KAUST Grant NumberKUK–C1–013–04
Permanent link to this recordhttp://hdl.handle.net/10754/600010
MetadataShow full item record
AbstractThis paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.
CitationOckendon JR, Tew RH (2012) Thin-Layer Solutions of the Helmholtz and Related Equations. SIAM Review 54: 3–51. Available: http://dx.doi.org/10.1137/090761641.
SponsorsThis work was based on research supported in part byaward KUK–C1–013–04 from King Abdullah University of Science and Technology.
CollectionsPublications Acknowledging KAUST Support
Showing items related by title, author, creator and subject.
High Weak Order Methods for Stochastic Differential Equations Based on Modified EquationsAbdulle, Assyr; Cohen, David; Vilmart, Gilles; Zygalakis, Konstantinos C. (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2012-01) [Article]© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
A transport equation for confined structures derived from the Boltzmann equationHeitzinger, Clemens; Ringhofer, Christian (Communications in Mathematical Sciences, International Press of Boston, 2011) [Article]
On the Existence and the Applications of Modified Equations for Stochastic Differential EquationsZygalakis, K. C. (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2011-01) [Article]In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. © 2011 Society for Industrial and Applied Mathematics.