Complex singularity of a stokes wave

Dyachenko, Sergey A.; Lushnikov, Pavel M.; Korotkevich, A. O.

doi:10.1134/s0021364013240077

Cited by 33 publications

(74 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…To address these singularities, we performed high-precision simulations (both a quad precision of 32 digits accuracy and a variable precision of 200 digits accuracy were used) of Stokes wave ranging from near-linear Stokes wave with H/λ → 0, c → 1, and v c /λ 1 to near-limiting Stokes wave with v c /λ 10 −7 and H → H max . In the previous work [37], we found that as H → H max , the branch point approaches real axis with the scaling law…”

Section: Introductionmentioning

confidence: 76%

“…Singularities of the operatorM −1 = (−c 2 |k| + 1) −1 are avoided in our simulations because the wavenumber |k| is integer, while for any Stokes wave c is within the range 1 < c 2 < 1.3. We found that the region of convergence of the Newton-CG/CR methods to nontrivial physical solution (37) is relatively (with respect to GPM) narrow and requires an initial guess y (0) to be quite close to the exact solution y. In practice, we first run GPM and then choose y (0) for Newton-CG/CR methods as the last available iterate of GPM.…”

Section: Newton Cg and Newton Cr Methodsmentioning

confidence: 99%

“…These oscillations represent a challenge for simulation, because propagation velocity is the only parameter in Eq. (37). Thus, it is impossible to go over the first maximum by changing continuously velocity of propagation c. This is because after the maximum is reached, we have to start decreasing the parameter c. However, decreasing of c causes iterations to converge to the less steep solution on the left from the maximum (which we already obtained on previous steps), instead of steeper solutions to the right from the maximum.…”

Section: Stokes Wave Velocity As a Function Of Steepnessmentioning

confidence: 97%

“…[47]. We used a version generalized Petviashvili method (GPM) [48,49] adjusted to Stokes wave as described in [37]. In practice, this method allowed to find highprecision solutions up to H/λ 0.1388.…”

Section: Numerical Simulation Of Stokes Wavementioning

confidence: 99%

“…Here, we assume that branch cut is a straight line connecting w = iv c and +i∞. Then, the asymptotic of |ẑ k | is given by [37]…”

Section: Stokes Wave Velocity As a Function Of Steepnessmentioning

confidence: 99%

See 4 more Smart Citations

Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation

2016

Self Cite

View full text Add to dashboard Cite

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with 2π/3 radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period λ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance v c from the real line corresponding to the fluid surface in conformal variables. The increase of the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value H max /λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called by the Stokes wave of the greatest height). Here H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10 −26 . The tables use from several poles for near-linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with v c /λ ∼ 10 −6 .

show abstract

Section: Introductionmentioning

confidence: 76%

Section: Newton Cg and Newton Cr Methodsmentioning

confidence: 99%

Section: Stokes Wave Velocity As a Function Of Steepnessmentioning

confidence: 97%