# A Modified PML Acoustic Wave Equation

## Abstract

**:**

## 1. Introduction

## 2. Regularized System

**Remark**

**1.**

#### 2.1. Energy Estimate of Weak Solution

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

#### 2.2. Existence and Uniqueness

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

## 3. Numerical Scheme

#### 3.1. Staggered Finite Differences

#### 3.2. Stability Analysis

**Remark**

**4.**

**Theorem**

**3.**

**Definition**

**1.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**3.**

**Remark**

**5.**

## 4. Numerical Result

## 5. Discussion

## Funding

## Conflicts of Interest

## Abbreviations

PML | Perfectly Matched Layers |

CFL | Courant-Friedrichs-Lewy |

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**Figure 1.**Comparison of errors: (

**a**) a fixed damping ${\sigma}_{0}=30$, (

**b**) a thickness $L=0.1875$ $(\beta =2)$.

**Figure 2.**Comparison of the difference at a point from different positions of source function with ${\sigma}_{0}=35$ and $L=0.1$.

**Figure 3.**$\mathcal{E}(t)$ with (

**a**) various damping values ${\sigma}_{0}=40,50,60,70$ for a fixed thickness $L=0.0625\phantom{\rule{3.33333pt}{0ex}}(\beta =0)$, (

**b**) various thickness $L=0.0625,0.1,0.125,0.15$ for a fixed damping ${\sigma}_{0}=50\phantom{\rule{3.33333pt}{0ex}}(\beta =0)$.

**Figure 4.**Snap shots of the regularized system at time ${t}_{n}=1,30,60,100,130,150,200,300,500$ with ${\sigma}_{0}=35,\beta =2,L=0.25$ (Red rectangular box represents the computational domain.)

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**MDPI and ACS Style**

Kim, D.
A Modified PML Acoustic Wave Equation. *Symmetry* **2019**, *11*, 177.
https://doi.org/10.3390/sym11020177

**AMA Style**

Kim D.
A Modified PML Acoustic Wave Equation. *Symmetry*. 2019; 11(2):177.
https://doi.org/10.3390/sym11020177

**Chicago/Turabian Style**

Kim, Dojin.
2019. "A Modified PML Acoustic Wave Equation" *Symmetry* 11, no. 2: 177.
https://doi.org/10.3390/sym11020177