Spectral Discontinuous Galerkin Method for Hyperbolic Problems

In this project we address the numerical approximation of hyperbolic equations and systems using the discontinuous Galerkin (DG) method in combination with higher order polynomial degrees. In short, this is called Spectral Discontinuous Galerkin (SDG) method. Our interest is to review the theoretical properties of the SDG-method, particularly for what concerns stability, convergence, dissipation and dispersion. Special emphases will be shed on the role of the two parameters,(the grid-size) and (the local polynomial degree). In this respect, we will carefully analyse the available theoretical results from the literature, then we extend some of them and implement several test cases with the purpose of assessing quantitatively the predicted theoretical properties

Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering

Well-balanced rr-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations

In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the $r$-adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems

On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added, shortened proo