Characteristic boundary value problems: estimates from H1 to L2

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space H1_tan, can be transformed into an L2 a priori estimate of the same problem

On a priori energy estimates for characteristic boundary value problems

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space H1_tan, can be transformed into an L2 a priori estimate of the same problem

Mixed impedance boundary value problems for the Laplace–Beltrami equation

This work is devoted to the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem (MIND~BVP) for the Laplace-Beltrami equation on a compact smooth surface $\mathcal{C}$ with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting $\mathbb{H}^1(\mathcal{C})$ when considering positive constants in the impedance condition. The main purpose is to consider the MIND~BVP in a nonclassical setting of the Bessel potential space $\mathbb{H}^s_p(\mathcal{C})$, for $s> 1/p$, $11/p$ and $1<p<\infty$ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting $\mathbb{H}^1(\mathcal{C})$ for arbitrary complex values of the nonzero constant in the impedance condition.publishe

A nonhydrostatic unstructured-mesh soundproof model for simulation of internal gravity waves

A semi-implicit edge-based unstructured-mesh model is developed that integrates nonhydrostatic soundproof equations, inclusive of anelastic and pseudo-incompressible systems of partial differential equations. The model builds on nonoscillatory forward-in-time MPDATA approach using finite-volume discretization and unstructured meshes with arbitrarily shaped cells. Implicit treatment of gravity waves benefits both accuracy and stability of the model. The unstructured-mesh solutions are compared to equivalent structured-grid results for intricate, multiscale internal-wave phenomenon of a non-Boussinesq amplification and breaking of deep stratospheric gravity waves. The departures of the anelastic and pseudo-incompressible results are quantified in reference to a recent asymptotic theory [Achatz et al. 2010, J. Fluid Mech., 663, 120-147)]

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

MULTI-VARIABLE GOULD-HOPPER AND LAGUERRE POLYNOMIALS

The monomiality principle was introduced by G. Dattoli, in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. In this article we show a general technique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite) are derived.The monomiality principle was introduced by G. Dattoli, in order toderive the properties of special or generalized polynomials starting fromthe corresponding ones of monomials. In this article we show a generaltechnique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite) are derived

An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /