Convolutional perfectly matched layer for elastic second-order wave equation

2010 ◽  
Vol 127 (3) ◽  
pp. 1318-1327 ◽  
Author(s):  
YiFeng Li ◽  
Olivier Bou Matar
Keyword(s):  
Wave Equation ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Second Order Wave Equation

scholarly journals Second-order scalar wave field modeling with a first-order perfectly matched layer

Solid Earth ◽  
2018 ◽  
Vol 9 (6) ◽  
pp. 1277-1298
Author(s):  
Xiaoyu Zhang ◽  
Dong Zhang ◽  
Qiong Chen ◽  
Yan Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Forward Modeling ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Staggered Grid ◽  
Scalar Wave ◽  
First Order ◽  
Second Order Wave Equation ◽  
Boundary Absorption

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

scholarly journals Second-order Scalar Wave Field Modeling with First-order Perfectly Matched Layer

2018 ◽  
Author(s):  
Xiaoyu Zhang ◽  
Dong Zhang ◽  
Qiong Chen ◽  
Yan Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Forward Modeling ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Staggered Grid ◽  
Scalar Wave ◽  
First Order ◽  
Second Order Wave Equation ◽  
Boundary Absorption

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional-grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered-grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

Singularities of electron kernel functions in an external electromagnetic field

1954 ◽  
Vol 222 (1148) ◽  
pp. 93-108 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Perturbation Theory ◽  
Wave Equation ◽  
Vacuum Polarization ◽  
Proper Time ◽  
External Electromagnetic Field ◽  
Kernel Functions ◽  
Second Order ◽  
Significant Part ◽  
Second Order Wave Equation

The electron kernel functions are derived from solutions of the second-order wave equation, using the proper-time parametrization. Iterated kernel functions are introduced and a gauge-independent perturbation theory is developed. The separation of singular parts proceeds in terms of the iterated kernel functions valid in the absence of an electromagnetic field, and the singular expressions which have to be compensated in order to determine the physically significant part of the vacuum polarization are obtained in a more transparent form than those given originally by Heisenberg.

Travelling wave solutions for a second order wave equation of KdV type

2007 ◽  
Vol 28 (11) ◽  
pp. 1455-1465 ◽  
Author(s):  
Yao Long ◽  
Ji-bin Li ◽  
Wei-guo Rui ◽  
Bin He
Keyword(s):  
Wave Equation ◽  
Travelling Wave ◽  
Second Order ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Second Order Wave Equation
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