A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

Fornberg, Bengt; Driscoll, Tobin A.

doi:10.1006/jcph.1999.6351

Cited by 115 publications

(109 citation statements)

References 15 publications

(23 reference statements)

Supporting

Mentioning

104

Contrasting

Order By: Relevance

“…Now in the uniform-flow case for almost all n holds the inequality ζ 1 π n/κ 1 R. As a result, the number of circular trajectories reduces to only four and two slowly decaying vertical sinusoidal lines (21) and (22) can be observed in the upper and lower half-planes as Im( k) → ±∞, respectively (Figure 7a). These lines smoothly merge with the horizontal surface-mode trajectories given by solution (23) which is valid when Re( k) → ±∞. Estimates of the Im( k) for surface modes based on formulae (23) give an approximate value of ±8.67 which is reasonably close to the computed value of ±9.05 for large Re( k).…”

Section: Soft-wall Solutionssupporting

confidence: 69%

“…These lines smoothly merge with the horizontal surface-mode trajectories given by solution (23) which is valid when Re( k) → ±∞. Estimates of the Im( k) for surface modes based on formulae (23) give an approximate value of ±8.67 which is reasonably close to the computed value of ±9.05 for large Re( k). However, already for the mean-flow velocity profile with a very thin near-wall sublayer shown in Figure 7b (a = 500) the eigenvalue pattern features substantial differences from its uniform-flow counterpart.…”

Section: Soft-wall Solutionssupporting

confidence: 69%

“…and the following two implicit second-order accurate schemes considered in Fornberg & Driscoll [23] for integration of stiff initial-value problems were used y j +1 − y j = 1 2 δ y j +1 + y j and y j +1 − y j = 1 2 δ 3 2 y j +1 + 1 2 y j −1 .…”

Section: Numerical Proceduresmentioning

confidence: 99%

See 2 more Smart Citations

Numerical study of acoustic modes in ducted shear flow

Vilenski

Rienstra

2007

Journal of Sound and Vibration

View full text Add to dashboard Cite

The propagation of small-amplitude modes in an inviscid but sheared mean flow inside a duct is studied numerically. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for the applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for the uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for the Helmholtz numbers up to 100 and the circumferential wavenumber as large as 50, typical Mach numbers being up to 0.65.In order to gain further insight into the possible structure of the modal solutions and to get an independent verification of the robustness of the numerical scheme, comparison to the asymptotic solution of the problem based on the WKB method is performed. The asymptotic solution is also used as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available.The bulk of the analysis concentrates on the influence of the wall lining. The proposed numerical procedure is adapted in order to include Ingard-Myers boundary conditions. In parallel with this, the WKB solution is used to check the numerical predictions of the typical behaviour of the axial wavenumber in the complex plane, when the wall impedance varies in the complex plane.Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that the use of Ingard-Myers condition in combination with an appropriate slipstream approximation instead of the actual no-slip mean flow profile gives valid results in the limit of vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases.

show abstract

Section: Soft-wall Solutionssupporting

confidence: 69%

Section: Soft-wall Solutionssupporting

confidence: 69%

See 1 more Smart Citation

Numerical study of acoustic modes in ducted shear flow

Vilenski

Rienstra

2007

Journal of Sound and Vibration

View full text Add to dashboard Cite

show abstract

“…However, our choice of time integration may not be the most advantageous match for either the finite-difference or the Chebyshev discretization. The methods presented by Fornberg and Driscoll [15] and by de Frutos and Sanz-Serna [11] might be a better match for the Chebyshev method than the mixed Adams-Bashforth and Crank-Nicolson method we have used. However, better as they may be, we do not expect these methods to change our benchmarking results in a qualitative way.…”

Section: Benchmarkingmentioning

confidence: 96%

A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Skogestad

Kalisch

2009

Mathematics and Computers in Simulation

View full text Add to dashboard Cite

show abstract

“…The domain sizes used are 40L × 40 −1/2 L to 60L × 60 −1/2 L, increasing as Γ approaches zero. The third numerical model, used to solve the KP equation (2.15), is a standard pseudo-spectral model similar to those described by Fornberg & Driscoll (1999) and Johnson & Vilenski (2004). Equation (2.15) is solved on a doubly periodic domain with size either…”

Section: Supercritical Flowmentioning

confidence: 99%

Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

Esler¹,

Rump²,

Johnson³

2007

J. Fluid Mech.

View full text Add to dashboard Cite

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be $c_d M^2/(F\sqrt{F^2-1})$, where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

show abstract

A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion

Cited by 115 publications

References 15 publications

Numerical study of acoustic modes in ducted shear flow

Numerical study of acoustic modes in ducted shear flow

A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

Contact Info

Product

Resources

About