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

Table V: Convergence of the dispersion error for the one-dimensional wave equation in a
periodic domain with wave numbers k = π for p = 1, 2; k = 2π for p = 3, 4; k = 4π for
p = 5, 6, 7; and k = 8π for p = 8, 9, 10. In each case the (p + 1)st-order SSP-RK scheme
is applied.
p = 1 p = 2
Error Order Error Order
N el = 2 1.1219E-00 5.4535E-02
N el = 4 1.8866E-01 2.5721 1.9858E-03 4.7794
N el = 8 3.9271E-02 2.2642 8.0576E-05 4.6232
N el = 16 9.1970E-03 2.0942 4.2539E-06 4.2435
N el = 32 2.2573E-03 2.0266 2.5327E-07 4.0700
N el = 64 5.6163E-04 2.0069 1.5631E-08 4.0182
p = 3 p = 4
Error Order Error Order
N el = 2 2.8401E-01 4.3239E-02
N el = 4 1.8427E-03 7.2680 8.7429E-05 8.9500
N el = 8 1.1103E-04 4.0528 9.1380E-08 9.9020
N el = 16 8.1803E-06 3.7626 3.1783E-09 4.8456
N el = 32 5.1772E-07 3.9819 5.1479E-11 5.9481
N el = 64 3.2403E-08 3.9980 8.0824E-13 5.9930
p = 5 p = 6 p = 7
Error Order Error Order Error Order
N el = 2 2.2250E-00 6.2763E-01 1.4307E-01
N el = 4 7.5731E-03 8.1987 4.3307E-04 10.501 1.8263E-05 12.936
N el = 8 4.8796E-06 10.600 4.9182E-08 13.104 3.2505E-10 15.778
N el = 16 2.1995E-08 7.7934 1.5744E-11 11.609 4.6185E-13 9.4590
N el = 32 3.2993E-10 6.0589
N el = 64 5.1621E-12 5.9981
p = 8 p = 9 p = 10
Error Order Error Order Error Order
N el = 2 3.8264E-00 8.3880E-01 1.5821E-00
N el = 4 4.9854E-02 6.2621 6.5271E-03 7.0057 6.6177E-04 11.223
N el = 8 1.1951E-06 15.348 3.1177E-08 17.676 6.6458E-10 19.925
N el = 16 6.6187E-12 17.462
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