Appendix A: Proof Of Theorem 4.1
Set \((v,\psi ,\phi ,\eta )=(u_{h}^{n},p_{h}^{n}-q_{h}^{n}, u_{h}^{n},r_{h}^{n})\) in (3.12), and define \(\theta (u_{h}^{n})=\int ^{u_{h}^{n}}f(s_{h}^{n})\mathrm{d}s_{h}^{n}\). Then the following result holds:
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}u_{h}^{n},u_{h}^{n}\right) _{D^{k}} -\varepsilon \big (p_{h}^{n}, u_{h}^{n}\big )_{D^{k}}\nonumber \\&\quad \quad -\,(\theta (u_{h}^{n}))^{-}_{k+\frac{1}{2}} +(\theta (u_{h}^{n}))^{+}_{k-\frac{1}{2}}+ \big ((\widehat{f}(u_{h}^{n})u^{-})_{k+\frac{1}{2}}\nonumber \\&\quad \quad -\,(\widehat{f}(u_{h}^{n}) (u_{h}^{n})^{+})_{k-\frac{1}{2}}\big )-\big (p_{h}^{n},q_{h}^{n}\big )_{D^{k}} +\big (p_{h}^{n},p_{h}^{n}\big )_{D^{k}}\nonumber \\&\quad \quad +\,\big (\Delta _{(\beta -2)/2}q^{n}_{h},q_{h}^{n}\big )_{D^{k}} -\big (\Delta _{(\beta -2)/2}q^{n}_{h},p_{h}^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\big (r_{h}^{n},r_{h}^{n}\big )_{D^{k}} +\big (u_{h}^{n},\frac{\partial r_{h}^{n}}{\partial x}\big )_{D^{k}}+\big (q_{h}^{n},u_{h}^{n}\big )_{D^{k}}\nonumber \\&\quad \quad +\,\big (r_{h}^{n},\frac{\partial u_{h}^{n}}{\partial x}\big )_{D^{k}} -\big ((\widehat{u}^{n}_{h}(r_{h}^{n})^{-})_{k+\frac{1}{2}} -(\widehat{u}^{n}_{h}(r_{h}^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad \quad -\,\big ((\widehat{r}^{n}_{h}(u_{h}^{n})^{-})_{k+\frac{1}{2}} -(\widehat{r}^{n}_{h}(u_{h}^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0. \end{aligned}$$
(A.1)
Summing over k, with the numerical fluxes (3.13), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}u_{h}^{n},u_{h}^{n}\right) -\varepsilon \big (p_{h}^{n}, u_{h}^{n}\big )\nonumber \\&-\sum _{k=1}^{K}\big ((\theta (u_{h}^{n}))^{-}_{k+\frac{1}{2}} -(\theta (u_{h}^{n}))^{+}_{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big ((\widehat{f}(u_{h}^{n})u^{-})_ {k+\frac{1}{2}}-(\widehat{f}(u_{h}^{n}) (u_{h}^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad -\,\big (p_{h}^{n},q_{h}^{n}\big ) +\big (p_{h}^{n},p_{h}^{n}\big )\nonumber \\&\quad +\,\big (\Delta _{(\beta -2)/2}q^{n}_{h},q_{h}^{n}\big ) -\big (\Delta _{(\beta -2)/2}q^{n}_{h},p_{h}^{n}\big )\nonumber \\&\quad +\,\big (r_{h}^{n},r_{h}^{n}\big )+\big (q_{h}^{n},u_{h}^{n}\big ) =0. \end{aligned}$$
(A.2)
Here \(\widehat{f}((u_{h}^{n})^{-},(u_{h}^{n})^{+})\) is monotone flux, we have
$$\begin{aligned}&\sum _{k=1}^{K}\left( (\widehat{f}(u_{h}^{n})u^{-})_{k+\frac{1}{2}} -(\widehat{f}(u_{h}^{n})(u_{h}^{n})^{+})_{k-\frac{1}{2}}\right) \nonumber \\&\quad -\, \sum _{k=1}^{K}\left( (\theta (u_{h}^{n}))^{-}_{k+\frac{1}{2}} -(\theta (u_{h}^{n}))^{+}_{k-\frac{1}{2}}\right) >0. \end{aligned}$$
(A.3)
This implies that
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}u_{h}^{n},u_{h}^{n}\right) +\big (r_{h}^{n},r_{h}^{n}\big )\nonumber \\&\quad +\,\big (p_{h}^{n},p_{h}^{n}\big )+\big (\Delta _{(\beta -2)/2} q^{n}_{h},q_{h}^{n}\big )\nonumber \\&\le \big (\Delta _{(\beta -2)/2}q^{n}_{h},p_{h}^{n}\big ) -\big (q_{h}^{n},u_{h}^{n}\big )\nonumber \\&\quad +\,\big (p_{h}^{n},q_{h}^{n}\big ) +\varepsilon \big (p_{h}^{n}, u_{h}^{n}\big ). \end{aligned}$$
(A.4)
Employing Young’s inequality and Lemma 4, we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} u_{h}^{n},u_{h}^{n}\right) +\Vert r_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&+\big (\Delta _{(\beta -2)/2}q^{n}_{h},q_{h}^{n}\big )\le c\Vert u_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}+c_{1}\Vert q_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}.\nonumber \\ \end{aligned}$$
(A.5)
Recalling Lemma 4, we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}u_{h}^{n},u_{h}^{n}\right) +\Vert r_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le c\Vert u_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}. \end{aligned}$$
(A.6)
It then follows that
$$\begin{aligned}&\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}u^{n},u^{n}_{h}\right) \nonumber \\&\quad \le \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})u^{l}_{h},u^{n}_{h}\right) \nonumber \\&\qquad +\, \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}} a_{n-1}^{\alpha _{j}}u^{0}_{h},u^{n}_{h}\right) + c\Vert u_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}.\nonumber \\ \end{aligned}$$
(A.7)
Using Cauchy–Schwarz inequality, we obtain
$$\begin{aligned}&\Vert u_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le c_{1}\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert u_{h}^{l}\Vert _{L^{2}(\varOmega )}\nonumber \\&\qquad \Vert u_{h}^{n}\Vert _{L^{2}(\varOmega )}+c_{2} \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\Vert u_{h}^{0}\Vert _{L^{2}(\varOmega )}\nonumber \\&\qquad \Vert u_{h}^{n}\Vert _{L^{2}(\varOmega )}+cQ\Vert u_{h}^{n}\Vert ^{2}_{L^{2}(\varOmega )}, \end{aligned}$$
(A.8)
where \(Q=\frac{1}{\sum _{j=1}^{S}\frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}}\) and provided c is sufficiently small such that \(1-cQ>0\), we obtain that
$$\begin{aligned}&\Vert u_{h}^{n}\Vert _{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert u_{h}^{l}\Vert _{L^{2}(\varOmega )}\right. \nonumber \\&\qquad \left. +\, \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}} Qa_{n-1}^{\alpha _{j}}\Vert u_{h}^{0}\Vert _{L^{2}(\varOmega )}\right) . \end{aligned}$$
(A.9)
Clearly the theorem is true for \(n = 1\). It is also true for \(n = 1, 2, \ldots , m-1\). So, by (A.9), we have
$$\begin{aligned}&\Vert u_{h}^{m}\Vert _{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert u_{h}^{l}\Vert _{L^{2}(\varOmega )}\right. \nonumber \\&\qquad \left. +\, \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}} Qa_{n-1}^{\alpha _{j}}\Vert u_{h}^{0}\Vert _{L^{2}(\varOmega )}\right) \nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\ |u_{h}^{0}\Vert _{L^{2}(\varOmega )}\right. \nonumber \\&\qquad \left. +\, \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\Vert u_{h}^{0}\Vert _{L^{2}(\varOmega )}\right) \nonumber \\&\quad =C\Vert u_{h}^{0}\Vert _{L^{2}(\varOmega )}. \end{aligned}$$
(A.10)
\(\square \)
Appendix B: Proof Of Theorem 2
From (4.3), we can get the error equation
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}(u^{n}-u^{n}_{h}),v\right) _{D^{k}} +\big (\gamma (x)^{n},v\big )_{D^{k}} \nonumber \\&\quad -\,\varepsilon \big (p^{n}-p^{n}_{h},v\big )_{D^{k}} +\big (p^{n}-p^{n}_{h},\psi \big )_{D^{k}} \nonumber \\&\quad +\,\big (q^{n}-q^{n}_{h},\phi \big )_{D^{k}}-\big (\Delta _{ (\beta -2)/2}(q^{n}-q^{n}_{h}),\psi \big )_{D^{k}}\nonumber \\&\quad +\,\big (r^{n}-r^{n}_{h},\frac{\partial \phi }{\partial x}\big )_{D^{k}}-((\widehat{r}^{n}-\widehat{r}^{n}_{h}) \phi ^{-})_{k+\frac{1}{2}}\nonumber \\&\quad +\,((\widehat{r}^{n}-\widehat{r}^{n}_{h})\phi ^{+}) _{k-\frac{1}{2}}+\big (r^{n}-r^{n}_{h},\eta \big )_{D^{k}}\nonumber \\&\quad +\,\big (u^{n}-u^{n}_{h},\frac{\partial \eta }{\partial x}\big )_{D^{k}} -((\widehat{u}^{n}-\widehat{u}^{n}_{h})\eta ^{-})_{k+\frac{1}{2}}\nonumber \\&\quad +\,((\widehat{u}^{n}-\widehat{u}^{n}_{h})\eta ^{+})_{k-\frac{1}{2}}=0. \end{aligned}$$
(B.1)
Using (4.7), the error equation (B.1) can be written
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j}) \Delta \tau _{j}\delta _{t}^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}),v\right) _{D^{k}} +\big (\gamma (x)^{n},v\big )_{D^{k}} \nonumber \\&\quad -\,\varepsilon \big (\sigma ^{n}-\sigma ^{e}_{n},v\big )_{D^{k}} +\big (\sigma ^{n}-\sigma ^{e}_{n},\psi \big )_{D^{k}} \nonumber \\&\quad -\, \big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}),\psi \big )_{D^{k}} +\big (\varphi ^{n}-\varphi ^{e}_{n},\phi \big )_{D^{k}} \nonumber \\&\quad +\,\big (\psi ^{n}-\psi ^{e}_{n},\frac{\partial \phi }{\partial x}\big )_{D^{k}} -((\psi ^{n}-\psi ^{e}_{n})^{-}\phi ^{-})_{k+\frac{1}{2}}\nonumber \\&\quad +\,((\psi ^{n}-\psi ^{e}_{n})^{-}\phi ^{+})_{k-\frac{1}{2}} +\big (\psi ^{n}-\psi ^{e}_{n},\eta \big )_{D^{k}} \nonumber \\&\quad +\,\big (\pi ^{n}-\pi ^{e}_{n},\frac{\partial \eta }{\partial x}\big ) _{D^{k}}-((\pi ^{n}-\pi ^{e}_{n})^{+}\eta ^{-})_{k+\frac{1}{2}} \nonumber \\&\quad +\,((\pi ^{n}-\pi ^{e}_{n})^{+}\eta ^{+})_{k-\frac{1}{2}}=0, \end{aligned}$$
(B.2)
and taking the test functions
$$\begin{aligned} v=\pi ^{n},\quad \psi =\sigma ^{n}-\varphi ^{n},\quad \phi =\pi ^{n},\quad \eta =\psi ^{n}, \end{aligned}$$
(B.3)
we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}),\pi ^{n}\right) +\big (\gamma (x)^{n},\pi ^{n}\big )\nonumber \\&\quad -\,\varepsilon \big (\sigma ^{n}-\sigma ^{e}_{n},\pi ^{n}\big ) +\big (\sigma ^{n}-\sigma ^{e}_{n},-\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad -\, \big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), -\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad -\,\big (\pi ^{e}_{n},\frac{\partial \psi ^{n}}{\partial x}\big ) +\big (\varphi ^{n}-\varphi ^{e}_{n},\pi ^{n}\big )\nonumber \\&\quad -\,\big (\psi ^{e}_{n},\frac{\partial \pi ^{n}}{\partial x}\big ) +\big (\psi ^{n}-\psi ^{e}_{n},\psi ^{n}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\psi ^{e}_{n})^{-}(\pi ^{n}) ^{-})_{k+\frac{1}{2}}-((\psi ^{e}_{n})^{-}(\pi ^{n})^{+}) _{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\pi ^{e}_{n})^{+}(\psi ^{n})^{-})_ {k+\frac{1}{2}}-((\pi ^{e}_{n})^{+}(\psi ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0, \end{aligned}$$
(B.4)
by the projection properties \(P^{+}\) and \(P^{-}\) we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} (\pi ^{n}-\pi ^{e}_{n}),\pi ^{n}\right) -\varepsilon \big (\sigma ^{n}-\sigma ^{e}_{n},\pi ^{n}\big )\nonumber \\&\quad +\,\big (\gamma (x)^{n},\pi ^{n}\big )+\big (\sigma ^{n}- \sigma ^{e}_{n},-\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad -\, \big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), -\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad +\,\big (\varphi ^{n}-\varphi ^{e}_{n},\pi ^{n}\big ) +\sum _{k=1}^{K}\big (((\psi ^{e}_{n})^{-}(\pi ^{n})^{-})_{k+\frac{1}{2}}\nonumber \\&\quad -\,((\psi ^{e}_{n})^{-}(\pi ^{n})^{+})_{k-\frac{1}{2}}\big ) +\big (\psi ^{n}-\psi ^{e}_{n},\psi ^{n}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\pi ^{e}_{n})^{+}(\psi ^{n})^{-})_{k+\frac{1}{2}} -((\pi ^{e}_{n})^{+}(\psi ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0. \end{aligned}$$
(B.5)
Employing Young’s inequality and Lemma 4 and the property of interpolation (4.8) and (4.4), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} \pi ^{n},\pi ^{n}\right) +\big (\sigma ^{n},\sigma ^{n}\big )\nonumber \\&\quad +\, \big (\Delta _{(\beta -2)/2}\varphi ^{n},\varphi ^{n}\big ) +\big (\psi ^{n},\psi ^{n}\big )\nonumber \\&\le C(h^{2N+2}+(\Delta t)^{4+\theta }+\theta ^{4})+c_{1}\Vert \psi ^{n}\Vert ^{2}_{L^{2} (\varOmega )}\nonumber \\&\quad +\,\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}\pi ^{e}_{n} ,\pi ^{n}\right) +c_{2}\Vert \sigma ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad +\, c_{3}\Vert \varphi ^{n}\Vert ^{2}_{L^{2}(\varOmega )} +c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}, \end{aligned}$$
(B.6)
by using Lemma 5, (4.7) and the interpolation property (4.8), we obtain
$$\begin{aligned}&\Vert \delta _{t}^{\alpha }(\mathcal {P}^{+}u(x,t_{n}) -u(x,t_{n}))\Vert _{L^{2}(\varOmega )}\nonumber \\&\quad \le C\big (h^{N+1}+(\Delta t)^{2-\alpha }\big ). \end{aligned}$$
(B.7)
From (3.1), (4.5) and (B.7), we obtain
$$\begin{aligned}&\left\| \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j} \delta _{t}^{\alpha _{j}}(\mathcal {P}^{+}u(x,t_{n})-u(x,t_{n})) \right\| _{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2}\right) . \end{aligned}$$
(B.8)
Hence
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j} \delta _{t}^{\alpha _{j}}\pi ^{n},\pi ^{n}\right) +\big (\sigma ^{n},\sigma ^{n}\big )\nonumber \\&\quad +\, \big (\Delta _{(\beta -2)/2}\varphi ^{n},\varphi ^{n}\big ) +\big (\psi ^{n},\psi ^{n}\big )\nonumber \\&\le C\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big ) +c_{2}\Vert \sigma ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad +\, c_{3}\Vert \varphi ^{n}\Vert ^{2}_{L^{2}(\varOmega )} +c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )} +c_{1}\Vert \psi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}.\nonumber \\ \end{aligned}$$
(B.9)
Recalling Lemma 2 and provided \(c_{i},\,\,i=1,2\) are sufficiently small such that \(c_{i}\le 1\), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}\pi ^{n},\pi ^{n}\right) \nonumber \\&\le C\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big )+c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}. \end{aligned}$$
(B.10)
It then follows that
$$\begin{aligned}&\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}\pi ^{n},\pi ^{n}_{h}\right) \nonumber \\&\quad \le \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\pi ^{l},\pi ^{n}\right) \nonumber \\&\qquad +\, \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}a_{n-1} ^{\alpha _{j}}\pi ^{0},\pi ^{n}\right) + c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\,C\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big ). \end{aligned}$$
(B.11)
Employing Young’s inequality, we obtain
$$\begin{aligned}&\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l}\Vert ^{2}_{L^{2} (\varOmega )}\nonumber \\&\qquad +\,\frac{1}{4}\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q(a_{0} ^{\alpha _{j}}-a_{n-1}^{\alpha _{j}}) \Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\, \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}} \Vert \pi ^{0}\Vert ^{2}_{L^{2}(\varOmega )} +cQ\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\sum _{j=1}^{S}\frac{W(\alpha _{j})\Delta \tau _{j}}{4\lambda _{j}} Qa_{n-1}^{\alpha _{j}}\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\,CQ\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big ). \end{aligned}$$
(B.12)
Notice the facts that
$$\begin{aligned} \Vert \pi ^{0}\Vert _{L^{2}(\varOmega )}\le Ch^{N+1}. \end{aligned}$$
(B.13)
Thus,
$$\begin{aligned}&\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l}\Vert ^{2}_{L^{2} (\varOmega )}\nonumber \\&\qquad +\,\left( cQ+\frac{1}{4}\right) \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\Vert \pi ^{n} \Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\,C\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}} h^{2N+2}\nonumber \\&\qquad +\,C\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\left( h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\right) ,\nonumber \\ \end{aligned}$$
(B.14)
provided c is sufficiently small such that \(\frac{3}{4}-cQ>0\), we obtain that
$$\begin{aligned}&\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}}) \Vert \pi ^{l}\Vert ^{2}_{L^{2}(\varOmega )}\right. \nonumber \\&\qquad \left. +\,\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}} \big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big )\right) .\nonumber \\ \end{aligned}$$
(B.15)
Clearly the theorem is true for \(n = 0\). It is also true for \(n = 1, 2, \ldots , m-1\). Then, by (B.15), we have
$$\begin{aligned}&\Vert \pi ^{m}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l}\Vert ^{2}_{L^{2}(\varOmega )}\right. \nonumber \\&\qquad \left. +\,\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big )\right) \nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})(h^{2N+2}\right. \nonumber \\&\qquad +\,(\Delta t)^{2+\theta }+\theta ^{4})+\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}}\nonumber \\&\left. \qquad \big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big )\right) \nonumber \\&\quad = C\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big ). \end{aligned}$$
(B.16)
Finally, by using triangle inequality and standard approximation theory, we get
$$\begin{aligned} \Vert u(x,t_{m})-u_{h}^{m}\Vert _{L^{2}(\varOmega )}\le C(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2}).\nonumber \\ \end{aligned}$$
(B.17)
\(\square \)
Appendix C: Proof Of Theorem 3
Using (4.7), the error equation (4.3) can be written
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j} \delta _{t}^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}),v\right) -\varepsilon \big (\sigma ^{n}-\sigma ^{e}_{n},v\big )\nonumber \\&\quad +\,\big (\gamma (x)^{n},v\big )-\sum _{k=1}^{K} \mathcal {H}_{k}(f;u,u_{h};v)\nonumber \\&\quad +\,\big (\sigma ^{n}-\sigma ^{e}_{n},\psi \big )- \big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}),\psi \big )\nonumber \\&\quad +\,\big (\varphi ^{n}-\varphi ^{e}_{n},\phi \big )+\big (\psi ^{n} -\psi ^{e}_{n},\frac{\partial \phi }{\partial x}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\psi ^{n}-\psi ^{e}_{n})^{-}\phi ^{-}) _{k+\frac{1}{2}} -((\psi ^{n}-\psi ^{e}_{n})^{-}\phi ^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\big (\psi ^{n}-\psi ^{e}_{n},\eta \big )+\big (\pi ^{n}- \pi ^{e}_{n},\frac{\partial \eta }{\partial x}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\pi ^{n}-\pi ^{e}_{n})^{+}\eta ^{-}) _{k+\frac{1}{2}} -((\pi ^{n}-\pi ^{e}_{n})^{+}\eta ^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0.\nonumber \\ \end{aligned}$$
(C.1)
Following the proof of Theorem 2, we take the test functions
$$\begin{aligned} v=\pi ^{n},\quad \psi =-\varphi ^{n}+\sigma ^{n},\quad \phi =\pi ^{n},\quad \eta =\psi ^{n},\nonumber \\ \end{aligned}$$
(C.2)
we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _ {t}^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}) ,\pi ^{n}\right) +\big (\gamma (x)^{n},\pi ^{n}\big )\nonumber \\&\quad -\,\varepsilon \big (\sigma ^{n}-\sigma ^{e}_{n},\pi ^{n}\big ) -\sum _{k=1}^{K}\mathcal {H}_{k}(f;u,u_{h};\pi ^{n})\nonumber \\&\quad +\,\big (\sigma ^{n}-\sigma ^{e}_{n},-\varphi ^{n}+\sigma ^{n}\big ) +\big (\varphi ^{n}-\varphi ^{e}_{n},\pi ^{n}\big )\nonumber \\&\quad -\, \big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), -\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad +\,\big (\psi ^{n}-\psi ^{e}_{n},\frac{\partial \pi ^{n}}{\partial x}\big ) -\sum _{k=1}^{K}\big (((\psi ^{n}-\psi ^{e}_{n})^{-}(\pi ^{n})^{-}) _{k+\frac{1}{2}}\nonumber \\&\quad -\,((\psi ^{n}-\psi ^{e}_{n})^{-}(\pi ^{n})^{+}) _{k-\frac{1}{2}}\big )+\big (\psi ^{n}-\psi ^{e}_{n},\psi ^{n}\big )\nonumber \\&\quad +\,\big (\pi ^{n}-\pi ^{e}_{n},\frac{\partial \psi ^{n}}{\partial x}\big )-\sum _{k=1}^{K}\big (((\pi ^{n}-\pi ^{e}_{n})^{+}(\psi ^{n})^{-}) _{k+\frac{1}{2}}\nonumber \\&\quad -\,((\pi ^{n}-\pi ^{e}_{n})^{+}(\psi ^{n})^{+})_{k-\frac{1}{2}}\big )=0, \end{aligned}$$
(C.3)
by the projection properties \(P^{+}\) and \(P^{-}\), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} (\pi ^{n}-\pi ^{e}_{n}),\pi ^{n}\right) -\varepsilon \big (\sigma ^{n}- \sigma ^{e}_{n},\pi ^{n}\big )\nonumber \\&\quad +\,\big (\gamma (x)^{n},\pi ^{n}\big )-\sum _{k=1}^{K}\mathcal {H} _{k}(f;u,u_{h};\pi ^{n})\nonumber \\&\quad +\,\big (\sigma ^{n}-\sigma ^{e}_{n},-\varphi ^{n}+\sigma ^{n}\big ) +\big (\varphi ^{n}-\varphi ^{e}_{n},\pi ^{n}\big )\nonumber \\&\quad -\,\big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), -\varphi ^{n}+\sigma ^{n}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\psi ^{e}_{n})^{-}(\pi ^{n})^{-})_{k+\frac{1}{2}} -((\psi ^{e}_{n})^{-}(\pi ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\big (\psi ^{n}-\psi ^{e}_{n},\psi ^{n}\big ) +\sum _{k=1}^{K}\big (((\pi ^{e}_{n})^{+}(\psi ^{n})^{-})_{k+\frac{1}{2}}\nonumber \\&\quad -\,((\pi ^{e}_{n})^{+}(\psi ^{n})^{+})_{k-\frac{1}{2}}\big )=0. \end{aligned}$$
(C.4)
Employing Young’s inequality and Lemma 4 and the property of interpolation (4.8), (B.7) and (B.8), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} \pi ^{n},\pi ^{n}\right) + \big (\Delta _{(\beta -2)/2}\varphi ^{n},\varphi ^{n}\big )\nonumber \\&\quad +\,\big (\sigma ^{n},\sigma ^{n}\big )+\big (\psi ^{n},\psi ^{n}\big )\nonumber \\&\le C\big (h^{2N+2}+(\Delta t)^{2+\theta }+\theta ^{4}\big )+c_{2} \Vert \sigma ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad +\, c_{3}\Vert \varphi ^{n}\Vert ^{2}_{L^{2}(\varOmega )} +c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}+c_{1}\Vert \psi ^{n}\Vert ^{2}_{L^{2}(\varOmega )} \nonumber \\&\quad -\,\frac{1}{4}\kappa (f^{*};u_{h}^{n})+(C+C_{*}(\Vert \pi ^{n}\Vert _{\infty }\nonumber \\&\quad +\,h^{-1}\Vert e_{u}\Vert _{\infty }^{2}))\Vert \pi ^{n}\Vert ^{2}+(C+C_{*}h^{-1}\Vert e_{u}\Vert _{\infty }^{2})h^{2N+1}.\nonumber \\ \end{aligned}$$
(C.5)
Recalling Lemma 2 and provided \(c_{i},\,\,i=1,2\) are sufficiently small such that \(c_{i}\le 1\), we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} \pi ^{n},\pi ^{n}\right) \nonumber \\&\quad \le C\big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big ) +c_{4}\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}.\nonumber \\ \end{aligned}$$
(C.6)
It then follows that
$$\begin{aligned}&\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}\pi ^{n},\pi ^{n}_{h}\right) \nonumber \\&\quad \le \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\pi ^{l},\pi ^{n}\right) \nonumber \\&\qquad +\, \left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}a_{n-1} ^{\alpha _{j}}\pi ^{0},\pi ^{n}\right) + c\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\qquad +\,C\big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big ). \end{aligned}$$
(C.7)
Employing Young’s inequality, we obtain
$$\begin{aligned}&\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega _{h})}\nonumber \\&\quad \le \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l} \Vert ^{2}_{L^{2}(\varOmega _{h})}\nonumber \\&\qquad +\,\left( cQ+\frac{1}{4}\right) \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\Vert \pi ^{n}\Vert ^{2} _{L^{2}(\varOmega _{h})}\nonumber \\&\qquad +\,C\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}} h^{2N+2}\nonumber \\&\qquad +\,C\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big ),\nonumber \\ \end{aligned}$$
(C.8)
provided c is sufficiently small such that \(\frac{3}{4}-cQ>0\), we obtain that
$$\begin{aligned}&\Vert \pi ^{n}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{n-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l}\Vert ^{2}_{L^{2} (\varOmega )}\right. \nonumber \\&\qquad \left. +\,\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1}^{\alpha _{j}} \big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big )\right) .\nonumber \\ \end{aligned}$$
(C.9)
Clearly the theorem is true for \(n = 0\). It is also valid for \(n = 1, 2, \ldots , m-1\). Then, by (C.9), we have
$$\begin{aligned}&\Vert \pi ^{m}\Vert ^{2}_{L^{2}(\varOmega )}\nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\Vert \pi ^{l}\Vert ^{2}_{L^{2} (\varOmega )}\right. \nonumber \\&\qquad \left. +\,\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big )\right) \nonumber \\&\quad \le C\left( \sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Q\sum _{l=1}^{m-1} (a_{n-l-1}^{\alpha _{j}}-a_{n-l}^{\alpha _{j}})\big (h^{2N+1}\right. \nonumber \\&\qquad +\,(\Delta t)^{2+\theta }+\theta ^{4}\big )\nonumber \\&\qquad \left. +\,\sum _{j=1}^{S} \frac{W(\alpha _{j})\Delta \tau _{j}}{\lambda _{j}}Qa_{n-1} ^{\alpha _{j}}\big (h^{2N+1}+(\Delta t)^{4+\theta }+\theta ^{4}\big )\right) \nonumber \\&\quad = C\big (h^{2N+1}+(\Delta t)^{2+\theta }+\theta ^{4}\big ). \end{aligned}$$
(C.10)
Finally, by using triangle inequality and standard approximation theory, we can get (4.15). \(\square \)
Appendix D: Proof Of Lemma 8
From the Galerkin orthogonality (5.11), we get
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}),\vartheta _{1}\right) _{D^{k}} +\big (\tau ^{n}-\tau ^{e}_{n},\phi _{x}\big )_{D^{k}} \nonumber \\&\quad -\,\big (\Delta _{(\beta -2)/2}(\epsilon ^{n}-\epsilon ^{e}_{n}),\rho \big ) _{D^{k}}+\big (\vartheta ^{n}-\vartheta ^{e}_{n},\psi _{x}\big )_{D^{k}}\nonumber \\&\quad +\,\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t}^{\alpha _{j}} (\sigma -\sigma ^{e}),\chi \right) _{D^{k}} +\big (\sigma ^{n}-\sigma ^{e}_{n},\varphi _{x}\big )_{D^{k}}\nonumber \\&\quad -\,\big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), \varrho \big )_{D^{k}}+\big (\pi ^{n}-\pi ^{e}_{n},\zeta _{x}\big )_{D^{k}}\nonumber \\&\quad +\,\varepsilon _{2} \big (\sigma ^{n}-\sigma ^{e}_{n},\vartheta _{1} \big )_{D^{k}}- \varepsilon _{2}\big (\pi ^{n}-\pi ^{e}_{n},\chi \big )_{D^{k}}\nonumber \\&\quad +\,\big (\epsilon ^{n}-\epsilon ^{e}_{n},\phi \big )_{D^{k}} +\big (\tau ^{n}-\tau ^{e}_{n},\varphi \big )_{D^{k}} +\big (\varpi ^{n}-\varpi ^{e}_{n},\varrho \big )_{D^{k}}\nonumber \\&\quad +\,\big (\phi ^{n}-\phi ^{e}_{n},\rho \big )_{D^{k}}+\big (\varphi ^{n} -\varphi ^{e}_{n},\psi \big )_{D^{k}} +\big (\vartheta ^{n} -\vartheta ^{e}_{n},\zeta \big )_{D^{k}}\nonumber \\&\quad +\,\varepsilon _{1}\big (\phi ^{n} -\phi ^{e}_{n},\vartheta _{1}\big )_{D^{k}}-\varepsilon _{1} \big (\varpi ^{n}-\varpi ^{e}_{n},\chi \big )_{D^{k}}\nonumber \\&\quad -\,\big (\gamma (q)^{n},\chi \big )+\big (\gamma (p)^{n},\vartheta _{1}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\tau ^{n}-\tau ^{e}_{n})^{+}(\phi )^{-}) _{k+\frac{1}{2}}\ -((\tau ^{n}-\tau ^{e}_{n})^{+}(\phi )^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\sigma ^{n}-\sigma ^{e}_{n})^{-} (\varphi )^{-})_{k+\frac{1}{2}} -((\sigma ^{n}-\sigma ^{e}_{n})^{-}(\varphi )^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\vartheta ^{n}-\vartheta ^{e}_{n})^{+}(\psi )^{-}) _{k+\frac{1}{2}} -((\vartheta ^{n}-\vartheta ^{e}_{n})^{+}(\psi )^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K} \big ((\pi ^{n}-\pi ^{e}_{n})^{-}(\zeta )^{-})_{k+\frac{1}{2}} -((\pi ^{n}-\pi ^{e}_{n})^{-}(\zeta )^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0. \end{aligned}$$
(D.1)
We take the test functions
$$\begin{aligned}&\vartheta _{1}=\pi ^{n},\quad \rho =\phi ^{n}-\epsilon ^{n},\quad \phi =\pi ^{n},\quad \varphi =-\vartheta ^{n},\nonumber \\&\chi =\sigma ^{n},\quad \varrho =\varpi ^{n}-\varphi ^{n},\quad \psi =-\sigma ^{n},\quad \zeta =\tau ^{n}, \end{aligned}$$
(D.2)
we obtain
$$\begin{aligned}&\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j} \delta _{t}^{\alpha _{j}}(\pi ^{n}-\pi ^{e}_{n}),\pi ^{n}\right) _{D^{k}} +\big (\tau ^{n}-\tau ^{e}_{n},\pi _{x}^{n}\big )_{D^{k}}\nonumber \\&\quad -\,\big (\Delta _{(\beta -2)/2}(\epsilon ^{n}-\epsilon ^{e}_{n}), \phi ^{n}-\epsilon ^{n}\big )_{D^{k}}+\big (\pi ^{n}-\pi ^{e}_{n}, \tau _{x}^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\left( \sum _{j=1}^{S}W(\alpha _{j})\Delta \tau _{j}\delta _{t} ^{\alpha _{j}}(\sigma ^{n}-\sigma ^{e}_{n}), \sigma ^{n}\right) _{D^{k}}-\big (\sigma ^{n}-\sigma ^{e}_{n}, \vartheta _{x}^{n}\big )_{D^{k}}\nonumber \\&\quad -\,\big (\Delta _{(\beta -2)/2}(\varphi ^{n}-\varphi ^{e}_{n}), \varpi ^{n}-\varphi ^{n}\big )_{D^{k}} -\big (\vartheta ^{n}-\vartheta ^{e}_{n},\sigma _{x}^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\big (\epsilon ^{n}-\epsilon ^{e}_{n},\pi \big )_{D^{k}} -\big (\tau ^{n}-\tau ^{e}_{n},\vartheta \big )_{D^{k}}\nonumber \\&\quad +\,\big (\phi ^{n}-\phi ^{e}_{n},\phi ^{n}-\epsilon ^{n}\big ) _{D^{k}}-\big (\varphi ^{n}-\varphi ^{e}_{n},\sigma ^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\big (\vartheta ^{n}-\vartheta ^{e}_{n},\tau ^{n}\big )_{D^{k}} +\big (\varpi ^{n}-\varpi ^{e}_{n},\varpi ^{n}-\varphi ^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\varepsilon _{1}\big (\phi ^{n} -\phi ^{e}_{n},\pi ^{n}\big )_{D^{k}}-\varepsilon _{1} \big (\varpi ^{n}-\varpi ^{e}_{n},\sigma ^{n}\big )_{D^{k}}\nonumber \\&\quad +\,\varepsilon _{2} \big (\sigma ^{n}-\sigma ^{e}_{n},\pi \big )_{D^{k}} -\varepsilon _{2} \big (\pi _{n}-\pi ^{e}_{n},\sigma \big )_{D^{k}}\nonumber \\&\quad -\,\big (\gamma (q)^{n},\sigma ^{n}\big )+\big (\gamma (p)^{n},\pi ^{n}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K}\big (((\tau ^{n}-\tau ^{e}_{n})^{+}(\pi ^{n})^{-}) _{k+\frac{1}{2}} -((\tau ^{n}-\tau ^{e}_{n})^{+}(\pi ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\sigma ^{n}-\sigma ^{e}_{n})^{-} (\vartheta ^{n})^{-})_{k+\frac{1}{2}} -((\sigma ^{n}-\sigma ^{e}_{n})^{-}(\vartheta ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad +\,\sum _{k=1}^{K}\big (((\vartheta ^{n}-\vartheta ^{e}_{n})^{+} (\sigma ^{n})^{-})_{k+\frac{1}{2}} -((\vartheta ^{n}-\vartheta ^{e}_{n})^{+}(\sigma ^{n})^{+}) _{k-\frac{1}{2}}\big )\nonumber \\&\quad -\,\sum _{k=1}^{K} \big ((\pi ^{n}-\pi ^{e}_{n})^{-}(\tau ^{n})^{-})_{k+\frac{1}{2}} -((\pi ^{n}-\pi ^{e}_{n})^{-}(\tau ^{n})^{+})_{k-\frac{1}{2}}\big )\nonumber \\&\quad =0.\nonumber \\ \end{aligned}$$
(D.3)
Summing over k, simplify by integration by parts and (5.6). \(\square \)
