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Table 13 Convergence orders of \(\pmb{\mathcal {O}(\Delta t^{\gamma})}\) of the decoupled Algorithm  3.3 at time \(\pmb{T=1.0}\) , with varying time step Δ t but fixed mesh \(\pmb{h=\frac{1}{32}}\)

From: Stability and convergence of some novel decoupled schemes for the non-stationary Stokes-Darcy model

Δ t \(\boldsymbol {\|{\mathbf{u}}_{3.3}^{m,\Delta t}-{\mathbf{u}}_{3.3}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,0}}\) \(\boldsymbol {\|{\mathbf{u}}_{3.3}^{m,\Delta t}-{\mathbf{u}}_{3.3}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,1}}\) \(\boldsymbol {\|p_{3.3}^{m,\Delta t}-p_{3.3}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{p_{f},\Delta t,0}}\)
0.1 0.00090236 1.93382 0.00891485 1.85837 0.0215839 1.89965
0.05 0.00046662 1.96917 0.00479714 2.00887 0.011362 1.95491
0.025 0.000236963 1.98516 0.00238798 1.68501 0.00581205 1.97876
0.0125 0.000119368   0.00141719   0.00293722  
Δ t \(\boldsymbol {\|\phi_{3.3}^{m,\Delta t}-\phi_{3.3}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{\phi,\Delta t,0}}\) \(\boldsymbol {\|\phi_{3.3}^{m,\Delta t}-\phi_{3.3}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{\phi,\Delta t,1}}\)
0.1 0.00433364 1.90828 0.0174115 1.88974
0.05 0.00227097 1.95936 0.00921372 1.94985
0.025 0.00115904 1.98099 0.00472534 1.92982
0.0125 0.00058508   0.00244859