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Table 14 Convergence orders of \(\pmb{\mathcal {O}(\Delta t^{\gamma})}\) of the decoupled Algorithm  3.4 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 \(\boldsymbol {\|{\mathbf{u}}_{3.4}^{m,\Delta t}-{\mathbf{u}}_{3.4}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,0}}\) \(\boldsymbol {\|{\mathbf{u}}_{3.4}^{m,\Delta t}-{\mathbf{u}}_{3.4}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,1}}\) \(\boldsymbol {\|p_{3.4}^{m,h}-p_{3.4}^{m,\frac{h}{2}}\|_{0}}\) \(\boldsymbol {\rho_{p_{f},\Delta t,0}}\)
0.1 0.000373673 1.97879 0.00354043 1.47841 0.0150435 1.94262
0.05 0.000188839 1.99068 0.00239475 1.77955 0.00774392 1.97437
0.025 9.48618e−005 1.99568 0.0013457 0.80887 0.00392223 1.98791
0.0125 4.75336e−005   0.00166368   0.00197304  
Δ t \(\boldsymbol {\|\phi_{3.4}^{m,\Delta t}-\phi_{3.4}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{\phi,\Delta t,0}}\) \(\boldsymbol {\|\phi_{3.4}^{m,\Delta t}-\phi_{3.4}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{\phi,\Delta t,1}}\)
0.1 0.00410297 1.87043 0.0244922 1.86572
0.05 0.0021936 1.94215 0.0131275 1.93868
0.025 0.00112947 1.9726 0.00677134 1.84128
0.0125 0.000572579   0.00367753