S. Kim. Wave SIMULATION

http://www.ms.uky.edu/~skim/WaveSimulation/#FullWave

WAVE SIMULATION (Last edited: October 2004)
Contents
Full Waveform Simulation
Frequency-Domain Solvers
Computer Programs: The source codes are available from GRADE or by asking skim@ms.uky.edu.
Lecture Note: S. Kim, Numerical Methods for Differential Equations NumerPDE.ps NumerPDE.pdf
Dr. Kim's Other Research Branches
Image Processing
Computational Seismology
Full Waveform Simulation
Scope & Goal
It is often desired to design efficient implicit algorithms for the time-domain wave propagation.
Goal: Development of O(N), unconditionally stable algorithms for the wave propagation in 2D/3D heterogeneous media.
On-going Projects: LOD and ADI-II, with applications to the acoustic wave propagation and anisotropic micro-heat transfer.
Publications
H. Lim, S. Kim, and J. Douglas, Jr.. Numerical methods for viscous and nonviscous wave equations. (submitted). Postscript
J. Douglas, Jr., S. Kim, and H. Lim (2003). An improved alternating-direction method for a viscous wave equation. Contemporary Mathematics 329, pp. 99-104. Postscript
J. Douglas, Jr. and S. Kim (2001). Improved accuracy for locally one-dimensional methods for parabolic equations. Math. Models and Methods in Appl. Sci. 11, pp. 1563-1579. Postscript [TopBottom]
Frequency-Domain Solvers
Scope & Goal
The frequency-domain (Helmholtz) problem is hard to solve numerically. In addition to having a complex-valued solution, it is neither Hermitian-symmetric nor coercive, in non-attenuative media. As a consequence, most standard iterative algorithms either fail to converge or converge so slowly as to be impractical. Since many applications need to simulate waves of 20-50 wavelengths and since stability for numerical methods requires at least 10-12 grid points per wavelength, the discrete problem on the coarsest possible mesh is still huge; in realistic simulations, it is required to choose 20-30 points per wavelength for an acceptable accuracy. As Zienkiewicz, a founding father of the finite element method, pointed out, the problem remains unsolved and a completely new method is needed.
Goal: Development of accurate and optimally convergent algorithms for the simulation of 20-50 wavelengths in 3D heterogeneous media; in particular, no objectionable phase lag!!
On-going Project: The High-Frequency Asymptotic Decomposition (HFAD) method and perfect absorbing boundary conditions.
Publications
S. Kim, C.-S. Shin, and J.B. Keller. Asymptotic decomposition methods for the numerical solution of the Helmholtz equation. Appl. Math. Letters (in press). Postscript
S. Kim (2003). Compact Schemes for Acoustics in the Frequency Domain. Mathematical and Computer Modelling 37, pp. 1335-1341. Postscript
S. Kim and Soohyun Kim (2002). Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media. SIAM J. Sci. Comput. 24, No. 2, pp. 684-701. PDF
S. Kim (1998). Domain decomposition iterative procedures for solving scalar waves in the frequency domain. Numerische Mathematik 79, pp. 231-259.
S. Kim and W.W. Symes (1998). Multigrid domain decomposition methods for the Helmholtz problem. In: Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation (Editor: J. A. DeSanto). SIAM, Philadelphia, pp. 617-619.
S. Kim and M. Lee (1996). Artificial damping techniques for scalar waves in the frequency domain. Computers Math. Applic. 31, No. 8, pp. 1-12.
S. Kim (1995). Parallel multidomain iterative algorithms for the Helmholtz wave equation. Appl. Numer. Math. 17, pp. 411-429.
S. Kim (1994). A parallelizable iterative procedure for the Helmholtz problem. Appl. Numer. Math. 14, pp. 435-449.
S. Kim (1994). Numerical Treatments for the Helmholtz Problem by Domain Decomposition Techniques. Contemp. Math. 180, pp. 245-250.
Some Numerical ExamplesIn the following, we compare the standard fourth-order Helmholtz solver and the HFAD technique incorporating a second-order discretization, for the wave propagation in an 1D heterogeneous medium. An impulse source is put at the left edge.
Coarse Mesh
Finer Mesh
Stadard Fourth-Order
HFAD
Note: In the above figure, the dotted curve denotes the true solution, while the solid curve is the real part of the numerical solution. One can see a large phase lag for the standard technique with nx=300 (top left), which corresponds to choosing an average of 10.7 points per wavelength. On the other hand, the solution of HFAD is accurate enough when nx=112 (bottom right), which corresponds to choosing an average of 4 points per wavelength.