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Table 12 Convergence orders of \(\pmb{\mathcal {O}(\Delta t^{\gamma})}\) of the decoupled Algorithm  3.2 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 \(\boldsymbol {\|{\mathbf{u}}_{3.2}^{m,\Delta t}-{\mathbf{u}}_{3.2}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,0}}\) \(\boldsymbol {\|{\mathbf{u}}_{3.2}^{m,\Delta t}-{\mathbf{u}}_{3.2}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{{\mathbf{u}}_{f},\Delta t,1}}\) \(\boldsymbol {\|p_{3.2}^{m,\Delta t}-p_{3.2}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{p_{f},\Delta t,0}}\)
0.1 0.000809376 1.95487 0.0080253 1.86026 0.0173397 1.9429
0.05 0.000414031 1.97844 0.00431408 2.02706 0.00892463 1.97378
0.025 0.000209272 1.98947 0.00212825 1.62937 0.00452159 1.98749
0.0125 0.00010519   0.00130618   0.00227502  
Δ t \(\boldsymbol {\|\phi_{3.2}^{m,h\Delta t}-\phi_{3.2}^{m,\frac{\Delta t}{2}}\|_{0}}\) \(\boldsymbol {\rho_{\phi,\Delta t,0}}\) \(\boldsymbol {\|\phi_{3.2}^{m,\Delta t}-\phi_{3.2}^{m,\frac{\Delta t}{2}}\|_{1}}\) \(\boldsymbol {\rho_{\phi,\Delta t,1}}\)
0.1 0.00257197 1.92495 0.0140113 1.89193
0.05 0.00133612 1.96705 0.00740581 1.94972
0.025 0.000679251 1.98468 0.0037984 1.90576
0.0125 0.000342248   0.00199312