Fig. 1
Deformational flow case: exact solution at (a) day 6, (c) day 20 and (e) day 60. The corresponding numerical solutions with EPI2 with a 12-h time-step are shown in (b), (d) and (f).
Table 1. Deformational flow case: normalised errors
RK4 Δt=2160 s1.386977×10−33.475878×10−31.894418×10−1EPI2 Δt=12 h1.386983×10−33.476770×10−31.894432×10−1EPI3 Δt=12 h1.387031×10−33.477584×10−31.894430×10−1
Fig. 2
Normalised (left) and (right) error measures for the height field in the unsteady flow case. Exponential methods EPI2 (top) and EPI3 (bottom) using various time-steps are compared against the reference RK4 and T–ABT. Note the logarithmic scale on the vertical axis.
Fig. 3
Normalised ℓ2 (left) and ℓ∞ (right) error measures for the height field in the Williamson et al. (1992) mountain case 5. Exponential methods EPI2 (top) and EPI3 (bottom) using various time-steps are compared against the reference RK4 and T–ABT. Note the logarithmic scale on the vertical axis.
Fig. 4
Normalised ℓ2 (left) and ℓ∞ (right) error measures for the height field in the Williamson et al. (1992) Rossby–Haurwitz wave case 6. Exponential methods EPI2 (top) and EPI3 (bottom) using various time-steps are compared against the reference RK4 and T–ABT. Note the logarithmic scale on the vertical axis.
Fig. 5
Relative vorticity field (northern hemisphere) at day 6 for the Galewsky et al. (2004) case. From top to bottom: RK4 Δt=240s, T–ABT Δt=900s, EPI2 Δt=3600s and EPI3 Δt=3600s.
Fig. 6
Relative vorticity field (northern hemisphere) at day 6 for the Galewsky et al. (2004) case with Δt=7200s for EPI2 (top) and EPI3 (bottom).
Fig. 7
Relative vorticity field (northern hemisphere) at day 6 for the Galewsky et al. (2004) case with explicit dissipation added. From top: RK4 Δt=240s, EPI2 Δt=7200s and EPI3 Δt=7200s.
Fig. 8
Divergence fields for the barotropic jet case after 12 h of simulation. Top: RK4 Δt=240s (left) and T–ABT Δt=900s (right). Middle: EPI2 (left) and EPI3 (right) with Δt=3600s. Bottom: EPI2 (left) and EPI3 (right) with Δt=7200s. Contour interval is 8×10−7s−1.
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Table 2. Average execution time (minutes) per simulated day, for the cases of the unsteady flow (Läuter et al., 2005) and the mountain and Rossby–Haurwitz wave of Williamson et al. (1992)
RK4 Δt=240s2.62.62.4T–ABT Δt=900s0.80.90.9EPI2 Δt=1 h3.23.33.6EPI2 Δt=2 h2.72.32.8EPI3 Δt=1 h3.23.33.4EPI3 Δt=2 h2.72.32.8
Table 3. Average time taken, in seconds, for the calculation of the Jacobi matrix, the forcing terms and the execution of the phipm routine at each step over the 14-d Rossby–Haurwitz wave simulation. The fourth column gives the average number of Krylov basis vectors per call to phipm
EPI2 Δt=1 h2.30.16.334.6EPI2 Δt=2 h2.20.111.859.5EPI3 Δt=1 h2.20.15.933.1EPI3 Δt=2 h2.20.111.658.3