Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

A hybridized discontinuous Galerkin method for nonlinear dispersive water waves

Simulation of water waves near the coast is an important problem in different branches of engineering and mathematics. For mathematical models to be valid in this region, they should include nonlinear and dispersive properties of the corresponding waves. Here, we study the numerical solution to three equations for modeling coastal water waves using the hybridized discontinuous Galerkin method (HDG). HDG is known to be a more efficient and in certain cases a more accurate alternative to some other discontinuous Galerkin methods, such as local DG. The first equation that we solve here is the Korteweg-de Vries equation. Similar to common HDG implementations, we first express the approximate variables and numerical fluxes in each element in terms of the approximate traces of the scalar variable, and its first derivative. These traces are assumed to be single-valued on each face. We next impose the conservation of the numerical fluxes via two sets of equations on the element boundaries. We solve this equation by Newton-Raphson method. We prove the stability of the proposed method for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with kth order elements, the computed variable and its first and second derivatives show optimal convergence at order k + 1 in both linear and nonlinear cases, which improves upon previously employed techniques. Next, we consider solving the fully nonlinear irrotational Green-Naghdi equation. This equation is often used to simulate water waves close to the shore, where there are significant dispersive and nonlinear effects involved. To solve this equation, we use an operator splitting method to decompose the problem into a dispersive part and a hyperbolic part. The dispersive part involves an implicit step, which has regularizing effects on the solution of the problem. On the other hand, for the hyperbolic sub-problem, we use an explicit hybridized DG method. Unlike the more common implicit version of the HDG, here we start by solving the flux conservation condition for the numerical traces. Afterwards, we use these traces in the original PDEs to obtain the internal unknowns. This process involves Newton iterations at each time step for computing the numerical traces. Next, we couple this solver with the dispersive solver to obtain the solution to the Green-Naghdi equation. We then solve a set of numerical examples to verify and validate the employed technique. In the first example we show the convergence properties of the numerical method. Next, we compare our results with a set of experimental data for nonlinear water waves in different situations. We observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data.Engineering Mechanic

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

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