Appendix: Proof of Theorem 3.1
Proof
We consider continuous variational formulations of (1.1) to derive error equations at time tn+𝜃 = (n + 𝜃)Δt.
$$ \begin{array}{@{}rcl@{}} &&{}({D}_{n+\theta}({ u}),{ v}^{h})+Pr ({F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u}),\nabla { v}^{h})+ b^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),{ v}^{h})\\ &&+Da^{-1}(F_{n+\theta}^{\varepsilon,\nu}({ u}),{ v}^{h}) -({F}_{n+\theta}^{\varepsilon,\nu}(p),\nabla \cdot { v}^{h})\\ &=&PrRa((H_{n+\theta}(T)-NH_{n+\theta}(S)),{ v}^{h})-PrHa^{2}(F_{n+\theta}^{\varepsilon, \nu}({ u}),{ v}^{h})\textbf{e}_{2}\\ &&+\mathcal{E}_{1}({ u}_{n+\theta},T_{n+\theta},S_{n+\theta},{ v}^{h}), \end{array} $$
(A.1)
$$ \begin{array}{@{}rcl@{}} &&({D}_{n+\theta}(T),\chi^{h})+({F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla T),\nabla \chi^{h})+c^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon_{1},\gamma}(T),\chi^{h})\\&&=\mathcal{E}_{2}(u_{n+\theta},T_{n+\theta},\chi^{h}), \end{array} $$
(A.2)
$$ \begin{array}{@{}rcl@{}} &&({D}_{n+\theta}(S),{\Phi}^{h})+Le^{-1}({F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla S),\nabla {\Phi}^{h})+d^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(S),{\Phi}^{h}),\\ &&=\mathcal{E}_{3}({ u}_{n+\theta},S_{n+\theta},{\Phi}^{h}). \end{array} $$
(A.3)
where:
$$ \begin{array}{@{}rcl@{}} && {} \mathcal{E}_{1}({ u}_{n+\theta},p_{n+\theta},T_{n+\theta},S_{n+\theta},{ v}^{h})=({D}_{n+\theta}({ u})-{ u}_{t}(t_{n+\theta}),{ v}^{h})+Pr ({F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})-\nabla { u}(t_{n+\theta}),\nabla { v}^{h})\\ &&+ b^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon,\nu}(u),{ v}^{h})-b^{*}(u(t_{n+\theta}),u(t_{n+\theta}),{ v}^{h})+Da^{-1}(F_{n+\theta}^{\varepsilon,\nu}(u)-u(t_{n+\theta}),{ v}^{h})\\ &&-({F}_{n+\theta}^{\varepsilon,\nu}(p)- p(t_{n+\theta}),\nabla \cdot { v}^{h})+PrRa(T(t_{n+\theta})-H_{n+\theta}(T),{ v}^{h})\\ &&-NPrRa(S(t_{n+\theta})-H_{n+\theta}(S),{ v}^{h})-PrHa^{2}(u(t_{n+\theta})-F_{n+\theta}^{\varepsilon, \nu}(u),{ v}^{h})\textbf{e}_{2}, \end{array} $$
(A.4)
$$ \begin{array}{@{}rcl@{}} &&\mathcal{E}_{2}({ u}_{n+\theta},T_{n+\theta},\chi^{h})= ({D}_{n+\theta}(T)-T_{t}(t_{n+\theta}),\chi^{h})+({F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla T)-\nabla T(t_{n+\theta}),\nabla \chi^{h})\\ &&+c^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon_{1},\gamma}(T),\chi^{h})-c^{*}(u(t_{n+\theta}), T(t_{n+\theta}),\chi^{h}), \end{array} $$
(A.5)
$$ \begin{array}{@{}rcl@{}} && \mathcal{E}_{3}({ u}_{n+\theta},S_{n+\theta},{\Phi}^{h})=({D}_{n+\theta}(S)-S_{t}(t_{n+\theta}),{\Phi}^{h})+Le^{-1}({F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla S)-\nabla S(t_{n+\theta}),\nabla {\Phi}^{h})\\ &&+d^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(S),{\Phi}^{h})-d^{*}(u(t_{n+\theta}), S(t_{n+\theta}),{\Phi}^{h}). \end{array} $$
(A.6)
for all (vh, χh,Φh) ∈ (Xh, Wh,Ψh). Subtracting (2.24)–(2.27) from (A.1) to (A.3) and decomposing the velocity, temperature, and concentration errors as:
$$ \begin{array}{@{}rcl@{}} e_{{ u},{n+\theta}}^{h}&=&{ u}_{n+\theta}-{ u}_{n+\theta}^{h}=({ u}_{n+\theta}-\mathcal{U})-({ u}_{n+\theta}^{h}-\mathcal{U})=\boldsymbol{\eta}_{{ u},n+\theta}-\boldsymbol{\phi}_{{u}, n+\theta}^{h},\\ e_{T,{n+\theta}}^{h}&=&T_{n+\theta}-T_{n+\theta}^{h}=(T_{n+\theta}-\mathcal{T})-(T_{n+\theta}^{h}-\mathcal{T})={\eta}_{T, n+\theta}-\phi_{T,n+\theta}^{h},\\ e_{S,{n+\theta}}^{h}&=&S_{n+\theta}-S_{n+\theta}^{h}=(S_{n+\theta}-\mathcal{S})-(S_{n+\theta}^{h}-\mathcal{S})={\eta}_{S,n+\theta}-\phi_{S,n+\theta}^{h}, \end{array} $$
(A.7)
where \(\mathcal {U}, \mathcal {T}, \text {and}~\mathcal {S}\) denote interpolants of u, T, and S in Vh, Wh,and Ψh, respectively, we have:
$$ \begin{array}{@{}rcl@{}} &&{}({D}_{n+\theta}(\boldsymbol{\phi}_{{ u}}^{h}),{ v}^{h})+Pr ({F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{{ u}}^{h}),\nabla { v}^{h}) +(Da^{-1}-PrHa^{2})(F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}),{ v}^{h})\\ &=&-PrRa((H_{n+\theta}({e_{T}^{h}})-NH_{n+\theta}({e_{S}^{h}})),{ v}^{h})+PrHa^{2}(F_{n+\theta}^{\varepsilon, \nu}(\boldsymbol{\eta}_{{ u}}),{ v}^{h})\textbf{e}_{2}\\ &&+({D}_{n+\theta}(\boldsymbol{\eta}_{{ u}}),{ v}^{h})+Pr ({F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\eta}_{{ u}}),\nabla { v}^{h}) +Da^{-1}(F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{{ u}}),{ v}^{h})\\ &&+b^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),{ v}^{h})-b^{*}({H}_{n+\theta}({ u}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}^{h}),{ v}^{h})\\ &&-({F}_{n+\theta}^{\varepsilon,\nu}(p-p^{h}),\nabla \cdot { v}^{h})-\mathcal{E}_{1}({ u}_{n+\theta},T_{n+\theta},S_{n+\theta},{ v}^{h}), \end{array} $$
(A.8)
$$ \begin{array}{@{}rcl@{}} &&({D}_{n+\theta}({\phi_{T}^{h}}),\chi^{h})+({F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla {\phi_{T}^{h}}),\nabla \chi^{h})\\ &=&({D}_{n+\theta}(\eta_{T}),\chi^{h})+({F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla \eta_{T}),\nabla \chi^{h})+c^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon_{1},\gamma}(T),\chi^{h})\\ &&-c^{*}({H}_{n+\theta}(u^{h}),{F}_{n+\theta}^{\varepsilon_{1},\gamma}(T^{h}),\chi^{h})-\mathcal{E}_{2}(u_{n+\theta},T_{n+\theta},\chi^{h}), \end{array} $$
(A.9)
$$ \begin{array}{@{}rcl@{}} &&({D}_{n+\theta}({\phi_{S}^{h}}),{\Phi}^{h})+Le^{-1}({F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla {\phi_{S}^{h}}),\nabla {\Phi}^{h}),\\ &=&({D}_{n+\theta}(\eta_{S}),{\Phi}^{h})+Le^{-1}({F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla \eta_{S}),\nabla {\Phi}^{h})+d^{*}({H}_{n+\theta}(u),{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(S),{\Phi}^{h})\\ &&-d^{*}({H}_{n+\theta}(u^{h}),{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(S^{h}),{\Phi}^{h})-\mathcal{E}_{3}(u_{n+\theta},S_{n+\theta},{\Phi}^{h}). \end{array} $$
(A.10)
Letting \(({ v}^{h},\chi ^{h},{\Phi }^{h})=(F^{\varepsilon _{1},\nu }_{n+\theta }(\mathbf {\phi }^{h}_{u}),{F}_{n+\theta }^{\varepsilon _{1},\gamma }({\phi _{T}^{h}}),{F}_{n+\theta }^{\varepsilon _{2},Le^{-1}}({\phi _{S}^{h}}))\) in (A.8)–(A.10) and utilizing (2.20) produce:
$$ \begin{array}{@{}rcl@{}} &&{}\frac{1}{\Delta t}\left\| \begin{bmatrix} \boldsymbol{\phi}_{{ u},{n+1}}^{h} \\ \boldsymbol{\phi}_{{ u},{n}}^{h}\quad \end{bmatrix}\right\|_{G}^{2} -\frac{1}{\Delta t}\left\| \begin{bmatrix} \boldsymbol{\phi}_{{ u},{n}}^{h}\quad\\ \boldsymbol{\phi}_{{ u},{n-1}}^{h} \end{bmatrix}\right\|_{G}^{2} +\frac{1}{4{\Delta} t}\left\|\boldsymbol{\phi}_{{ u},{n+1}}^{h}-2\boldsymbol{\phi}_{{ u},{n}}^{h}+\boldsymbol{\phi}_{{ u},{n-1}}^{h}\right\|_{F}^{2}\\ &&+Pr \|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{u}^{h})\|^{2} +Da^{-1}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|^{2}\\ &=&-PrRa((H_{n+\theta}({e_{T}^{h}})-NH_{n+\theta}({e_{S}^{h}})),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))+PrHa^{2}(F_{n+\theta}^{\varepsilon, \nu}(e_{u}^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))\textbf{e}_{2}\\ &&+({D}_{n+\theta}(\boldsymbol{\eta}_{{ u}}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))+Pr ({F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\eta}_{{ u}}), F_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{{ u}}^{h})) \\&&+Da^{-1}(F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{{ u}}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))+b^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))\\ &&-b^{*}({H}_{n+\theta}({ u}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))+({F}_{n+\theta}^{\varepsilon,\nu}(p-p^{h}),\nabla \cdot F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))\\ &&-\mathcal{E}_{1}({ u}_{n+\theta},T_{n+\theta},S_{n+\theta},F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})). \end{array} $$
(A.11)
Applying the Cauchy-Schwarz, Young’s inequalities, and Taylor series expansion with remainder and (A.7), the terms on the right hand side of (A.11) are bounded by:
$$ \begin{array}{@{}rcl@{}} |PrRa((H_{n+\theta}({e_{T}^{h}})-NH_{n+\theta}({e_{S}^{h}})),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))| &\leq&CPrRa^{2}\left( \|H_{n+\theta}(\eta_{T})\|^{2}+\|H_{n+\theta}({\phi_{T}^{h}})\|^{2}\right.\\ &&\left.+N^{2}(\|H_{n+\theta}(\eta_{S})\|^{2}+\|H_{n+\theta}({\phi_{S}^{h}})\|^{2})\right)\\ &&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}, \end{array} $$
(A.12)
$$ \begin{array}{@{}rcl@{}} PrHa^{2}|(F_{n+\theta}^{\varepsilon, \nu}(e_{{ u}}^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))\textbf{e}_{2}|&\leq&CPrHa^{4}\|F_{n+\theta}^{\varepsilon, \nu}(\boldsymbol{\eta}_{u})\|^{2}+PrHa^{2}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h})\|^{2}\\ &&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{u}^{h})\|^{2}, \end{array} $$
(A.13)
and
$$ \begin{array}{@{}rcl@{}} |({D}_{n+\theta}(\boldsymbol{\eta}_{{ u}}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))| & \leq & C{Pr^{-1}}\|{D}_{n+\theta}(\boldsymbol{\eta}_{{ u}})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|^{2}, \end{array} $$
(A.14)
$$ \begin{array}{@{}rcl@{}} Pr |({F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\eta}_{u}), F_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{u}^{h}))| & \leq & CPr\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\eta}_{u})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{u}^{h})\|^{2}, \end{array} $$
(A.15)
$$ \begin{array}{@{}rcl@{}} Da^{-1}|(F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))| & \leq & CDa^{-1}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{u})\|^{2}+\frac{Da^{-1}}{2}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h})\|^{2}, \end{array} $$
(A.16)
$$ \begin{array}{@{}rcl@{}} |({F}_{n+\theta}^{\varepsilon,\nu}(p-p^{h}),\nabla \cdot F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))| & \leq & C{Pr}^{-1}\|{F}_{n+\theta}^{\varepsilon,\nu}(p-p^{h})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{u}^{h})\|^{2}. \end{array} $$
(A.17)
Note that \(b^{*}({H}_{n+\theta }({ u}^{h}),{F}_{n+\theta }^{\varepsilon ,\nu }(\boldsymbol {\phi }_{{ u}}^{h}),F_{n+\theta }^{\varepsilon ,\nu }(\boldsymbol {\phi }_{{ u}}^{h}))=0\), then the nonlinear terms can be written as:
$$ \begin{array}{@{}rcl@{}} &&{}|-b^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))+b^{*}({H}_{n+\theta}({ u}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &\leq&|b^{*}({H}_{n+\theta}(\boldsymbol{\phi}_{{ u}}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|+|b^{*}({H}_{n+\theta}(\boldsymbol{\eta}_{{ u}}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &&+|b^{*}({H}_{n+\theta}({ u}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{{ u}}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|. \end{array} $$
(A.18)
For the terms on the right hand side of (A.18), we use (2.7) and Poincaré-Friedrichs’ inequality, which yields:
$$ \begin{array}{@{}rcl@{}} &&{}|b^{*}({H}_{n+\theta}(\boldsymbol{\phi}_{{ u}}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &\leq&\frac{1}{2}\left( \|{H}_{n+\theta}(\boldsymbol{\phi}_{{ u}}^{h})\|\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})\|_{\infty}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|+\|{H}_{n+\theta}(\boldsymbol{\phi}_{{ u}}^{h})\|\|{F}_{n+\theta}^{\varepsilon,\nu}({ u})\|_{\infty}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|\right)\\ &\leq&C{Pr}^{-1}\left( \|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})\|^{2}_{\infty}+\|{F}_{n+\theta}^{\varepsilon,\nu}({ u})\|^{2}_{\infty}\right)((\theta+1)^{2}\|\boldsymbol{\phi}_{{ u},n}^{h}\|^{2}+\theta^{2}\|\boldsymbol{\phi}_{{ u},n-1}^{h}\|^{2})+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2},\\ &&|b^{*}({H}_{n+\theta}(\boldsymbol{\eta}_{{ u}}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &\leq&C{Pr}^{-1}\|{H}_{n+\theta}(\nabla\boldsymbol{\eta}_{{ u}})\|^{2}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2},\\ &&|b^{*}({H}_{n+\theta}({ u}^{h}),{F}_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\eta}_{{ u}}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &\leq&C{Pr}^{-1}\|{H}_{n+\theta}(\nabla{ u}^{h})\|^{2}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\eta}_{{ u}})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}. \end{array} $$
(A.19)
We then bound the all terms of \(\mathcal {E}_{1}({ u}_{n+\theta },T_{n+\theta },S_{n+\theta },F_{n+\theta }^{\varepsilon ,\nu }(\boldsymbol {\phi }_{{ u}}^{h}))\) using Cauchy-Schwarz, Young’s inequality and Taylor series expansion with remainder. Then, one gets:
$$ \begin{array}{@{}rcl@{}} |({D}_{n+\theta}({ u})-{ u}_{t}(t_{n+\theta}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|&\leq&C{Pr}^{-1}\|{D}_{n+\theta}({ u})-{ u}_{t}(t_{n+\theta})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}\\ &\leq&C{Pr}^{-1}\theta^{6}{\Delta} t^{3}{\int}_{t_{n-1}}^{t_{n+1}}\|{ u}_{ttt}\|^{2}\mathrm{d} t\\ &&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}, \end{array} $$
(A.20)
$$ \begin{array}{@{}rcl@{}} Pr |({F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})-\nabla { u}(t_{n+\theta}),\nabla F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))|&\leq&CPr \|\nabla(\theta u_{n+1}+(1-\theta)u_{n}-u_{n+\theta})\|^{2}\\ &&+C{Pr}^{-1}\epsilon^{2}\theta^{2}\|\nabla ({ u}_{n+1}-2{ u}_{n}+{ u}_{n-1})\|^{2}\\ &&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}, \end{array} $$
(A.21)
$$ \begin{array}{@{}rcl@{}} Da^{-1}|(F_{n+\theta}^{\varepsilon,\nu}(u)-u(t_{n+\theta}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h}))|&\leq&\|F_{n+\theta}^{\varepsilon,\nu}(u)-u(t_{n+\theta})\|^{2}+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{u}^{h})\|^{2}, \end{array} $$
(A.22)
$$ \begin{array}{@{}rcl@{}} |({F}_{n+\theta}^{\varepsilon,\nu}(p)- p(t_{n+\theta}),\nabla \cdot F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|&\leq&C{Pr}^{-1}\|{F}_{n+\theta}^{\varepsilon,\nu}(p)- p(t_{n+\theta})\|^{2}\\ &&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{u}^{h})\|^{2}. \end{array} $$
(A.23)
To estimate remaining nonlinear terms in (A.4), we use (2.5)–(2.9) and Taylor series expansion with remainder as:
$$ \begin{array}{@{}rcl@{}} &&{}|b^{*}({H}_{n+\theta}({ u}),{F}_{n+\theta}^{\varepsilon,\nu}({ u}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))-b^{*}({ u}(t_{n+\theta}),{ u}(t_{n+\theta}),{ v}^{h})|\\ &\leq&C{Pr}^{-1}\left( \theta^{2}(1+\theta^{2})\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla{ u})\|^{2}+(\theta^{2}(1-\theta^{2})+\epsilon^{2}\theta^{2})\|\nabla { u}_{n+\theta}\|^{2}\right){\Delta} t^{3}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla u_{tt}\|^{2} \mathrm{d} t\\&&+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{u}^{h})\|^{2}. \end{array} $$
(A.24)
The other terms on the right hand side of (A.4) can be bounded as:
$$ \begin{array}{@{}rcl@{}} &&{}|PrRa(T(t_{n+\theta})-H_{n+\theta}(T^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|+|NPrRa(S(t_{n+\theta})-H_{n+\theta}(S^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))|\\ &&+|PrHa^{2}({ u}(t_{n+\theta})-F_{n+\theta}^{\varepsilon, \nu}({ u}^{h}),F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h}))\textbf{e}|\\ &\leq&CPr\theta^{2}(1+\theta)^{2}{\Delta} t^{3}\left( Ra^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla T_{tt}\|^{2} \mathrm{d} t+N^{2}Ra^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla S_{tt}\|^{2} \mathrm{d} t \right.\\ &&\left.+Ha^{4}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla { u}_{tt}\|^{2} \mathrm{d} t\right)+\frac{Pr}{28}\|F_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\phi}_{{ u}}^{h})\|^{2}. \end{array} $$
(A.25)
Inserting all bounds in (A.11) and multiplying by Δt, we get:
$$ \begin{array}{@{}rcl@{}} &&\left\| \begin{bmatrix} \boldsymbol{\phi}_{{ u},{n+1}}^{h} \\ \boldsymbol{\phi}_{{ u},{n}}^{h}\quad \end{bmatrix}\right\|_{G}^{2} -\left\| \begin{bmatrix} \boldsymbol{\phi}_{{ u},{n}}^{h}\quad\\ \boldsymbol{\phi}_{{ u},{n-1}}^{h} \end{bmatrix}\right\|_{G}^{2} +\frac{1}{4}\left\|\boldsymbol{\phi}_{{ u},{n+1}}^{h}-2\boldsymbol{\phi}_{{ u},{n}}^{h}+\boldsymbol{\phi}_{{ u},{n-1}}^{h}\right\|_{F}^{2}\\&&+\frac{Pr{\Delta} t}{2} \|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{u}^{h})\|^{2} +\frac{Da^{-1}{\Delta} t}{2}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|^{2} \\ &\leq&C{\Delta} t\left( \|H_{n+\theta}(\eta_{T})\|^{2}+\|NH_{n+\theta}(\eta_{S})\|^{2}+PrHa^{2}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|^{2}+(1+Da^{-1})\|F_{n+\theta}^{\varepsilon, \nu}(\boldsymbol{\eta}_{u})\|^{2}\right.\\ &&+{Pr}^{-1}\theta^{6}{\Delta} t^{3}{\int}_{t_{n-1}}^{t_{n+1}}\|{ u}_{ttt}\|^{2}\mathrm{d} t+Pr \|\nabla(\theta{ u}_{n+1}+(1-\theta){ u}_{n}-{ u}_{n+\theta})\|^{2}\\ &&+{Pr}^{-1}\epsilon^{2}\theta^{2}\|\nabla ({ u}_{n+1}-2{ u}_{n}+{ u}_{n-1})\|^{2}+\|{D}_{n+\theta}(\boldsymbol{\eta}_{{ u}})\|^{2}+Pr\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\eta}_{u})\|^{2}\\ &&+\|{H}_{n+\theta}(\nabla\boldsymbol{\eta}_{{ u}})\|^{2}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla{ u})\|^{2}+\|{H}_{n+\theta}(\nabla{ u}^{h})\|^{2}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla\boldsymbol{\eta}_{{ u}})\|^{2}\\ &&+\|F_{n+\theta}^{\varepsilon,\nu}({ u})-{ u}(t_{n+\theta})\|^{2}+\theta^{2}(1+\theta^{2}){\Delta} t^{3}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla{ u})\|^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla{ u}_{tt}\|^{2} \mathrm{d} t\\ &&+(\theta^{2}(1-\theta^{2})+\epsilon^{2}\theta^{2}){\Delta} t^{3}\|\nabla { u}_{n+\theta}\|^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla{ u}_{tt}\|^{2} \mathrm{d} t+\|{F}_{n+\theta}^{\varepsilon,\nu}(p-p^{h})\|^{2}\\ &&+Pr\theta^{2}(1+\theta)^{2}{\Delta} t^{3}\left( Ra^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla T_{tt}\|^{2} \mathrm{d} t+N^{2}Ra^{2}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla S_{tt}\|^{2} \mathrm{d} t\right.\\ &&\left.\left.+Ha^{4}{\int}_{t_{n-1}}^{t_{n+1}}\|\nabla { u}_{tt}\|^{2} \mathrm{d} t\right)\right)+C\nu^{-1}{\Delta} t\left( \|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})\|^{2}_{\infty}+\|{F}_{n+\theta}^{\varepsilon,\nu}({ u})\|^{2}_{\infty}\right)\\ &&\times((\theta+1)^{2}\|\boldsymbol{\phi}_{{ u},n}^{h}\|^{2}+\theta^{2}\|\boldsymbol{\phi}_{{ u},n-1}^{h}\|^{2})+C{\Delta} t(\|H_{n+\theta}({\phi_{T}^{h}})\|^{2}+\|NH_{n+\theta}({\phi_{S}^{h}})\|^{2}). \end{array} $$
(A.26)
Using (2.11) and (2.17), summing over the time steps from n = 0 to n = M − 1 and utilizing induction, one gets:
$$ \begin{array}{@{}rcl@{}} &&{}\left\|\boldsymbol{\phi}_{{ u},{M}}^{h}\right\|^{2}+ \frac{1}{2\theta+1}\sum\limits_{n=0}^{M-1}\left\|\boldsymbol{\phi}_{{ u},{n+1}}^{h}-2\boldsymbol{\phi}_{{ u},{n}}^{h}+\boldsymbol{\phi}_{{ u},{n-1}}^{h}\right\|_{F}^{2}\\&& +\frac{2Pr{\Delta} t}{2\theta+1} \sum\limits_{n=0}^{M-1}\|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla \boldsymbol{\phi}_{{ u}}^{h})\|^{2}+\frac{2Da^{-1}{\Delta} t}{2\theta+1}\sum\limits_{n=0}^{M-1}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{u}^{h})\|^{2} \\ &\leq&\left( \frac{2\theta-1}{2\theta+1}\right)^{M}\left\|\boldsymbol{\phi}_{{ u},{0}}^{h}\right\|^{2}+2\left\| \begin{bmatrix} \boldsymbol{\phi}_{{ u},{0}}^{h}\quad\\ \boldsymbol{\phi}_{{ u},{-1}}^{h} \end{bmatrix}\right\|_{G}^{2}+C(h^{2k}+{\Delta} t^{4})\\ &&+2Pr^{-1}{\Delta} t\sum\limits_{n=0}^{M-1}\left( \|{F}_{n+\theta}^{\varepsilon,\nu}(\nabla { u})\|^{2}_{\infty}+\|{F}_{n+\theta}^{\varepsilon,\nu}({ u})\|^{2}_{\infty}\right)((\theta+1)^{2}\|\boldsymbol{\phi}_{{ u},n}^{h}\|^{2}+\theta^{2}\|\boldsymbol{\phi}_{{ u},n-1}^{h}\|^{2}) \\&&+PrHa^{2}{\Delta} t\sum\limits_{n=0}^{M-1}\|F_{n+\theta}^{\varepsilon,\nu}(\boldsymbol{\phi}_{{ u}}^{h})\|^{2}+2{\Delta} t\sum\limits_{n=0}^{M-1}(\|H_{n+\theta}({\phi_{T}^{h}})\|^{2}+2\|NH_{n+\theta}({\phi_{S}^{h}})\|^{2}). \end{array} $$
(A.27)
In a similar manner, setting \(\chi ^{h}={F}_{n+\theta }^{\varepsilon _{1},\gamma }({\phi _{T}^{h}})\) in (A.9) and \({\Phi }^{h}={F}_{n+\theta }^{\varepsilon _{2},Le^{-1}}({\phi _{S}^{h}})\) in (A.10) and using similar arguments of (A.27) give the error estimation for the temperature and the concentration as:
$$ \begin{array}{@{}rcl@{}} &&{}\|{\phi}_{T,{M}}^{h}\|^{2} +\frac{1}{2\theta+1}\sum\limits_{n=0}^{M-1}\left\|\phi_{T,{n+1}}^{h}-2\phi_{T,{n}}^{h}+\phi_{T,{n-1}}^{h}\right\|_{F}^{2}+\frac{2{\Delta} t}{2\theta+1}\sum\limits_{n=0}^{M-1}\|{F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla {\phi_{T}^{h}})\|^{2}\\&\leq&\left( \frac{2\theta-1}{2\theta+1}\right)^{M}\left\|{{\phi}_{T,{0}}^{h}}\right\|^{2}+2\left\|{ \begin{bmatrix} {\phi}_{T,{0}}^{h}\quad\\ {\phi}_{T,{-1}}^{h} \end{bmatrix}}\right\|_{G}^{2}+C(h^{2k}+{\Delta} t^{4})\\&&+C{\Delta} t\sum\limits_{n=0}^{M-1}\left( \|{F}_{n+\theta}^{\varepsilon_{1},\gamma}(\nabla T)\|^{2}_{\infty}+\|{F}_{n+\theta}^{\varepsilon_{1},\gamma}(T)\|^{2}_{\infty}\right)((\theta+1)^{2}\|{\phi}_{u,{n}}^{h}\|^{2}+\theta^{2}\|{\phi}_{u,{n-1}}^{h}\|^{2}), \end{array} $$
(A.28)
$$ \begin{array}{@{}rcl@{}} &&{}\|{\phi}_{S,{M}}^{h}\|^{2} +\frac{1}{2\theta+1}\sum\limits_{n=0}^{M-1}\left\|\phi_{S,{n+1}}^{h}-2\phi_{S,{n}}^{h}+\phi_{S,{n-1}}^{h}\right\|_{F}^{2}+\frac{2Le^{-1}{\Delta} t}{2\theta+1}\sum\limits_{n=0}^{M-1}\|{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla {\phi_{S}^{h}})\|^{2}\\&\leq&\left( \frac{2\theta-1}{2\theta+1}\right)^{M}\left\|{{\phi}_{S,{0}}^{h}}\right\|^{2}+2\left\| { \begin{bmatrix} {\phi}_{S,{0}}^{h}\quad\\ {\phi}_{S,{-1}}^{h} \end{bmatrix}}\right\|_{G}^{2}+C(h^{2k}+{\Delta} t^{4})\\&&+C{\Delta} t\sum\limits_{n=0}^{M-1}\left( \|{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(\nabla S)\|^{2}_{\infty}+\|{F}_{n+\theta}^{\varepsilon_{2},Le^{-1}}(S)\|^{2}_{\infty}\right)((\theta+1)^{2}\|{\phi}_{u,{n}}^{h}\|^{2}\\&&+\theta^{2}\|{\phi}_{u,{n-1}}^{h}\|^{2}). \end{array} $$
(A.29)
Summing (A.27), (A.28) and (A.29), applying discrete Gronwall inequality (2.19) and triangle inequality produce the stated result. □
