461 research outputs found
A Polynomial Spectral Calculus for Analysis of DG Spectral Element Methods
We introduce a polynomial spectral calculus that follows from the summation
by parts property of the Legendre-Gauss-Lobatto quadrature. We use the calculus
to simplify the analysis of two multidimensional discontinuous Galerkin
spectral element approximations
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Discontinuous Galerkin (DG) methods have a long history in computational
physics and engineering to approximate solutions of partial differential
equations due to their high-order accuracy and geometric flexibility. However,
DG is not perfect and there remain some issues. Concerning robustness, DG has
undergone an extensive transformation over the past seven years into its modern
form that provides statements on solution boundedness for linear and nonlinear
problems.
This chapter takes a constructive approach to introduce a modern incarnation
of the DG spectral element method for the compressible Navier-Stokes equations
in a three-dimensional curvilinear context. The groundwork of the numerical
scheme comes from classic principles of spectral methods including polynomial
approximations and Gauss-type quadratures. We identify aliasing as one
underlying cause of the robustness issues for classical DG spectral methods.
Removing said aliasing errors requires a particular differentiation matrix and
careful discretization of the advective flux terms in the governing equations.Comment: 85 pages, 2 figures, book chapte
On Multiscale Methods in Petrov-Galerkin formulation
In this work we investigate the advantages of multiscale methods in
Petrov-Galerkin (PG) formulation in a general framework. The framework is based
on a localized orthogonal decomposition of a high dimensional solution space
into a low dimensional multiscale space with good approximation properties and
a high dimensional remainder space{, which only contains negligible fine scale
information}. The multiscale space can then be used to obtain accurate Galerkin
approximations. As a model problem we consider the Poisson equation. We prove
that a Petrov-Galerkin formulation does not suffer from a significant loss of
accuracy, and still preserve the convergence order of the original multiscale
method. We also prove inf-sup stability of a PG Continuous and a Discontinuous
Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the
Petrov-Galerkin method can decrease the computational complexity significantly,
allowing for more efficient solution algorithms. As another application of the
framework, we show how the Petrov-Galerkin framework can be used to construct a
locally mass conservative solver for two-phase flow simulation that employs the
Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous
Galerkin Finite Element method with an upwind scheme for a hyperbolic
conservation law
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
- …