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By Ed: Nikolaos A. Kampanis, Vassilios Dougalis, John A. Ekaterinaris

As a result of elevate in computational strength and new discoveries in propagation phenomena for linear and nonlinear waves, the world of computational wave propagation has develop into extra major in recent times. Exploring the newest advancements within the box, powerful Computational tools for Wave Propagation offers a number of sleek, invaluable computational tools used to explain wave propagation phenomena in chosen components of physics and expertise. that includes contributions from the world over identified specialists, the booklet is split into 4 components. It starts off with the simulation of nonlinear dispersive waves from nonlinear optics and the idea and numerical research of Boussinesq structures. the subsequent part makes a speciality of computational methods, together with a finite aspect technique and parabolic equation innovations, for mathematical types of underwater sound propagation and scattering. The booklet then deals a entire advent to trendy numerical tools for time-dependent elastic wave propagation. the ultimate half offers an summary of high-order, low diffusion numerical equipment for complicated, compressible flows of aerodynamics. focusing on physics and know-how, this quantity presents the mandatory computational how to successfully take on the resources of difficulties that contain a few kind of wave movement.

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Fiz, 18:1745, 1972, Sov. Physics JETP, 35:908, 1972. Chapter 2 Numerical Solution of the Nonlinear Helmholtz Equation G. il S. 1 Introduction Background The objective of this chapter is to describe a new numerical algorithm for studying nonlinear self-focusing of time-harmonic electromagnetic waves. The physical mechanism that leads to self-focusing is known as the Kerr effect. At the microscopic level, the Kerr effect may originate from electrostriction, nonresonant electrons, or from molecular orientation.

16: Cross sectional view of the color density plot for the same solution as in the previous figure. 17: Local grid distribution near the line singularity. Acknowledgments I would like to thank G. Fibich for some helpful discussions. References [1] G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney. Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schr¨ odinger equation. SIAM J. Sci. , 25(1):186– 212, 2003. [2] M. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics.

Russell. Moving mesh strategy based upon gradient flow equation for two dimensional problems. SIAM J. Sci. , (to appear). [17] M. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang. Stability of isotropic singularities for the nonlinear Schr¨ odinger equation. Physica D, 47:393–415, 1991. 34 COMPUTATIONAL METHODS FOR WAVE PROPAGATION [18] B. LeMesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem. Physica, 31D:78, 1988. [19] R. Li, T. -W. Zhang. A moving mesh finite element algorithm for singular problems in two and three space dimensions.

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