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

0.4
1
0.6
1
0.6
0.8
5
RK45CK
5
SSP−RK3
4
4
3
1
0.8
0.4
0.2
3
2
0.8
0.6
0.2
0.2
0.4
2
0.2
0.4
1
Imaginary axis
1
0
−1
1
0.6
1
0.6
0.4
0.2
0.8
1
Imaginary axis
1
0
−1
1
0.8
0.2
0.6
0.4
0.8
0.6
−2
−3
0.8
1
0.4
0.6
0.6
0.4
0.8
0.2
1
−2
−3
0.2
0.8
0.4
1
−4
−4
−5
−6 −4 −2 0 2
Real axis
−5
−6 −4 −2 0 2
Real axis
5
SSP−RK4
5
SSP−RK5
4
4
3
3
0.6
0.8
0.4
1
2
1
0.4
0.6
0.8
2
1
0.2
0.6
0.8
Imaginary axis
1
0
−1
−2
0.8
1
0.4
0.6
0.8
0.2
0.2
0.6
0.4
0.6
0.8
0.4
1
1
Imaginary axis
1
0
−1
−2
0.8
0.6
0.8
1
0.4
0.2
0.4
0.2
0.4
1
0.6
0.8
−3
−3
0.6
0.8
1
0.4
−4
−4
−5
−6 −4 −2 0 2
Real axis
−5
−6 −4 −2 0 2
Real axis
5
SSP−RK6
5
SSP−RK7
Imaginary axis
4
3
2
1
0
−1
−2
−3
−4
1
0.8
0.6
0.4
0.4
0.2
0.6
0.8
0.2
1
0.2
0.2
0.4
0.4
0.6
1
0.6
0.8
1
0.8
0.6 0.6
0.8
0.8
1
1
Imaginary axis
4
3
2
1
0
−1
−2
−3
−4
1
0.6
0.8
0.4
0.4
0.8
0.6
0.2
0.2
1
0.6
1
0.2
0.4
0.8
0.4
0.2
0.2
0.8
0.6
0.4
0.6
1
1
1
1
−5
−6 −4 −2 0 2
Real axis
−5
−6 −4 −2 0 2
Real axis
Figure 2: Stability regions for the five-stage fourth-order Runge-Kutta method (top left)
and for five different SSP-RK methods of order m = 3, 4, 5, 6, 7
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