Enhancing the PML absorbing boundary conditions for the wave equation

2005 ◽  
Vol 53 (3) ◽  
pp. 1242-1246 ◽  
Author(s):  
Y.S. Rickard ◽  
N.K. Nikolova
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary

Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1153-1163 ◽  
Author(s):  
R. A. Renaut ◽  
J. Petersen
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Wide Angle ◽  
Courant Number ◽  
Absorbing Boundary ◽  
Wide Range ◽  
Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

scholarly journals Absorbing boundary conditions for wave‐equation migration

Geophysics ◽  
1980 ◽  
Vol 45 (5) ◽  
pp. 895-904 ◽  
Author(s):  
Robert W. Clayton ◽  
Björn Engquist
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Final Section ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Seismic Section ◽  
Difference Wave ◽  
Different Types ◽  
Almost All ◽  
The Cost

The standard boundary conditions used at the sides of a seismic section in wave‐equation migration generate artificial reflections. These reflections from the edges of the computational grid appear as artifacts in the final section. Padding the section with zero traces on either side adds to the cost of migration and simply delays the inevitable reflections. We develop stable absorbing boundary conditions that annihilate almost all of the artificial reflections. This is demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented can be used with any of the different types of finite‐difference wave‐equation migration, at essentially no extra cost.

scholarly journals ABSORBING BOUNDARY CONDITIONS FOR A WAVE EQUATION WITH A TEMPERATURE-DEPENDENT SPEED OF SOUND

2013 ◽  
Vol 21 (02) ◽  
pp. 1250028 ◽  
Author(s):  
IGOR SHEVCHENKO ◽  
MANFRED KALTENBACHER ◽  
BARBARA WOHLMUTH
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Speed Of Sound ◽  
Wave Equations ◽  
Focused Ultrasound ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
High Intensity Focused Ultrasound ◽  
Temperature Dependent ◽  
Absorbing Boundary

In this work, new absorbing boundary conditions (ABCs) for a wave equation with a temperature-dependent speed of sound are proposed. Based on the theory of pseudo-differential calculus, first- and second-order ABCs for the one- and two-dimensional wave equations are derived. Both boundary conditions are local in space and time. The well-posedness of the wave equation with the developed ABCs is shown through the reduction of the original problem to an equivalent one for which the uniqueness and existence of the solution has already been established. Although the second-order ABC is more accurate, the numerical realization is more challenging. Here we use a Lagrange multiplier approach which fits into the abstract framework of saddle point formulations and yields stable results. Numerical examples illustrating stability, accuracy and flexibility of the ABCs are given. As a test setting, we perform computations for a high-intensity focused ultrasound (HIFU) application, which is a typical thermo-acoustic multi-physics problem.

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