Theodoros Katsaounis

Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.

High Frequency limit of the Helmholtz Equations

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.

Upwinding Sources at Interfaces in conservation laws.

Abstract Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical flows because of the gravity, and their numerical approximation leads to specific difficulties. In the context of finite-volume schemes, many authors have proposed to upwind sources at interfaces, the U.S.I. method, while a cell-centered treatment seems more natural. This note gives a general mathematical formalism for such schemes. We define consistency and give a stability condition for the U.S.I. method. We relate the notion of consistency to the “well-balanced” property, but its stability remains open, and we also study second-order approximations, as well as error estimates. The general case of a nonuniform spatial mesh is particularly interesting, motivated by two-dimensional problems set on unstructured grids.

A Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

Approximations to solutions of the inhomogeneous boundary value problem for the Navier-Stokes equations are constructed via the discontinuous Galerkin method. The velocity field is approximated using piecewise polynomial functions that are totally discontinuous across interelement boundaries and which are pointwise divergence-free on each element (locally solenoidal). The pressure is approximated by standard continuous piecewise polynomial functions.

Finite volume relaxation schemes for multidimensional conservation laws

We consider finite volume relaxation schemes for multidimensional scalar conservation laws. These schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1/4 in L∞([0,T],L loc 1 (R d )).

COMPUTATION OF HIGH FREQUENCY FIELDS NEAR CAUSTICS

It is well known that although the usual harmonic ansatz of geometrical optics fails near a caustic, uniform expansions can be found which remain valid in the neighborhood of the caustic, and reduce asymptotically to the usual geometric field far enough from it. Such expansions can be constructed by several methods which make essentially use of the symplectic structure of the phase space. In this paper we efficiently apply the Kravtsov–Ludwig method of relevant functions, in conjunction with Hamiltonian ray tracing to define the topology of the caustics and compute high-frequency scalar wave fields near smooth and cusp caustics. We use an adaptive Runge–Kutta method to successfully retrieve the complete ray field in the case of piecewise smooth refraction indices. We efficiently match the geometric and modified amplitudes of the multi-valued field to obtain numerically the correct asymptotic behavior of the solution. Comparisons of the numerical results with analytical calculations in model problems show excellent accuracy in calculating the modified amplitudes using the Kravtsov–Ludwig formulas.

A MODIFIED STRUCTURED CENTRAL SCHEME FOR 2D HYPERBOLIC CONSERVATION LAWS

Abstract We present a new central scheme for approximating solutions of two-dimensional systems of hyperbolic conservation laws. This method is based on a modification of the staggered grid proposed in [1] which prevents the crossings of discontinuities in the normal direction, while retaining the simplicity of the central framework. Our method satisfies a local maximum principle which is based on a more compact stencil. Unlike the previous method, it enables a natural extension to adaptive methods on structured grids.

Dispersive wave runup on non-uniform shores

Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically a representative of Boussinesq type equations in view of important applications to the coastal hydrodynamics. Numerical results of the runup of a moderate wave onto a non-uniform beach are presented along with great lines of the employed numerical method (see D. Dutykh et al. (2011) [6] for more details).

Effective Equations for Localization and Shear Band Formation

We develop a quantitative criterion determining the onset of localization and shear band formation at high strain-rate deformations of metals. We introduce an asymptotic procedure motivated by the theory of relaxation and the Chapman–Enskog expansion and derive an effective equation for the evolution of the strain rate, consisting of a second order nonlinear diffusion regularized by fourth order effects and with parameters determined by the degree of thermal softening, strain hardening, and strain-rate sensitivity. The nonlinear diffusion equation changes type across a threshold in the parameter space from forward parabolic to backward parabolic, what highlights the stable and unstable parameter regimes. The fourth order effects play a regularizing role in the unstable region of the parameter range.