Properties of Finite-Difference Operators for the Steady-Wave Problem

1993 ◽  
Vol 37 (01) ◽  
pp. 1-7
Author(s):  
John S. Letcher
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Radiation Condition ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Difference Operators ◽  
Surface Boundary ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

A feature of most implementations of Dawson's boundary-integral method for steady free-surface flows is the use of upstream finite-difference operators for the streamwise derivative occurring in the linearized free-surface boundary condition. An algebraic analysis of a family of candidate operators reveals their essential damping and dispersion error characteristics, which correlate well with their observed performance in two-dimensional example flows. Some new operators are found which perform somewhat better than Dawson's, but the general outlook for accurate results using difference operators is nevertheless bleak. It is shown that the calculation necessarily diverges as panel size is reduced, and a breakdown at higher speeds is also inevitable. More promise appears to lie in satisfying the radiation condition by several alternative ways, which are briefly discussed.

scholarly journals An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T1-T9 ◽  
Author(s):  
Chong Zeng ◽  
Jianghai Xia ◽  
Richard D. Miller ◽  
Georgios P. Tsoflias
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Surface Boundary ◽  
Near Surface ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Grid Nodes

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

Bow and stern flows with constant vorticity

1999 ◽  
Vol 399 ◽  
pp. 277-300 ◽  
Author(s):  
SCOTT W. McCUE ◽  
LAWRENCE K. FORBES
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Radiation Condition ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Infinite Body ◽  
Constant Vorticity

Free surface flows of a rotational fluid past a two-dimensional semi-infinite body are considered. The fluid is assumed to be inviscid, incompressible, and of finite depth. A boundary integral method is used to solve the problem for the case where the free surface meets the body at a stagnation point. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterized by a train of waves upstream. It is shown numerically that the amplitude of these waves increases as each of the Froude number, vorticity and height of the body above the bottom increases.

A numerical free‐surface condition for elastic/viscoelastic finite‐difference modeling in the presence of topography

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1921-1934 ◽  
Author(s):  
Johan O. A. Robertsson
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Condition ◽  
Finite Difference Schemes ◽  
Surface Boundary ◽  
Computationally Efficient ◽  
Free Surfaces ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Irregular Topography

An accurate free‐surface boundary condition is important for solving a wide variety of seismic modeling problems. In particular, for earthquake site studies or shallow environmental investigations the surface of the earth may have a significant impact on the outcome of simulations. Computations based on several elastic/viscoelastic flat horizontal free‐surface conditions are compared and benchmarked against an analytical solution. An accurate and simple condition is found and then generalized to allow for irregular free surfaces. This new method is simple to implement in conventional staggered finite‐difference schemes, is computationally efficient and enables modeling of highly irregular topography. The accuracy of the method is investigated and criteria for sampling of the wavefield are derived.

scholarly journals A stable free‐surface boundary condition for two‐dimensional elastic finite‐difference wave simulation

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2247-2249 ◽  
Author(s):  
John E. Vidale ◽  
Robert W. Clayton
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Difference Approximation ◽  
Surface Boundary ◽  
Stability Range ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
The One ◽  
Lateral Variations

Two of the persistent problems in finite‐difference solutions of the elastic wave equation are the limited stability range of the free‐surface boundary condition and the boundary condition’s treatment of lateral variations in velocity and density. The centered‐difference approximation presented by Alterman and Karal (1968), for example, remains stable only for β/α greater than 0.30, where β and α are the shear [Formula: see text] and compressional [Formula: see text] wave velocities. The one‐sided approximation (Alterman and Rotenberg, 1969) and composed approximation (Ilan et al., 1975) have similar restrictions. The revised‐composed approximation of Ilan and Loewenthal (1976) overcomes this restriction, but cannot handle laterally varying media properly.

Sign in / Sign up

Export Citation Format

Share Document