Abstract
In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series' radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.
| Original language | English |
|---|---|
| Article number | 082104 |
| Journal | Physics of Fluids |
| Volume | 22 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - Aug 2010 |
| Externally published | Yes |
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Keywords
- Dispersion relations
- Numerical solutions
- Bernoulli's principle
- Exact solutions
- Solution processes
ASJC Scopus subject areas
- Condensed Matter Physics
Cite this
Accurate series solutions for gravity-driven Stokes waves. / Dallaston, Michael C.; McCue, Scott W.
In: Physics of Fluids, Vol. 22, No. 8, 082104, 08.2010.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Accurate series solutions for gravity-driven Stokes waves
AU - Dallaston, Michael C.
AU - McCue, Scott W.
PY - 2010/8
Y1 - 2010/8
N2 - In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series' radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.
AB - In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series' radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.
KW - Dispersion relations
KW - Numerical solutions
KW - Bernoulli's principle
KW - Exact solutions
KW - Solution processes
UR - http://www.scopus.com/inward/record.url?scp=77956638311&partnerID=8YFLogxK
U2 - 10.1063/1.3480394
DO - 10.1063/1.3480394
M3 - Article
VL - 22
JO - Physics of Fluid
JF - Physics of Fluid
SN - 1070-6631
IS - 8
M1 - 082104
ER -