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Theory and Modern Applications

Table 10 Convergence orders of \(\pmb{\mathcal {O}(h^{\mu})}\) of the decoupled Algorithm  3.3 at time \(\pmb{T=1.0}\) , with varying mesh h but fixed time step \(\pmb{\Delta t=0.01}\)

From: Stability and convergence of some novel decoupled schemes for the non-stationary Stokes-Darcy model

1/ h

\(\boldsymbol {\|{\mathbf{u}}_{3.3}^{m,h}-{\mathbf{u}}_{3.3}^{m,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{{\mathbf{u}}_{f},h,0}}\)

\(\boldsymbol {\|{\mathbf{u}}_{3.3}^{m,h}-{\mathbf{u}}_{3.3}^{m,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{{\mathbf{u}}_{f},h,1}}\)

\(\boldsymbol {\|p_{3.3}^{m,h}-p_{3.3}^{m,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p_{f},h,0}}\)

2

0.215099

3.80338

1.6502

1.91072

0.940517

1.50494

4

0.0565547

3.86854

0.863652

1.93151

0.624954

2.43437

8

0.0146192

4.04581

0.447138

2.13698

0.256721

2.85629

16

0.00361342

 

0.209238

 

0.0898792

 

1/ h

\(\boldsymbol {\|\phi_{3.3}^{m,h}-\phi_{3.3}^{m,\frac{h}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\phi,h,0}}\)

\(\boldsymbol {\|\phi_{3.3}^{m,h}-\phi_{3.3}^{m,\frac{h}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\phi,h,1}}\)

2

0.135335

3.3074

1.30801

1.68713

4

0.0409188

4.07757

0.775285

1.90679

8

0.0100351

4.18889

0.406592

1.98307

16

0.00239564

 

0.205032