Numerical solution of the two-dimensional Poincaré equation

• Published in: Journal of computational and applied mathematics Vol. 200; no. 1; pp. 317 - 341
• Main Authors: Swart, Arno, Sleijpen, Gerard L.G., Maas, Leo R.M., Brandts, Jan
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-03-2007; Elsevier
• Subjects: Exact sciences and technology; Ill-posed problems; Internal waves; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Poincaré equation; Regularisation; Sciences and techniques of general use; Poincaré equation; Ill-posed problems; 67B55; 65F22; Regularisation; 65N22; Internal waves; 65N22; 65F22; 67B55; Two dimensional equation; Poincaré equation; Regularisation; Ill-posed problems; Internal waves; Numerical analysis; Applied mathematics; Internal wave; Numerical solution; Ill posed problem; Regularization; Exact solution
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2005.12.024

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.