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Theory and Modern Applications

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