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Nodal Discontinuous Galerkin Methods

Algorithms, Analysis, and Applications

verfasst von: Jan S. Hesthaven, Tim Warburton

Verlag: Springer New York

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mathematicsisplayinganevermoreimportantroleinthephysicalandbiol- ical sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and tea- ing, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose ofthistextbookseriesistomeetthecurrentandfutureneedsoftheseadvances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface The algorithms, methods, and Matlab implementations described in this text have been developed during almost a decade of collaboration. During this time we have worked to simplify the basic methods and make the ideas more accessible to a broad audience. Many people, both students and colleagues, have helped during the development of this project and we are grateful for all their suggestions and input.

Inhaltsverzeichnis

Frontmatter

1. Introduction

When faced with the task of solving a partial differential equation computationally, one quickly realizes that there is quite a number of different methods for doing so. Among these are the widely used finite difference, finite element, and finite volume methods, which are all techniques used to derive discrete representations of the spatial derivative operators. If one also needs to advance the equations in time, there is likewise a wide variety of methods for the integration of systems of ordinary differential equations available to choose among. With such a variety of successful and well tested methods, one is tempted to ask why there is a need to consider yet another method.

2. The key ideas

While we initially strive to stay away from complex notation a few fundamentals are needed. We consider problems posed on the physical domain Ω with boundary ∂Ω and assume that this domain is well approximated by the computational domain Ω_h. This is a space filling triangulation composed of a collection of K geometry-conforming nonoverlapping elements, D^k. The shape of these elements can be arbitrary although we will mostly consider cases where they are d-dimensional simplexes.

3. Making it work in one dimension

As simple as the formulations in the last chapter appear, there is often a leap between mathematical formulations and an actual implementation of the algorithms. This is particularly true when one considers important issues such as efficiency, flexibility, and robustness of the resulting methods.

In this chapter we address these issues by first discussing details such as the form of the local basis and, subsequently, how one implements the nodal DG-FEMs in a flexible way. To keep things simple, we continue the emphasis on one-dimensional linear problems, although this results in a few apparently unnecessarily complex constructions. We ask the reader to bear with us, as this slightly more general approach will pay off when we begin to consider more complex nonlinear and/or higher-dimensional problems.

4. Insight through theory

While the last chapters have focused on basic ideas and their implementation as a computational method, further insight into the performance of the method can be gained by revisiting some issues in more detail. To keep things relatively simple, we focus on the properties of the scheme for one-dimensional linear problems. However, as we will see later, many of the results obtained here carry over to multidimensional problems and even nonlinear problems with just a few modifications.

5. Nonlinear problems

So far, we have focused entirely on linear problems with constant or piecewise constant coefficients. As we have seen, the methods and analysis for these cases is relatively complete.

In this chapter we expand the discussion to include more complex problems — in particular, problems with smoothly varying coefficients and genuinely nonlinear problems. As we will see, this introduces new elements that need attention, and the analysis of the methods for such problems is more complex. In fact, we will often not attempt to give a complete analysis but merely outline the key results. However, the extension to strongly nonlinear problems displays many unique features and the power and robustness of the discontinuous Galerkin methods.

6. Beyond one dimension

We have so far focused almost entirely on the formulation, analysis, and implementation of the discontinuous Galerkin schemes (DG-FEM) for onedimensional problems. In this chapter we consider in more detail the extension to multidimensional problems and the challenges introduced by this. We will quickly realize that what may have seemed unnecessarily complicated in the one-dimensional case now enables us to expand the formulation to multiple dimensions with only minor changes.

7. Higher-order equations

So far, we have only considered problems with first order spatial derivatives (e.g., as in conservation laws), and shown the methods to perform well and in agreement with the strong theoretical foundation. It is natural to ask whether one can extend the formulation to include more general problem types. This is the topic of this chapter and, as we will see shortly, the generalization to deal with higher-order spatial operators is less direct than one would expect. Let us consider the following simple example, taken from [288].

8. Spectral properties of discontinuous Galerkin operators

In the preceding chapters, we have developed schemes and discussed basic properties such as stability and convergence. We have not, however, discussed the spectral properties of the operators apart from the discussion of the numerical dispersion relations in Section 4.6. In that discussion it became clear that the eigenvalues of the operators play a key role when determining the properties of the discrete operators and, thus, the behavior of the scheme.

9. Curvilinear elements and nonconforming discretizations

All previous chapters have focused almost exclusively on the simplest cases of geometric discretizations where all boundaries are assumed to be piecewise linear and all elements share faces of equal size and order of approximation. However, one of the major advantages of discontinuous Galerkin (DG) methods lies in their flexibility to go beyond these cases and support more complex situation.

In this chapter we discuss the modifications required to extend the linear conforming elements to include the treatment of meshes containing curvilinear elements and/or non-conforming elements. As we will see, the required changes are limited, but the advantages of doing so can be dramatic in terms of improvements in accuracy and reductions in computational effort.

10. Into the third dimension

In Chapter 6 we demonstrated the extension of the discontinuous Galerkin (DG) methods to problems in two spatial dimensions. This involved the introduction of nodal sets for the triangle and an orthonormal polynomial basis that we used as a reference basis for interpolation, differentiation, and the computation of inner products. In this chapter we will go further and consider the additional details required to extend this approach to three-dimensional domains.

Backmatter

Titel: Nodal Discontinuous Galerkin Methods
verfasst von: Jan S. Hesthaven
Tim Warburton
Verlag: Springer New York
Electronic ISBN: 978-0-387-72067-8
Print ISBN: 978-0-387-72065-4
DOI: https://doi.org/10.1007/978-0-387-72067-8

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