Conservative arbitrary order finite difference schemes for shallow-water flows

AbstractThe classical nonlinear shallow-water model (SWM) of an ideal fluid is considered. For the model, a new method for the construction of mass and total energy conserving finite difference schemes is suggested. In fact, it produces an infinite family of finite difference schemes, which are either linear or nonlinear depending on the choice of certain parameters. The developed schemes can be applied in a variety of domains on the plane and on the sphere. The method essentially involves splitting of the model operator by geometric coordinates and by physical processes, which provides substantial benefits in the computational cost of solution. Besides, in case of the whole sphere it allows applying the same algorithms as in a doubly periodic domain on the plane and constructing finite difference schemes of arbitrary approximation order in space. Results of numerical experiments illustrate the skillfulness of the schemes in describing the shallow-water dynamics

Dynamical Supersymmetry Breaking

Supersymmetry is one of the most plausible and theoretically motivated frameworks for extending the Standard Model. However, any supersymmetry in Nature must be a broken symmetry. Dynamical supersymmetry breaking (DSB) is an attractive idea for incorporating supersymmetry into a successful description of Nature. The study of DSB has recently enjoyed dramatic progress, fueled by advances in our understanding of the dynamics of supersymmetric field theories. These advances have allowed for direct analysis of DSB in strongly coupled theories, and for the discovery of new DSB theories, some of which contradict early criteria for DSB. We review these criteria, emphasizing recently discovered exceptions. We also describe, through many examples, various techniques for directly establishing DSB by studying the infrared theory, including both older techniques in regions of weak coupling, and new techniques in regions of strong coupling. Finally, we present a list of representative DSB models, their main properties, and the relations between them.Comment: 113 pages, Revtex. Minor changes, references added and corrected. To appear in Reviews of Modern Physic

Mathematical problems of the dynamics of incompressible fluid on a rotating sphere

This book presents selected mathematical problems involving the dynamics of a two-dimensional viscous and ideal incompressible fluid on a rotating sphere. In this case, the fluid motion is completely governed by the barotropic vorticity equation (BVE), and the viscosity term in the vorticity equation is taken in its general form, which contains the derivative of real degree of the spherical Laplace operator. This work builds a bridge between basic concepts and concrete outcomes by pursuing a rich combination of theoretical, analytical and numerical approaches, and is recommended for specialists developing mathematical methods for application to problems in physics, hydrodynamics, meteorology and geophysics, as well for upper undergraduate or graduate students in the areas of dynamics of incompressible fluid on a rotating sphere, theory of functions on a sphere, and flow stability

A balanced and absolutely stable numerical thermodynamic model for closed and open oceanic basins

Por medio de condiciones especiales de frontera. el problema bien establecido se formula para el modelo oceánico termodinámico de Adem en una región de océano abierta. cuando existe un flujo anómalo de calor a través de las fronteras laterales. Se muestran la unicidad y estabilidad de las soluciones del modelo. Se estima la velocidad de disipación de las anomalías de temperatura en presencia de la difusión y ausencia del forzamiento. Se muestra que el operador del modelo es positivo definido, positivo semidefinido o antisimétrico, dependiendo del tipo de condiciones de frontera y de la difusión. EI método de separación se aplica para construir un esquema implícito en diferencias finitas con aproximación de segundo orden, el cual es económico, balanceado e incondicionalmente estable. Cada uno de los problemas separados es de dimensión 1, y se resuelve fácilmente por el método de factorización. Se justifica la aplicación del método de separación. El algoritmo numérico se puede generalizar con facilidad para el modelo de 3-dimensiones

A numerical study of nonlinear diffusion phenomena in heterogeneous media: energy transfer at diverse blow-up modes and self-organisation processes

A detailed analysis of a new method for numerical simulation of nonlinear diffusion phenomena is carried out. The method is based on operator splitting performed in time and space, and yields highly accurate solutions in complex 2D and 3D computational domains. After providing a circumstantial mathematical description of the developed method, we test it in several numerical experiments aimed, firstly, to model energy transfer at diverse modes of evolution of the dynamical system, and, secondly, to simulate self-organisation processes typical for real-world applications. A discussion of the outcomes of the numerical experiments is given. This is a follow-up paper of our recent original results presented at the 19th European conference on mathematics for industry