Table 8 Convergence orders of the Euler implicit/explicit scheme at time \(\pmb{T=1.0}\) with time step \(\pmb{\Delta t=0.01}\)
From: The Euler implicit/explicit scheme for the Boussinesq equations
\(\boldsymbol {\frac{1}{h}}\) | \(\boldsymbol {\|u^{E,n,h}-u^{E,n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{u,h,0}}\) | \(\boldsymbol {\|u^{E,n,h}-u^{E,n,\frac{h}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{u,h,1}}\) | \(\boldsymbol {\|p^{E,n,h}-p^{E,n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{p,h,0}}\) |
|---|---|---|---|---|---|---|
2 | 0.0172184 | 2.65201 | 0.230411 | 2.14169 | 0.631987 | 4.86024 |
4 | 0.00649258 | 3.37906 | 0.107584 | 2.20925 | 0.130032 | 3.19873 |
8 | 0.00192142 | 3.98294 | 0.0486971 | 2.12866 | 0.0406512 | 3.29108 |
16 | 0.000482412 | 0.0228769 | 0.0123519 |
\(\boldsymbol {\frac{1}{h}}\) | \(\boldsymbol {\|\theta^{E,n,h}-\theta^{E,n,\frac{h}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{\theta,h,0}}\) | \(\boldsymbol {\|\theta^{E,n,h}-\theta^{E,n,\frac{h}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{\theta,h,1}}\) |
|---|---|---|---|---|
2 | 0.01664 | 3.77398 | 0.131698 | 1.96258 |
4 | 0.00440914 | 3.80302 | 0.0671049 | 1.91516 |
8 | 0.00115938 | 3.92765 | 0.0350387 | 1.99231 |
16 | 0.000295184 | 0.017587 |