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

∆ t / ∆ t max
∆ t / ∆ t max
1
0.9
0.8
0.7
0.6
0.5
−0.003
−0.004
−0.002
DoF / λ
10 9 8 7
−0.003
−0.004
−0.006
−0.006
−0.01
−0.008
−0.01
−0.008
0.4
0.4
4 3.5 3
λ / h
(a) p = 1
DoF / λ
1
15
10
5
1
0.8
0.6
−3e−07
−1e−06
−1e−06
−3e−06
−0.002
−3e−06
−1e−05
−0.003
−1e−05
−0.004
−0.0001
−0.001
−0.006
1
0.9
0.8
0.7
0.6
0.5
0.8
0.6
∆ t / ∆ t ∆ t / ∆ t max
max
∆ t / ∆ t max
∆ t / ∆ t max
1
0.8
0.6
−3e−05
0.4
0.2
1
0.8
0.6
14 12
−0.0001
−3e−05
−0.0001
−0.0003
DoF / λ
10 8
−0.0003
−0.001
−0.001
−0.01
−0.01
4 3.5 3 2.5 2 1.5
λ / h
(b) p = 2
DoF / λ
20 15 10
−1e−08
−1e−07
−1e−08
−1e−07
6
−1e−06
−1e−05
−0.0001
−0.03
−0.01
4
1
−0.03
−0.0001
−0.001
0.8
0.6
0.4
0.2
1
5
1
−0.001
0.8
0.6
∆ t / ∆ t ∆ t / ∆ t max
max
0.4
0.2
−3e−07
4
3.5
3
2.5
λ / h
−0.0001
2
−0.001
1.5 1
0.4
0.2
0.4
−1e−09
0.2
4
3.5
3
−1e−06
−1e−05
2.5 2 1.5
λ / h
1
0.4
0.2
(c) p = 3
(d) p = 4
∆ t / ∆ t max
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
−1e−10
3.5
20
−1e−09
−1e−09
3
−1e−08
DoF / λ
15
−1e−08
−1e−07
−1e−07
−1e−06
−1e−06
10
−1e−05
−0.0001
−1e−05
−0.0001
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
2.5 2 1.5 1
λ / h
∆ t / ∆ t max
∆ t / ∆ t max
DoF / λ
10
1
9 8 7 6 5
0.9
0.8
0.7
0.6
0.5
0.4
−1e−06
−3e−07
−1e−06
1.2
−3e−06
−1e−05
−3e−06
1
−1e−05
−3e−05
−0.0001
−3e−05
−0.0001
0.8
λ / h
−0.001
−0.001
0.6
4
1
0.9
−0.01
−0.01
0.8
0.7
0.6
0.5
0.4
∆ t / ∆ t max
(e) p = 5
(f) p = 6
Figure 6: 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 = 1, 2, 3, 4, 5, 6 on a uniform mesh
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