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Table 1 Numerical results of the standard Galerkin FEM at time \(\pmb{T=1.0}\) with varying time step Δ t but fixed mesh size \(\pmb{h=\frac{1}{32}}\)

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

Δ t

\(\boldsymbol {\frac{\|u-u^{n}\|_{0}}{\|u\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(u-u^{n})\|_{0}}{\|\nabla u\|_{0}}}\)

\(\boldsymbol {\frac{\|p-p^{n}\|_{0}}{\|p\|_{0}}}\)

\(\boldsymbol {\frac{\|\theta-\theta^{n}\|_{0}}{\|\theta\|_{0}}}\)

\(\boldsymbol {\frac{\|\nabla(\theta-\theta^{n})\|_{0}}{\|\nabla\theta\|_{0}}}\)

CPU(S)

0.1

0.000171956

0.0127757

0.00415879

0.000115897

0.0104146

189.625

0.05

0.000162403

0.0127748

0.00416427

0.000107522

0.0104136

306.596

0.025

0.00015784

0.0127745

0.00416706

0.000103686

0.0104134

476.16

0.0125

0.000155623

0.0127744

0.0041685

0.000101859

0.0104133

965.143

0.00625

0.000154529

0.0127744

0.00416949

0.000100969

0.0104133

1,807.32