Table 6 Convergence orders of the Euler implicit/explicit scheme 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 | \(\boldsymbol {\|u^{E,n,\Delta t}-u^{E,n,\frac{\Delta t}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{u,\Delta t,0}}\) | \(\boldsymbol {\|u^{E,n,\Delta t}-u^{E,n,\frac{\Delta t}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{u,\Delta t,1}}\) | \(\boldsymbol {\|p^{E,n,\Delta t}-p^{E,n,\frac{\Delta t}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{p,\Delta t,0}}\) |
|---|---|---|---|---|---|---|
0.1 | 1.09855e−005 | 2.06806 | 7.97559e−005 | 2.06786 | 2.10207e−005 | 2.04139 |
0.05 | 5.31197e−006 | 2.03533 | 3.85693e−005 | 2.03005 | 1.02973e−005 | 1.57437 |
0.025 | 2.60989e−006 | 2.01716 | 1.89991e−005 | 1.91612 | 6.54057e−006 | 0.654705 |
0.0125 | 1.29384e−006 | 9.91544e−006 | 9.9901e−006 |
Δ t | \(\boldsymbol {\|\theta^{E,n,\Delta t}-\theta^{E,n,\frac{\Delta t}{2}}\|_{0}}\) | \(\boldsymbol {\rho_{\theta,\Delta t,0}}\) | \(\boldsymbol {\|\theta^{E,n,\Delta t}-\theta^{E,n,\frac{\Delta t}{2}}\|_{1}}\) | \(\boldsymbol {\rho_{\theta,\Delta t,1}}\) |
|---|---|---|---|---|
0.1 | 1.14669e−005 | 2.06806 | 8.10171e−005 | 2.06809 |
0.05 | 5.54475e−006 | 2.0351 | 3.91749e−005 | 2.0351 |
0.025 | 2.72457e−006 | 2.01847 | 1.92496e−005 | 2.01848 |
0.0125 | 1.34982e−006 | 9.53665e−006 |