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Table 5 Convergence orders of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with fixed mesh \(\pmb{h=\frac{1}{32}}\)

From: The Euler implicit/explicit scheme for the Boussinesq equations

Δ t

\(\boldsymbol {\|u^{n,\Delta t}-u^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{u,\Delta t,0}}\)

\(\boldsymbol {\|u^{n,\Delta t}-u^{n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{u,\Delta t,1}}\)

\(\boldsymbol {\|p^{n,\Delta t}-p^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{p,\Delta t,0}}\)

0.1

1.09819e−005

2.06348

7.97307e−005

2.06314

2.09931e−005

2.0025

0.05

5.32202e−006

2.04257

3.86453e−005

2.03837

1.04834e−005

1.64187

0.025

2.60556e−006

2.01799

1.89589e−005

1.91156

6.38507e−006

0.618958

0.0125

1.29116e−006

 

9.91804e−006

 

1.03158e−005

 

Δ t

\(\boldsymbol {\|\theta^{n,\Delta t}-\theta^{n,\frac{\Delta t}{2}}\|_{0}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,0}}\)

\(\boldsymbol {\|\theta^{n,\Delta t}-\theta^{n,\frac{\Delta t}{2}}\|_{1}}\)

\(\boldsymbol {\rho_{\theta,\Delta t,1}}\)

0.1

1.14663e−005

2.06739

8.10132e−005

2.06742

0.05

5.54628e−006

2.03617

3.91857e−005

2.03617

0.025

2.72388e−006

2.01728

1.92448e−005

2.01729

0.0125

1.35028e−006

 

9.53992e−006