# Stability Analysis of the Explicit Difference Scheme for Richards Equation

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## Abstract

**:**

## 1. Introduction

## 2. Richards Equation and the Explicit Difference Scheme

- h-based$$C\left(h\right)\frac{\partial h}{\partial t}-\nabla \xb7K\left(h\right)\nabla h-\frac{\partial K}{\partial z}=0,$$
- $\theta $-based$$\frac{\partial \theta}{\partial t}-\nabla \xb7D\left(\theta \right)\nabla \theta -\frac{\partial K}{\partial z}=0,$$
- mixed$$\frac{\partial \theta}{\partial t}-\nabla \xb7K\left(h\right)\nabla h-\frac{\partial K}{\partial z}=0,$$

**Remark**

**1.**

## 3. Stability Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Numerical Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathit{\tau}$ | $\mathit{\tau}=0.4$ | $\mathit{\tau}=0.2$ | $\mathit{\tau}=0.1$ | $\mathit{\tau}=0.05$ | $\mathit{\tau}=0.025$ | $\mathit{\tau}=0.0125$ | ||
---|---|---|---|---|---|---|---|---|

Accuracy | ||||||||

${\mathit{\u03f5}}_{2}$ | ||||||||

${\u03f5}_{2}=0$ | Unstable | Unstable | Unstable | Unstable | $5.60\times {10}^{-4}$ | $3.26\times {10}^{-5}$ | ||

${\u03f5}_{2}=0.0001$ | Unstable | Unstable | Unstable | Unstable | $1.38\times {10}^{-4}$ | $3.75\times {10}^{-5}$ | ||

${\u03f5}_{2}=0.0005$ | Unstable | $1.88\times {10}^{-1}$ | $5.47\times {10}^{-2}$ | $5.00\times {10}^{-3}$ | $3.78\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | ||

${\u03f5}_{2}=0.001$ | $9.10\times {10}^{-2}$ | $1.44\times {10}^{-2}$ | $4.00\times {10}^{-3}$ | $1.50\times {10}^{-3}$ | $7.54\times {10}^{-4}$ | $3.79\times {10}^{-4}$ |

N | 1000 | 2000 | 4000 | 8000 | 16,000 |
---|---|---|---|---|---|

${|h-H|}_{\infty ,h}$ | $1.90\times {10}^{-3}$ | $9.65\times {10}^{-4}$ | $4.82\times {10}^{-4}$ | $2.41\times {10}^{-4}$ | $1.21\times {10}^{-4}$ |

Ratio | Non | 0.98 | 1.00 | 1.00 | 0.99 |

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**MDPI and ACS Style**

Liu, F.; Fukumoto, Y.; Zhao, X. Stability Analysis of the Explicit Difference Scheme for Richards Equation. *Entropy* **2020**, *22*, 352.
https://doi.org/10.3390/e22030352

**AMA Style**

Liu F, Fukumoto Y, Zhao X. Stability Analysis of the Explicit Difference Scheme for Richards Equation. *Entropy*. 2020; 22(3):352.
https://doi.org/10.3390/e22030352

**Chicago/Turabian Style**

Liu, Fengnan, Yasuhide Fukumoto, and Xiaopeng Zhao. 2020. "Stability Analysis of the Explicit Difference Scheme for Richards Equation" *Entropy* 22, no. 3: 352.
https://doi.org/10.3390/e22030352