Explicit fidelity schemes for quadratic regularization

$$\text{gradient\hspace{0.17em}descent\hspace{0.17em}}\{\begin{array}{l}\Delta u_{\alpha }^n=-\Delta t\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)\hfill \\ \Delta t\le \frac{2\Delta x^2}{4Nc+\lambda \Delta x^2}\hfill \end{array}$$ (42)

$$\begin{array}{l}\text{\hspace{0.17em}1-order\hspace{0.17em}accelerated\hspace{0.17em}}\{\begin{array}{l}\Delta u_{\alpha }^n=\frac{1}{1+a\Delta t}\Delta u_{\alpha }^{n-1}-\frac{\Delta t^2}{1+a\Delta t}\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)\hfill \\ \Delta t\le \Delta x\sqrt{\frac{4}{4Nc+\lambda \Delta x^2}+{\left(\frac{a\Delta x}{4Nc+\lambda \Delta x^2}\right)}^2}+\frac{a\Delta x^2}{4Nc+\lambda \Delta x^2}\hfill \end{array}\hfill \end{array}$$ (43)

$$\text{2-order\hspace{0.17em}accelerated}\{\begin{array}{l}\Delta u_{\alpha }^n=\frac{2-a\Delta t}{2+a\Delta t}\Delta u_{\alpha }^{n-1}-\frac{2\Delta t^2}{2+a\Delta t}\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)\hfill \\ \Delta t\le \frac{2\Delta \pi }{\sqrt{4N_c+\lambda \Delta x^2}}\hfill \end{array}$$ (44)

$$\text{\hspace{0.17em}semi-implicit\hspace{0.17em}}\{\begin{array}{l}{v}_{\alpha }^n=u_{\alpha }^n+\frac{2-a\Delta t}{2+a\Delta t}\Delta u^{n-1}\hfill \\ u_{\alpha }^{n+1}=v_{\alpha }^n-\frac{2\Delta t^2}{2+a\Delta t}\left(\lambda \left(v_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{v_{\alpha +e_k}^n-2v_{\alpha }^n+v_{\alpha -e_k}^n}{\Delta x^2}\right)\hfill \\ \Delta t\le \frac{2\Delta x}{\sqrt{3\left(4Nc+\lambda \Delta x^2\right)}}\text{\hspace{0.17em}(sufficent\hspace{0.17em}but\hspace{0.17em}not\hspace{0.17em}necessary\hspace{0.17em}when\hspace{0.17em}}a>0)\hfill \end{array}$$ (45)