# Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

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## Abstract

**:**

## 1. Introduction

- Additional degrees of freedom are needed for dynamic problems, in general, to handle missing information.
- We note that restrictive time steps scale with the coarse mesh size, and thus, are much coarser.

## 2. Problem Setting

- The second variational derivatives ${\delta}^{2}F$ and ${\delta}^{2}G$ satisfy$${\delta}^{2}F(u)(v,v)\ge c(u){\parallel v\parallel}_{V}^{2}\phantom{\rule{0.277778em}{0ex}}\forall u,v\in V$$$${\delta}^{2}G(u)(v,v)\ge b(u){\parallel v\parallel}^{2}\phantom{\rule{0.277778em}{0ex}}\forall u,v\in V,$$
- The second variational derivatives ${\delta}^{2}F$ and ${\delta}^{2}G$ are bounded. That is,$$|{\delta}^{2}{F(u)(w,v)|\le C(u)\parallel v\parallel}_{V}{\parallel w\parallel}_{V}\phantom{\rule{0.277778em}{0ex}}\forall u,v,w\in V$$$$|{\delta}^{2}{G(u)(w,v)|\le B\parallel v\parallel}_{{L}_{2}}{\parallel w\parallel}_{{L}_{2}}\phantom{\rule{0.277778em}{0ex}}\forall u,v,w\in V,$$

## 3. Discretization

## 4. Partially Explicit Scheme with Space Splitting

#### Energy Stability

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 5. Discussions

#### 5.1. $G=0$ Case

#### 5.2. $G\ne 0$ Case

## 6. Numerical Results

- First, we used implicit CEM to compute the solution without additional degrees of freedom (called “Implicit CEM” in our graphs).
- Secondly, we computed the solution with additional degrees of freedom using implicit CEM (called “Implicit CEM with additional basis” in our graphs).
- Finally, we computed the solution with additional degrees of freedom using our proposed partially explicit approach (called “Partially Explicit Splitting CEM” in our graphs).

#### 6.1. ${V}_{H,1}$ and ${V}_{H,2}$ Constructions

#### 6.1.1. CEM Method

#### 6.1.2. Construction of ${V}_{H,2}$

#### 6.2. Linear $F(U)$

#### 6.3. Nonlinear $F(U)$

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 5.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 8.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 11.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 14.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 17.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 20.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 23.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 26.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

**Figure 29.**(

**Left**) Reference solution at $t=T$. (

**Middle**) Implicit CEM solution (with additional basis) at $t=T$. (

**Right**) Partially explicit solution at $t=T$.

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**MDPI and ACS Style**

Chung, E.T.; Efendiev, Y.; Leung, W.T.; Li, W.
Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems. *Mathematics* **2021**, *9*, 3000.
https://doi.org/10.3390/math9233000

**AMA Style**

Chung ET, Efendiev Y, Leung WT, Li W.
Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems. *Mathematics*. 2021; 9(23):3000.
https://doi.org/10.3390/math9233000

**Chicago/Turabian Style**

Chung, Eric T., Yalchin Efendiev, Wing Tat Leung, and Wenyuan Li.
2021. "Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems" *Mathematics* 9, no. 23: 3000.
https://doi.org/10.3390/math9233000