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# Two alternatives for solving hyperbolic boundary value problems of geophysical fluid dynamics

## Abstract

Linear internal waves in inviscid bounded fluids generally give a mathematically ill-posed problem since hyperbolic equations are combined with elliptic boundary conditions. Such problems are difficult to solve. Two new approaches are added to the existing methods: the first solves the two-dimensional spatial wave equation by iteratively adjusting Cauchy data such that boundary conditions are satisfied along a predefined boundary. After specifying the data in this way, solutions can be computed using the d'Alembert formula.

The second new approach can numerically solve a wider class of two dimensional linear hyperbolic boundary value problems by using a ‘boundary collocation’ technique. This method gives solutions in the form of a partial sum of analytic functions that are, from a practical point of view, more easy to handle than solutions obtained from characteristics. Collocation points have to be prescribed along certain segments of the boundary but also in the so-called fundamental intervals, regions along the boundary where Cauchy data can be given arbitrarily without over-or under-determining the problem. Three prototypical hyperbolic boundary value problems are solved with this method: the Poincaré, the Telegraph, and the Tricomi boundary value problem. All solutions show boundary-detached internal shear layers, typical for hyperbolic boundary value problems. For the Tricomi problem it is found that the matrix that has to be inverted to find solutions from the collocation approach is ill-conditioned; thus solutions depend on the distribution of the collocation points and need to be regularized.

## References

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Aster, R. C., Borchers, B. & Thurber, C. H. 2005 Parameter Estimation and Inverse Problems. Elsevier.
Brown, S. N. & Stewartson, K. 1976 Asymptotic methods in the theory of rotating fluids. In Asymptotic Methods and Singular Perturbations (ed. O'Malley, R. E.), pp. 1–21. SIAM-AMS Proceedings, vol. 10, American Mathematical Society.
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. 180, 187219.
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271297.
Friedlander, S. 1982 Turning surface behaviour for internal waves subject to general gravitational field. Geophys. Astrophys. Fluid Dyn. 21, 189200.
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a rotating stratified fluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. 19, 267291.
Harlander, U. & Maas, L. R. M. 2006 Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves. Meteorologische Z. 15, 439450.
Hollerbach, R. & Kerswell, R. R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327339.
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Int. J. Bifurcation Chaos, 15, 27572782.
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F.-P. A. 1997 Observation of an internal wave attractor in a confined, stable stratified fluid. Nature 388, 557561.
Maas, L. R. M. & Harlander, U. 2007 Equatorial wave attractors and inertial oscillations. J. Fluid Mech. 570, 4767.
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.
Manwell, A. R. 1979 The Tricomi Equation with Applications to the Theory of Plane Transonic Flow. Pitman.
Moore, E. H. 1920 On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394395.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, Vol. I & II. McGraw-Hill.
Myint-U, T. 1987 Partial Differential Equations for Scientists and Engineers, 3rd End. North Holland.
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 1944.
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477509.
Payne, L. E. 1975 Improperly Posed Problems in Partial Differential Equations. SIAM.
Penrose, R. 1955 A generalized inverse for matrices. Proce. Camb. Phil. Soc. 51, 406413.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett. 85, 42774280.
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 7799.
Slavyanov, S. Y. & Lay, W. 2000 Special Functions: A Unified Theory based on Singularities. Oxford University Press.
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 5, 577592.
Swart, A., Sleijpen, G. L. G., Maas, L. R. M. & Brandts, J. 2007 Numerical solution of the two-dimensional Poincaré equation. J. Comput. Appl. Math. 200, 317341.
Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. SIAM.
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# Two alternatives for solving hyperbolic boundary value problems of geophysical fluid dynamics

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