74 research outputs found
A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Korteweg-de Vries Equation
We construct, analyze and numerically validate a posteriori error estimates for conservative discontinuous Galerkin (DG) schemes for the Generalized Korteweg-de Vries (GKdV) equation. We develop the concept of dispersive reconstruction, i.e., a piecewise polynomial function which satisfies the GKdV equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error. Both semidiscrete and fully discrete approximations are treated
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A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Korteweg-de Vries Equation
We construct, analyze and numerically validate a posteriori error estimates for conservative discontinuous Galerkin (DG) schemes for the Generalized Korteweg-de Vries (GKdV) equation. We develop the concept of dispersive reconstruction, i.e., a piecewise polynomial function which satisfies the GKdV equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error. Both semidiscrete and fully discrete approximations are treated
The design of conservative finite element discretisations for the vectorial modified KdV equation
We design a consistent Galerkin scheme for the approximation of the vectorial
modified Korteweg-de Vries equation. We demonstrate that the scheme conserves
energy up to machine precision. In this sense the method is consistent with the
energy balance of the continuous system. This energy balance ensures there is
no numerical dissipation allowing for extremely accurate long time simulations
free from numerical artifacts. Various numerical experiments are shown
demonstrating the asymptotic convergence of the method with respect to the
discretisation parameters. Some simulations are also presented that correctly
capture the unusual interactions between solitons in the vectorial setting
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
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