8 research outputs found
A conservative fully-discrete numerical method for the regularised shallow water wave equations
The paper proposes a new, conservative fully-discrete scheme for the
numerical solution of the regularised shallow water Boussinesq system of
equations in the cases of periodic and reflective boundary conditions. The
particular system is one of a class of equations derived recently and can be
used in practical simulations to describe the propagation of weakly nonlinear
and weakly dispersive long water waves, such as tsunamis. Studies of
small-amplitude long waves usually require long-time simulations in order to
investigate scenarios such as the overtaking collision of two solitary waves or
the propagation of transoceanic tsunamis. For long-time simulations of
non-dissipative waves such as solitary waves, the preservation of the total
energy by the numerical method can be crucial in the quality of the
approximation. The new conservative fully-discrete method consists of a
Galerkin finite element method for spatial semidiscretisation and an explicit
relaxation Runge--Kutta scheme for integration in time. The Galerkin method is
expressed and implemented in the framework of mixed finite element methods. The
paper provides an extended experimental study of the accuracy and convergence
properties of the new numerical method. The experiments reveal a new
convergence pattern compared to standard Galerkin methods
Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation
In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme
Recommended from our members
Nonlocal models with a finite range of nonlocal interactions
Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations.
To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described