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Property-Preserving Numerical Schemes for Conservation Laws cover
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High-order numerical methods for hyperbolic conservation laws do not guarantee the validity of constraints that physically meaningful approximations are supposed to satisfy. The finite volume and finite element schemes summarized in this book use limiting techniques to enforce discrete maximum principles and entropy inequalities. Spurious oscillations are prevented using artificial viscosity operators and/or essentially nonoscillatory reconstructions.

An introduction to classical nonlinear stabilization approaches is given in the simple context of one-dimensional finite volume discretizations. Subsequent chapters of Part I are focused on recent extensions to continuous and discontinuous Galerkin methods. Many of the algorithms presented in these chapters were developed by the authors and their collaborators. Part II gives a deeper insight into the mathematical theory of property-preserving numerical schemes. It begins with a review of the convergence theory for finite volume methods and ends with analysis of algebraic flux correction schemes for finite elements. In addition to providing ready-to-use algorithms, this text explains the design principles behind such algorithms and shows how to put theory into practice. Although the book is based on lecture notes written for an advanced graduate-level course, it is also aimed at senior researchers who develop and analyze numerical methods for hyperbolic problems.

Sample Chapter(s)
Preface
Chapter 1: Introduction and motivation

Contents:

  • Algorithms:
    • Introduction and Motivation
    • Numerical Methods for 1D Hyperbolic Problems
    • Edge-Based Flux Correction for Continuous Galerkin Methods
    • Element-Based Algorithms for Continuous Galerkin Methods
    • Flux and Slope Limiting for Discontinuous Galerkin Methods
    • High-Order Finite Elements and Time Discretizations
  • Theory:
    • Analysis of Finite Volume Methods for Hyperbolic Problems
    • Analysis of Finite Element Schemes for Hyperbolic Problems

Readership: Instructors, advanced graduate students, researchers in the field of numerical methods for conservation laws, practitioners in the field of computational fluid dynamics.

Free Access
FRONT MATTER
  • Pages:i–xx

https://doi.org/10.1142/9789811278198_fmatter

Preview Abstract

The following sections are included:

  • Dedication
  • Preface
  • List of acronyms
  • Contents

Algorithms

Free Access
Chapter 1: Introduction and motivation
  • Pages:3–9

https://doi.org/10.1142/9789811278198_0001

Preview Abstract

Many mathematical models of continuum mechanics are derived from integral conservation laws. Examples of such models include the Euler and Navier–Stokes equations of fluid mechanics, as well as the shallow water equations and scalar convection-diffusion-reaction equations. The main focus of this book is on hyperbolic initial value problems of the form

∂u∂t+∇⋅f(u)=0  inΩ×ℝ+,(1.1a)
u(⋅,0)=u0inΩ,(1.1b)
where Ω ⊂ ℝd, d ∈ {1, 2, 3} is a bounded domain. Unless mentioned otherwise, we impose periodic boundary conditions on ∂Ω…

No Access
Chapter 2: Numerical methods for 1D hyperbolic problems
  • Pages:11–66

https://doi.org/10.1142/9789811278198_0002

Preview Abstract

To introduce the basic ideas in a simple setting, we begin with a review of numerical methods for the one-dimensional hyperbolic problem

∂u∂t+∂f(u)∂x=0  in ℝ×ℝ+,(2.1a)
u(⋅,0)=u0,(2.1b)
where f(u) is a possibly nonlinear function of a scalar conserved quantity u(x, t) ∈ ℝ. Instead of giving an in-depth introduction to finite volume methods for problems of the form (2.1), we focus on algorithms that can be extended to finite element schemes and general meshes…

No Access
Chapter 3: Edge-based flux correction for continuous Galerkin methods
  • Pages:67–148

https://doi.org/10.1142/9789811278198_0003

Preview Abstract

Numerical solution of multidimensional problems in complex geometries requires the use of more sophisticated numerical algorithms than finite volume schemes for 1D problems and straightforward extensions based on operator splitting. In this chapter, we discretize the governing equations in space using linear (ℙ1) or multilinear (ℚ1) finite elements on meshes that can be structured or unstructured. In contrast to finite volume schemes, there is no direct way to manipulate a continuous Galerkin (CG) space discretization by choosing another numerical flux or applying a slope limiter. However, discrete maximum principles (DMP) and entropy stability conditions can be enforced using algebraic flux correction (AFC) tools [264, 286]…

No Access
Chapter 4: Element-based algorithms for continuous Galerkin methods
  • Pages:149–183

https://doi.org/10.1142/9789811278198_0004

Preview Abstract

Most implementations of continuous Galerkin methods in finite element libraries use element-based data structures and assemble algebraic systems element-by-element. The use of numerical fluxes associated with pairs of neighboring nodes or cells is more common for finite volume and discontinuous Galerkin methods. The algebraic flux correction schemes presented in Chap. 3 modify global matrices, residuals or right-hand sides of the baseline discretization in a black-box manner, which makes them easy to implement in an existing code. However, preferred high- and/or low-order stabilization techniques might produce element vectors rather than fluxes. If such element contributions can be decomposed into fluxes, then edge-based limiting techniques are applicable. On the other hand, it is possible to modify element vectors directly in a way that preserves the sums of their components and thereby provides the discrete conservation property. In fact, the classical FEM-FCT schemes introduced by Löhner et al. [323, 324] were designed to limit element contributions. Edge-based AFC tools became more popular in the 1990s [17, 331, 343, 384, 413] but an element-based limiting strategy might still be a better choice for certain purposes…

No Access
Chapter 5: Flux and slope limiting for discontinuous Galerkin methods
  • Pages:185–213

https://doi.org/10.1142/9789811278198_0005

Preview Abstract

Many finite element methods for hyperbolic problems are based on discontinuous Galerkin (DG) approximations. Attractive features of DG approaches include the local conservation property and a great flexibility in the choice of limiting strategies. The algorithms proposed in [14,20,182,382] enforce preservation of local bounds using generalizations of the algebraic flux correction (AFC) tools presented in Chap. 3 and 4. More commonly, generalizations of flux and/or slope limiters for finite volume schemes (as presented in Chap. 2) are employed in the DG setting [272, 364, 473]…

No Access
Chapter 6: High-order finite elements and time discretizations
  • Pages:215–249

https://doi.org/10.1142/9789811278198_0006

Preview Abstract

The design of accuracy-preserving limiting techniques for high-order finite element discretizations requires careful revisions of algorithms developed for ℙ1/ℚ1 elements. Straightforward extensions to polynomials of degree p > 1 may fail to provide the desired properties. Moreover, higher accuracy can often be achieved using a piecewise-polynomial approximation of degree p = 1 with the same total number of degrees of freedom (DoFs)…

Theory

No Access
Chapter 7: Analysis of finite volume methods for hyperbolic problems
  • Pages:253–361

https://doi.org/10.1142/9789811278198_0007

Preview Abstract

In this chapter, we review theoretical foundations of finite volume schemes for hyperbolic conservation laws. So far, our presentation was focused on the design of numerical algorithms that possess certain properties. Some sufficient conditions for stability, positivity preservation, and validity of discrete maximum principles were formulated as theorems. The aspects of existence, uniqueness, and convergence received little attention in previous chapters. In Part II of this book, we present the theory that remained behind the scenes in Part I. This theory provides a deeper insight into the design criteria on which numerical methods for hyperbolic PDEs are based. Moreover, the results of detailed theoretical studies make it possible to develop more accurate and more reliable numerical approximations.

No Access
Chapter 8: Analysis of finite element schemes for hyperbolic problems
  • Pages:363–429

https://doi.org/10.1142/9789811278198_0008

Preview Abstract

Numerical analysis of finite element methods for linear elliptic and parabolic problems is a well-explored research field, which is comprehensively covered in many textbooks (see, e.g, [247, 301, 391, 436]). Theoretical investigations of FEM-based approaches for hyperbolic and convection-dominated transport problems are more challenging, especially in the nonlinear case. In this chapter, we focus on theoretical foundations of (stabilized) continuous Galerkin methods for scalar hyperbolic conservation laws. The preliminary analysis of algebraic flux correction schemes in Chap. 3 is extended to implicit time stepping and to stationary problems. The topics to be discussed include criteria for well-posedness of continuous and discrete problems, proofs of discrete maximum principles, and derivation of a priori error estimates under worst-case assumptions. Our review is mainly based on the text by Lohmann [319], which in turn frequently refers to Di Pietro and Ern [114] and Barrenechea et al. [24, 26, 27, 29]. We recommend these publications, as well as [28, 72, 200, 264, 301, 398], for further reading.

Free Access
BACK MATTER
  • Pages:431–470

https://doi.org/10.1142/9789811278198_bmatter

Preview Abstract

The following sections are included:

  • Bibliography
  • Index

Dmitri Kuzmin is a professor at TU Dortmund University and a well-known expert in the field of limiting techniques for finite element methods. His work is published in more than one hundred twenty research articles and three books including this one.

 

Hennes Hajduk is a postdoctoral associate. He received his PhD degree and the Best Thesis Award from the TU Dortmund University in 2022. His professional experience includes working at the Lawrence Livermore National Laboratory as an intern and visiting researcher.

Supplementary Materials (733 KB)

Sample Chapter(s)
Preface
Chapter 1: Introduction and motivation