Numerical schemes for nonlinear systems

 

Bingzhen Zhou

 

Hunan Normal University

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Model problem

Consider energy Functional $$ E[\phi(\boldsymbol{x})]=\int_{\Omega}\left[\frac{\lambda}{2}|\nabla \phi|^{2}+H(\phi)\right] d \boldsymbol{x} .$$ The $H^{-1}$ gradient (Cahn-Hilliard): \[ \begin{array}{l}{\frac{\partial \phi}{\partial t}=m\Delta \mu} \\ {\mu=\delta E / \delta \phi=-\lambda\Delta \phi+h(\phi)}.\end{array} \] subject to either periodic boundary conditions or $$ \left.\frac{\partial \phi}{\partial \boldsymbol{n}}\right|_{\partial \Omega}=\left.\frac{\partial \mu}{\partial \boldsymbol{n}}\right|_{\partial \Omega}=0 .$$ Where $h = H'$ and $H(\phi)=\frac{\lambda}{4 \eta^{2}}\left(\phi^{2}-1\right)^{2}$. Satisfied: $$ \frac{d}{d t} E[\phi(\boldsymbol{x})]=-m\|\nabla \mu\|^{2}. $$

Direct approach

First-order semi-implicit schemes \begin{equation*} \begin{aligned} \frac{1}{\Delta t}\left(\phi^{n+1}-\phi^{n}\right)&=\Delta \mu^{n+1}, \\ \mu^{n+1}&=-\Delta \phi^{n+1}+h\left(\phi^{n}\right). \end{aligned} \end{equation*} Under the condition \begin{equation*} \Delta t \leq \frac{4 \varepsilon^{4}}{L^2}, \end{equation*} Where $L=||h'(\phi)||_{\infty}$ and $\varepsilon$ is coefficient related to $h$. The following energy law holds \begin{equation*} E\left(u^{n+1}\right) \leq E\left(u^{n}\right), \quad \forall n \geq 0. \end{equation*} Shen'2010

Stabilized approach (Convex splitting)

The stabilized first-order semi-implicit schemes \begin{equation}\label{1111} \begin{aligned} \frac{1}{\Delta t}\left(\phi^{n+1}-\phi^{n}\right) & =\Delta \mu^{n+1} \\ \mu^{n+1}& =-\Delta \phi^{n+1}+S\left(\phi^{n+1}-\phi^{n}\right)+h\left(\phi^{n}\right) . \end{aligned} \end{equation} where $S$ is a stabilizing parameter.
For $\color{red} S \geq \frac{L}{2}$, The following energy law holds \begin{equation*} E\left(u^{n+1}\right) \leq E\left(u^{n}\right), \quad \forall n \geq 0. \end{equation*} Remove $\mu^{n+1}$ lead to $\frac{1}{\Delta t}\phi^{n+1} + \Delta^2 \phi^{n+1} - S\Delta \phi^{n+1} = R(\phi^{n})$.
It can decoupled into two sequential second-order equations \begin{equation} \begin{aligned} \Delta \psi^{n+1} - (\alpha + S )\psi^{n+1} & =R(\phi^{n})\\ \Delta \phi^{n+1} + \alpha \phi^{n+1}& =\psi^{n+1}. \end{aligned} \end{equation}

SAV approach

Assume $E_1(\phi) = \int_{\Omega} H(\phi)d x + C_0\geq 0$, and let $$ r(t)=\sqrt{E_{1}(\phi)}. $$ then we have $$ \begin{array}{l}{\phi_t=\Delta \mu} \\ {\mu=-\Delta \phi+\frac{r(t)}{\sqrt{E_{1}[\phi]}} h(\phi)} \\ {r_{t}=\frac{1}{2 \sqrt{E_{1}[\phi]}} \int_{\Omega} h(\phi) \phi_{t} d x}.\end{array} $$ unconditionally stable with modified energy $$ \widetilde{E}=\int_{\Omega}\frac{1}{2}\left|\nabla \phi\right|^{2} d x+r^{2} $$

Numerical scheme of SAV

Modified scheme $$ \begin{cases}{\phi_t=\Delta \mu} \\[0.5em] {\mu=-\Delta \phi+ \xi h(\phi)} \\[0.5em] {\xi = \frac{r(t)}{\sqrt{E_{1}[\phi]}},\quad r_{t}=\frac{1}{2 \sqrt{E_{1}[\phi]}} \int_{\Omega} h(\phi) \phi_{t} d x}.\end{cases} $$ Numerical scheme \begin{equation*} \begin{cases}{\frac{1}{\Delta t}(\phi^{n+1} - \phi^n)= \Delta \mu^{n+1}} \\[0.5em] {\mu^{n+1}=-\Delta \phi^{n+1}+ \xi^{n+1} h(\phi^{n})} \\[0.5em] {\xi^{n+1} = \frac{r^{n+1}}{\sqrt{E_{1}^{n}}},\quad r^{n+1} - r^n = \frac{1}{2 \sqrt{E_{1}^{n}}} \int_{\Omega} h(\phi^n) (\phi^{n+1} - \phi^n) d x}.\end{cases} \end{equation*} unconditionally stable with modified energy $$ \widetilde{E}^n=\int_{\Omega}\frac{1}{2}\left|\nabla \phi^{n}\right|^{2} d x+\left(r^{n}\right)^{2} $$

Numerical scheme details of SAV

\begin{equation*} \begin{cases}{\frac{1}{\Delta t}(\phi^{n+1} - \phi^n)= \Delta \mu^{n+1}} \\[0.5em] {\mu^{n+1}=-\Delta \phi^{n+1}+ \xi^{n+1} h(\phi^{n})} \\[0.5em] {\xi^{n+1} = \frac{r^{n+1}}{\sqrt{E_{1}^{n}}},\quad r^{n+1} - r^n = \frac{1}{2 \sqrt{E_{1}^{n}}} \int_{\Omega} h(\phi^n) (\phi^{n+1} - \phi^n) d x}.\end{cases} \end{equation*} Remove $\mu^{n+1}$ we get, \begin{equation*} A\phi^{n+1} = B + \xi^{n+1}C.\quad A = I+\Delta t \Delta^2,\;\;\; B = \phi^n,\;\;\; C = \Delta t \Delta h(\phi^n). \end{equation*} Since $A$ is linear operator, then we have $$\phi^{n+1} = A^{-1}B + \xi^{n+1}A^{-1}C := \phi^{n+1}_1+ \xi^{n+1}\phi^{n+1}_2.$$ we can update $\xi^{n+1}$ by \begin{equation*} \xi^{n+1}\sqrt{E_{1}^{n}} - \xi^{n}\sqrt{E_{1}^{n-1}} = \frac{1}{2 \sqrt{E_{1}^{n}}}\Big(h(\phi^n),\; \phi^{n+1}_1+ \xi^{n+1}\phi^{n+1}_2 - \phi^n\Big). \end{equation*}

Error analysis of SAV

Assume that $\phi^{0} \in H^{4}, \;\phi \in L^{\infty}\left(0, T ; W^{1, \infty}\right)$ and \begin{equation*} \phi_{t} \in L^{\infty}\left(0, T ; H^{-1}\right) \cap L^{2}\left(0, T ; H^{1}\right), \quad \phi_{t t} \in L^{2}\left(0, T ; H^{-1}\right) \end{equation*} then we have \begin{equation*} \begin{split}&{\frac{1}{2}\left\|\nabla (\phi^n-\phi(t^n))\right\|^{2}+\frac{\lambda}{2}\left\|(\phi^n-\phi(t^n))\right\|^{2}+\left|r^{n}-r(t^n)\right|^{2}} \\ &{ \leq C \exp \left((1-C \Delta t)^{-1} t^{n}\right) \Delta t^{2} \int_{0}^{t^{n}}\left(\left\|\phi_{t t}(s)\right\|_{H^{-1}}^{2}+\left\|\phi_{t}(s)\right\|_{H^{1}}^{2}\right) d s}.\end{split} \end{equation*} $C$ dependent $T, \phi^{0}, \Omega,\|\phi\|_{L^{\infty}\left(0, T ; W^{1, \infty}\right)},$ and $\left\|\phi_{t}\right\|_{L^{\infty}\left(0, T ; H^{-1}\right)}$.

and \[ |\xi^{n} - 1|\leq C\Delta t . \]

JIE SHEN, JIE XU, 2018.

Another numerical scheme of SAV

\begin{equation*} \begin{cases}{\frac{1}{\Delta t}(\phi^{n+1} - \phi^n)= \Delta \mu^{n+1}} \\[0.5em] {\mu^{n+1}=-\Delta \phi^{n+1}+ S(\phi^{n+1}- \phi^n) + \xi^{n+1} h(\phi^{n})} \\[0.5em] {\xi^{n+1} = \frac{r^{n+1}}{\sqrt{E_{1}[\widetilde \phi^{n+1}]}},\;\; r^{n+1} \!-\! r^n = \frac{1}{2 \sqrt{E_{1}[\widetilde \phi^{n+1}]}} \int_{\Omega} h(\phi^n) (\widetilde \phi^{n+1}\! -\! \widetilde \phi^{n}) d x}.\end{cases} \end{equation*} Remove $\mu^{n+1}$, we get \begin{equation*} A\phi^{n+1} = B + \xi^{n+1}C.\quad A = I+\Delta t \Delta^2,\;\;\; B = \phi^n,\;\;\; C = \Delta t \Delta h(\phi^n). \end{equation*} Since $A$ is linear operator, then we have $$\phi^{n+1} = A^{-1}B + \xi^{n+1}A^{-1}C := \phi^{n+1}_1+ \xi^{n+1}\phi^{n+1}_2.$$ and $\widetilde \phi^{n+1}$ defined by $\widetilde \phi^{n+1} = \phi^{n+1}_1 + \phi^{n+1}_2$.
Suchuan Dong, Zhiguo Yang, Lianlei Lin, 2018.

gPAV scheme

Remember $\dfrac{d E}{d t} = -\int_{\Omega} m|\nabla \mu|^{2} d \Omega$, modified $\widetilde E = E + C_0 \geq 0$, and notes $E:= \widetilde E$, let $R = E$ then \begin{equation*} \begin{cases} \phi_t=m \nabla^{2} \mu , \\[0.4em] \mu = -\lambda \nabla^{2} \phi+S(\phi-\phi)+\xi h(\phi) , \\[0.4em] \xi = \dfrac{R}{E}, \quad \dfrac{d R}{d t} = \xi \displaystyle\Big(-\int_{\Omega} m|\nabla \mu|^{2} d \Omega \Big). \end{cases} \end{equation*} The numerical scheme \begin{equation*} \begin{cases} \dfrac{ \phi^{n+1} - \phi^{n} }{\Delta t}=m\nabla^2 \mu^{n+1} \\[0.2em] \mu^{n+1}=-\lambda \nabla^{2} \phi^{n+1}+S(\phi^{n+1}-{\phi}^{n})+\xi^{n+1} h({\phi}^{n})\\[0.2em] \xi^{n+1} = \dfrac{R^{n+1}}{E[\widetilde{\phi}^{n+1}]}, \quad \dfrac{ R^{n+1} - R^{n} }{\Delta t}=- \xi^{n+1} \displaystyle\int_{\Omega} m|\nabla \widetilde \mu^{n+1}|^{2} d \Omega. \end{cases} \end{equation*}

gPAV numerical scheme

\begin{equation*} \begin{cases} \phi^{n+1} - \phi^{n} =m\Delta t\nabla^2 \mu^{n+1} \\[0.2em] \mu^{n+1}=-\lambda \nabla^{2} \phi^{n+1}+S(\phi^{n+1}-{\phi}^{n})+\xi^{n+1} h({\phi}^{n})\\[0.2em] \xi^{n+1} = \dfrac{R^{n+1}}{E[\widetilde{\phi}^{n+1}]}, \quad \dfrac{ R^{n+1} - R^{n} }{\Delta t}=- \xi^{n+1} \displaystyle\int_{\Omega} m|\nabla \widetilde \mu^{n+1}|^{2} d \Omega. \end{cases} \end{equation*} Remove $\mu^{n+1}$, then \begin{equation*} \nabla^{2}\left(\nabla^{2} \phi^{n+1}\right)-\frac{S}{\lambda} \nabla^{2} \phi^{n+1}+\frac{1}{m \lambda \Delta t} \phi^{n+1}= T_1(\phi^n)+\xi^{n+1}T_2(\phi^n), \end{equation*} Decoupled into \begin{equation*} \begin{split} &\nabla^{2} \psi^{n+1}-(\alpha+\frac{S}{\lambda}) \psi^{n+1}=T_1(\phi^n)+\xi^{n+1}T_2(\phi^n), \\[0.2em] &\nabla^{2} \phi^{n+1}+\alpha \phi^{n+1}=\psi^{n+1}. \end{split} \end{equation*} The $\tilde{\mu}^{n+1}$ follows \begin{equation*} \tilde{\mu}^{n+1}=-\lambda \nabla^{2} \tilde{\phi}^{n+1}+h(\tilde{\phi}^{n+1})=-\lambda(\tilde{\psi}^{n+1}-\alpha \tilde{\phi}^{n+1})+h(\tilde{\phi}^{n+1}). \end{equation*}

Stability of gPAV method

\begin{equation*} \begin{cases} \phi^{n+1} - \phi^{n} =m\Delta t\nabla^2 \mu^{n+1} \\[0.2em] \mu^{n+1}=-\lambda \nabla^{2} \phi^{n+1}+S(\phi^{n+1}-{\phi}^{n})+\xi^{n+1} h({\phi}^{n})\\[0.2em] \xi^{n+1} = \dfrac{R^{n+1}}{E[\widetilde{\phi}^{n+1}]}, \quad \dfrac{ R^{n+1} - R^{n} }{\Delta t}=- \xi^{n+1} \displaystyle\int_{\Omega} m|\nabla \widetilde \mu^{n+1}|^{2} d \Omega. \end{cases} \end{equation*} From the last-line equations, we have \begin{equation*} \xi^{n+1}=\frac{\xi^{n}E[\widetilde{\phi}^{n}]}{E[\tilde{\phi}^{n+1}]+\Delta t \int_{\Omega} m|\nabla \widetilde{\mu}^{n+1}|^{2} d \Omega } . \end{equation*}

Since $E \geq 0$, if we set $\xi^0 = 1$, then \[ \xi^{n+1} \geq 0, \] and the modified energy satisfied \[ R^{n+1} - R^{n} \leq 0. \]

 

  1. Zhiguo Yang, Suchuan Dong, 2019.
  2. Jie Shen , Jie Xu , Jiang Yang, 2018.
  3. JIE SHEN, JIE XU, 2018.
  4. gPAV BDF-2 notes.

 

Bingzhen Zhou

zbzhen@smail
.hunnu.edu.cn

Hunan Normal University