APPENDIX A
LDG DISCRETIZATIONS FOR \(u_{h}^{1},u_{h}^{2}\)
In order to better introduce the third order Runge–Kutta method, we set
$${{F}^{ + }}(\boldsymbol{\phi} ,{v}) = - (\boldsymbol{\phi} ,\nabla {v}) + \sum\limits_K {{({{\boldsymbol{\phi} }^{R}} \cdot \boldsymbol{\nu} ,{v})}_{{\partial K}}},$$
$$ {{F}^{ - }}({v},\boldsymbol{\phi} ) = - ({v},\nabla \cdot \boldsymbol{\phi} ) + \sum\limits_K {{({{{v}}^{L}},\boldsymbol{\nu} \cdot \boldsymbol{\phi} )}_{{\partial K}}}.$$
Assume that we know the numerical solution at time level n, the solution at time level n + 1 is obtained by the following equations [23].
$$\left( {\frac{{u_{h}^{{n + 1,1}} - u_{h}^{n}}}{\tau },{v}} \right) = \gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right),$$
(A.1a)
$$\left( {\frac{{u_{h}^{{n + 1,2}} - u_{h}^{n}}}{\tau },{v}} \right) = \frac{{1 - \gamma }}{2}\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right)$$
$$ + \,\gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,2}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,2}}),{v}} \right)} \right),$$
(A.1b)
$$\left( {\frac{{u_{h}^{{n + 1}} - u_{h}^{n}}}{\tau },{v}} \right) = (1 - {{b}_{2}} - \gamma )\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right)$$
$$ + \,{{b}_{2}}\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,2}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,2}}),{v}} \right)} \right)$$
$$ + \,\gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1}}),{v}} \right)} \right),$$
(A.1c)
$$({\mathbf{q}}_{h}^{{n + 1,i}},{\mathbf{p}}) = {{F}^{ - }}(u_{h}^{{n + 1,i}},{\mathbf{p}}),\quad i = 1,2,$$
(A.1d)
$$({\mathbf{q}}_{h}^{{n + 1}},{\mathbf{p}}) = {{F}^{ - }}(u_{h}^{{n + 1}},{\mathbf{p}}),$$
(A.1e)
where \(\gamma = 0.435866521508,\;{{b}_{2}} = 0.25(5 - 20\gamma + 6{{\gamma }^{2}})\).
APPENDIX B
THE PROOF OF LEMMA 4
1.1 B.1. ERRORS EQUATIONS FOR \(u_{h}^{1},u_{h}^{2}\)
We give some notations to better present the error equations.
$${{E}_{1}}{{u}^{{n + 1}}} = \frac{{{{u}^{{n + 1,1}}} - {{u}^{n}}}}{\tau },\quad {{E}_{2}}{{u}^{{n + 1}}} = \frac{{{{u}^{{n + 1,2}}} - 2{{u}^{{n + 1,1}}} + {{u}^{n}}}}{\tau },\quad {{E}_{3}}{{u}^{{n + 1}}} = \frac{{2{{u}^{{n + 1}}} + {{u}^{{n + 1,2}}} - 3{{u}^{{n,1}}}}}{\tau }.$$
Motivated by the analysis in [25], we take the simple variations of –2*(1) + (2) and ‒3*(1) + (2) + 2*(4), the first three equations in (A.1) can be rewritten as follows:
$$\left( {{{E}_{1}}u_{h}^{{n + 1}},{v}} \right) = \gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right),$$
(B.1a)
$$\left( {{{E}_{2}}u_{h}^{{n + 1}},{v}} \right) = \frac{{1 - \gamma }}{2}\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right)$$
$$ + \,\gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,2}} - 2{\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,2}}) - 2f(u_{h}^{{n + 1,1}}),{v}} \right)} \right),$$
(B.1b)
$$\left( {{{E}_{3}}u_{h}^{{n + 1}},{v}} \right) = \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,1}}),{v}} \right)} \right)$$
$$ + \,(2{{b}_{2}} + \gamma )\left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1,2}} - 2{\mathbf{q}}_{h}^{{n + 1,1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1,2}}) - 2f(u_{h}^{{n + 1,1}}),{v}} \right)} \right)$$
$$ + \,2\gamma \left( {{{F}^{ + }}({\mathbf{q}}_{h}^{{n + 1}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f(u_{h}^{{n + 1}}),{v}} \right)} \right).$$
(B.1c)
For the error analysis, we write Eq. (1) in a similar form as in (A.1d)–(A.1e) and (B.1), then we have
$$\left( {{{E}_{1}}{{u}^{{n + 1}}},{v}} \right) = \gamma \left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1,1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}),{v}} \right)} \right),$$
(B.2a)
$$\left( {{{E}_{2}}{{u}^{{n + 1}}},{v}} \right) = \frac{{1 - \gamma }}{2}\left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1,1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}),{v}} \right)} \right)$$
$$\, + \gamma \left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1,2}}} - 2{{{\mathbf{q}}}^{{n + 1,1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,2}}}) - 2f({{u}^{{n + 1,1}}}),{v}} \right)} \right),$$
(B.2b)
$$\left( {{{E}_{3}}{{u}^{{n + 1}}},{v}} \right) = \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1,1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}),{v}} \right)} \right)$$
$$ + \,(2{{b}_{2}} + \gamma )\left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1,2}}} - 2{{{\mathbf{q}}}^{{n + 1,1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,2}}}) - 2f({{u}^{{n + 1,1}}}),{v}} \right)} \right)$$
$$ + \,2\gamma \left( {{{F}^{ + }}({{{\mathbf{q}}}^{{n + 1}}};{v}) - \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1}}}),{v}} \right)} \right) + 2({{\tilde {\iota }}^{{n + 1}}},{v}),$$
(B.2c)
$$({{{\mathbf{q}}}^{{n + 1,i}}},{\mathbf{p}}) = {{F}^{ - }}({{u}^{{n + 1,i}}},{\mathbf{p}}),\quad i = 1,2,$$
(B.2d)
$$({{{\mathbf{q}}}^{{n + 1}}},{\mathbf{p}}) = {{F}^{ - }}({{u}^{{n + 1}}},{\mathbf{p}}),$$
(B.2e)
where \({\text{||}}{{\tilde {\iota }}^{{n + 1}}}{\text{||}} \leqslant C{{\tau }^{3}}\) is the local truncation error for the third order Runge–Kutta method [23, 25].
Subtracting (A.1d)–(A.1e), (B.1) from (B.1), we have the error equations
$$\left( {{{E}_{1}}e_{u}^{{n + 1}},{v}} \right) = \gamma {{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1,1}};{v}) - \frac{\gamma }{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),{v}} \right),$$
(B.3a)
$$\left( {{{E}_{2}}e_{u}^{{n + 1}},{v}} \right) = \frac{{1 - \gamma }}{2}{{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1,1}};{v}) + \gamma {{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1,2}} - 2e_{{\mathbf{q}}}^{{n + 1,1}};{v})$$
$$ - \frac{{1 - \gamma }}{{2{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),{v}} \right)$$
(B.3b)
$$ - \frac{\gamma }{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,2}}}) - 2f({{u}^{{n + 1,1}}}) - (f(u_{h}^{{n + 1,2}}) - 2f(u_{h}^{{n + 1,1}})),{v}} \right),$$
$$\left( {{{E}_{3}}e_{u}^{{n + 1}},{v}} \right) = \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right){{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1,1}};{v}) + (2{{b}_{2}} + \gamma ){{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1,2}} - 2e_{{\mathbf{q}}}^{{n + 1,1}};{v})$$
$$ + \;2\gamma {{F}^{ + }}(e_{{\mathbf{q}}}^{{n + 1}};{v}) - \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),{v}} \right)$$
$$ - \;\frac{{(2{{b}_{2}} + \gamma )}}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,2}}}) - f(u_{h}^{{n + 1,2}}) - (2f({{u}^{{n + 1,1}}}) - 2f(u_{h}^{{n + 1,1}})),{v}} \right)$$
$$ - \;\frac{{2\gamma }}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1}}}) - f(u_{h}^{{n + 1}}),{v}} \right) + 2({{\tilde {\iota }}^{{n + 1}}},{v}),$$
(B.3c)
$$(e_{{\mathbf{q}}}^{{n + 1,i}},{\mathbf{p}}) = {{F}^{ - }}(e_{u}^{{n + 1,i}},{\mathbf{p}}),\quad i = 1,2,$$
(B.3d)
$$(e_{{\mathbf{q}}}^{{n + 1}},{\mathbf{p}}) = {{F}^{ - }}(e_{u}^{{n + 1}},{\mathbf{p}}).$$
(B.3e)
1.2 B.2. ERROR ESTIMATES FOR \(u_{h}^{1},u_{h}^{2}\)
Let \({v} = P{{e}_{{{{u}^{{n + 1,1}}}}}},P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}},P{{e}_{{{{u}^{{n + 1}}}}}}\) in error Eqs. (B.3a)–(B.3c) respectively, with (31)–(32) and the property of \(\Pi \) in Subsection 3.3, we have that
$$\frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}} - P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} - \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ + \;\frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}} + P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} - \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ + \;{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}} + P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}} - P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ - \;\frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}: = \sum\limits_{i = 1}^4 {{\mathcal{G}}_{i}},$$
(B.4)
where
$${{\mathcal{G}}_{1}} = - \tau \left( {{{E}_{1}}{{u}^{{n + 1}}} - {{E}_{1}}P{{u}^{{n + 1}}},P{{e}_{{{{u}^{{n + 1,1}}}}}}} \right) - \tau \left( {{{E}_{2}}{{u}^{{n + 1}}} - {{E}_{2}}P{{u}^{{n + 1}}},P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}} \right)$$
$$ - \,\tau \left( {{{E}_{3}}{{u}^{{n + 1}}} - {{E}_{3}}P{{u}^{{n + 1}}},P{{e}_{{{{u}^{{n + 1}}}}}}} \right) + 2\tau ({{\tilde {\iota }}^{{n + 1}}},P{{e}_{{{{u}^{{n + 1}}}}}}),$$
$${{\mathcal{G}}_{2}} = \gamma \tau {{F}^{ - }}({{u}^{{n + 1,1}}} - P{{u}^{{n + 1,1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}) + \frac{{1 - \gamma }}{2}\tau {{F}^{ - }}({{u}^{{n + 1,2}}} - P{{u}^{{n + 1,2}}} - 2({{u}^{{n + 1,1}}} - P{{u}^{{n + 1,1}}}),\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}})$$
$$ + \,\gamma \tau {{F}^{ - }}({{u}^{{n + 1,2}}} - P{{u}^{{n + 1,2}}} - 2({{u}^{{n + 1,1}}} - P{{u}^{{n + 1,1}}}),\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}})$$
$$ + \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\tau {{F}^{ - }}({{u}^{{n + 1}}} - P{{u}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}) + \gamma \tau {{F}^{ - }}({{u}^{{n + 1}}} - P{{u}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}})$$
$$ + \,(2{{b}_{2}} + \gamma )\tau {{F}^{ - }}({{u}^{{n + 1}}} - P{{u}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}),$$
$${{\mathcal{G}}_{3}} = - \frac{\gamma }{{{{\varepsilon }^{2}}}}\tau (f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),P{{e}_{{{{u}^{{n + 1,1}}}}}}) - \frac{{1 - \gamma }}{{2{{\varepsilon }^{2}}}}\tau \left( {f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}} \right),$$
$$ - \frac{\gamma }{{{{\varepsilon }^{2}}}}\tau \left( {f({{u}^{{n + 1,2}}}) - 2f({{u}^{{n + 1,1}}}) - (f(u_{h}^{{n + 1,2}}) - 2f(u_{h}^{{n + 1,1}})),P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}} \right)$$
$$ - \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\tau \frac{1}{{{{\varepsilon }^{2}}}}\left( {f({{u}^{{n + 1,1}}}) - f(u_{h}^{{n + 1,1}}),P{{e}_{{{{u}^{{n + 1}}}}}}} \right) - \frac{{2\gamma }}{{{{\varepsilon }^{2}}}}\tau \left( {f({{u}^{{n + 1}}}) - f(u_{h}^{{n + 1}}),P{{e}_{{{{u}^{{n + 1}}}}}}} \right)$$
$$ - \frac{{(2{{b}_{2}} + \gamma )}}{{{{\varepsilon }^{2}}}}\tau \left( {f({{u}^{{n + 1,2}}}) - f(u_{h}^{{n + 1,2}}) - (2f({{u}^{{n + 1,1}}}) - 2f(u_{h}^{{n + 1,1}})),P{{e}_{{{{u}^{{n + 1}}}}}}} \right),$$
$${{\mathcal{G}}_{4}} = - \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ - \frac{{1 - \gamma }}{2}\tau (\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}) - \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\tau (\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}})$$
$$\, - (2{{b}_{2}} + \gamma )\tau (\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}),$$
$${{\mathcal{G}}_{5}} = - \gamma \tau ({{{\mathbf{q}}}^{{n + 1,1}}} - \Pi {{{\mathbf{q}}}^{{n + 1,1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}) - \frac{{1 - \gamma }}{2}\tau ({{{\mathbf{q}}}^{{n + 1,2}}} - \Pi {{{\mathbf{q}}}^{{n + 1,2}}} - 2({{{\mathbf{q}}}^{{n + 1,1}}} - \Pi {{{\mathbf{q}}}^{{n + 1,1}}}),\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}})$$
$$ + \,\gamma \tau ({{{\mathbf{q}}}^{{n + 1,2}}} - \Pi {{{\mathbf{q}}}^{{n + 1,2}}} - 2({{{\mathbf{q}}}^{{n + 1,1}}} - \Pi {{{\mathbf{q}}}^{{n + 1,1}}}),\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}})$$
$$ - \left( {\frac{5}{2} + 2{{b}_{2}} - \frac{7}{2}\gamma } \right)\tau {{F}^{ - }}({{{\mathbf{q}}}^{{n + 1}}} - \Pi {{{\mathbf{q}}}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}) - \gamma \tau {{F}^{ - }}({{{\mathbf{q}}}^{{n + 1}}} - \Pi {{{\mathbf{q}}}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}})$$
$$ - \,(2{{b}_{2}} + \gamma )\tau {{F}^{ - }}({{{\mathbf{q}}}^{{n + 1}}} - \Pi {{{\mathbf{q}}}^{{n + 1}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}).$$
From the property of P in Subsection 3.3, we can see that
$${\text{|}}{{\mathcal{G}}_{1}}{\text{|}} \leqslant C\tau ({{h}^{{2k + 2}}} + {{\tau }^{6}}) + \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}).$$
And in one-dimension, \({{\mathcal{G}}_{2}} = 0\). In multi-dimension
$${\text{|}}{{\mathcal{G}}_{2}}{\text{|}} \leqslant C{{h}^{{2k + 2}}}\tau + \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}})$$
with Lemma 3.
Using (19), (20) and the Lipschitz continuity property (2) of f yields
$${\text{|}}{{\mathcal{G}}_{3}} + {{\mathcal{G}}_{5}}{\text{|}} \leqslant \widetilde C\tau ({{h}^{{2k + 2}}} + {{\tau }^{6}}) + \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}})$$
$$ + \,\widetilde C\tau ({\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}})$$
$$ \leqslant \widetilde C\tau ({{h}^{{2k + 2}}} + {{\tau }^{6}}) + \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}}\, + {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}})$$
$$ + \widetilde {\,C}\tau ({\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}} - P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}} + P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}),$$
where \(\widetilde C\) depends on \(\varepsilon \) and \(\epsilon \) is a small enough positive constant generated by Young’s inequality.
Inserting the estimates of \({{\mathcal{G}}_{i}},i = 1,2,3,5\) and the expression of \({{\mathcal{G}}_{4}}\) into (B.4), we have that
$${\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - \;{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}} - P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}} + P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ + \;\frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}} + P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \frac{1}{2}{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}} - P{{e}_{{{{u}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - {{\mathcal{G}}_{4}}$$
$$ \leqslant \widetilde C\tau ({{h}^{{2k + 2}}} + {{\tau }^{6}}) + \widetilde C\tau ({\text{||}}P{{e}_{{{{u}^{{n + 1,1}}}}}} - P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{{n + 1,2}}}}}} - 2P{{e}_{{{{u}^{{n + 1,1}}}}}} + P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}$$
$$ + \;{\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}\; + \;{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}) + \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \;{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \;{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}).$$
(B.5)
We set \({\mathbf{X}} = (\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}},\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}})\), and \( - {{\mathcal{G}}_{4}} = \int_\Omega {\mathbf{X}}M{{{\mathbf{X}}}^{T}}dx\) with
$$M = \tau \left( {\begin{array}{*{20}{c}} \gamma &{\frac{{1 - \gamma }}{4}}&{\frac{5}{4} + {{b}_{2}} - \frac{7}{4}\gamma } \\ {\frac{{1 - \gamma }}{4}}&\gamma &{{{b}_{2}} + \frac{\gamma }{2}} \\ {\frac{5}{4} + {{b}_{2}} - \frac{7}{4}}&{{{b}_{2}} + \frac{\gamma }{2}}&\gamma \end{array}} \right).$$
(B.6)
With the values of \(\gamma \) and \({{b}_{2}}\) in (A.1), we can check that M is positive definite, which indicates \( - {{\mathcal{G}}_{4}} > 0\) and
$$ - {{\mathcal{G}}_{4}} - \epsilon \tau ({\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \;{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,2}}}}}} - 2\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1,1}}}}}}{\text{|}}{{{\text{|}}}^{2}}\; + \;{\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}}) \geqslant 0,$$
with \(\epsilon \) a small enough positive constant.
The from (B.5), we have that
$${\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} - \,{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}} + \gamma \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} \leqslant \widetilde C\tau ({{h}^{{2k + 2}}} + {{\tau }^{6}}) + \widetilde C\tau ({\text{||}}P{{e}_{{{{u}^{{n + 1}}}}}}{\text{|}}{{{\text{|}}}^{2}} + \,{\text{||}}P{{e}_{{{{u}^{n}}}}}{\text{|}}{{{\text{|}}}^{2}}).$$
(B.7)
If \(\tau \leqslant {{\tau }_{0}}\), of which depends on \(\varepsilon \) but is independent of h, with (B.7) and the initial condition of \(u_{h}^{0}\) in (26), we can get
$${\text{||}}P{{e}_{{{{u}^{1}}}}}{\text{||}} + \,{\text{||}}P{{e}_{{{{u}^{2}}}}}{\text{||}} + \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{2}}}}}{\text{||}} + \tau {\text{||}}\Pi {{e}_{{{{{\mathbf{q}}}^{1}}}}}{\text{||}} \leqslant \widetilde C{{h}^{{k + 1}}} + \widetilde C{{\tau }^{3}},$$
(B.8)
which indicates the result (28) in Lemma 4.
