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Theory and Modern Applications

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