Theory and Modern Applications
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 |