−1e−07
15
−1e−06
10
−1e−05
−0.0001
5
1
−0.001
0.9
1
0.9
20
−1e−09
−1e−08
15
−1e−07
10
1
0.9
∆ t / ∆ t max
0.8
0.7
0.6
0.5
0.4
0.3
−1e−09
0.2
35
30
−1e−08
25
−1e−07
20
λ / h
−1e−06
15
−1e−05
10
−0.0001
5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
∆ t / ∆ t max
∆ t / ∆ t max
0.8
0.7
0.6
0.5
−1e−10
0.4
0.3
20
−1e−09
15
−1e−08
−1e−08
λ / h
−1e−07
10
−1e−06
−1e−05
−1e−06
−1e−05
5
−0.0001
−0.0001
0.8
0.7
0.6
0.5
0.4
0.3
∆ t / ∆ t max
(c) p = 4
(d) p = 5
Figure 8: Dissipation error as a function of wave length per mesh size (λ/h), degrees of
freedom per wavelength (DoF/λ) and relative time step (∆t/∆t max ) for polynomial orders
p = 2, 3, 4, 5 on a non-uniform random mesh
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