Issue 61

N. Razali et alii, Frattura ed Integrità Strutturale, 61 (2022) 214-229; DOI: 10.3221/IGF-ESIS.61.14

3) CFL=10.0

Figure 23: Two-step active symmetrised IMR method results for CFL=10.0

Figure 24: Two-step active symmetrised IMR method mesh results for CFL=10.0

The above results and incorporating this two-step active IMR method into the problem of an advection equation reveal that the results do not explode (oscillatory), as shown in Figs. 19–24. No convergence of the exact solution at any time is observed. The results of all CFL values become diffuse from the exact results, even if CFL=10.0. Although this method does not focus on accurate results, it is stable for all Δ t and Δ x . Fig. 25 shows the solutions calculated by each method at different u and CFL=1. It shows the damping and dissipative behaviour of 1 active IMR, 2 active IMR and Lax methods as compared to the FTCS. It shows the oscillatory behaviour of the FTCS as the amplitude increases. In summary, for the square-wave (FTCS) method, the solution explodes (oscillatory) and becomes unstable for a CFL value greater than 0.01. For the step-wave Lax method, when CFL>1 → the solution is unstable, when CFL<1 → the solution diffuses (obtaining smaller time steps worsens) and when CFL=1 → , the solution converges to the exact solution, for the one-step active symmetrised IMR. The results do not explode (oscillate). The results of all CFL values become diffuse from the accurate results, even if CFL=10.0. Although this method does not focus on accurate results, it is stable for all Δ t and Δ x, for the two-step active symmetrised IMR. The results do not explode (oscillate). No convergence of the exact solution for CFL values is observed, even if CFL=10.0. Although this method does not focus on accurate results, it is stable for all Δ t and Δ x.

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