##
**Spectral/\(hp\) element methods for computational fluid dynamics.
2nd ed.**
*(English)*
Zbl 1116.76002

Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 0-19-852869-8/hbk). xviii, 657 p. (2005).

There has been significant progress in the development of multi-domain spectral methods both at the fundamental as well as at the application level in the last few years. Thus the ‘nontrivial’ task of updating the book in order to include these new developments. The authors also wanted to make these methods easier to comprehend and implement by students, responding directly to many requests and feedback they received after the publication of the first edition.

The book is the revised version of the first edition published in 1997 [see for the review Zbl 0954.76001]. The motivation for the revision are new developments which are primarily in the discontinuous Galerkin methods, in non-tensorial nodal spectral element methods in complex domains, and in stabilization and filtering techniques. From practical print of view, high-order solutions in complex geometries require high-order meshes and high-order post-processing, a subject that is often neglected in everyday ‘production’ computing and simulation. Such subjects are now addressed in detail in this new edition of the book. The authors had also seen the spectral/\(hp\) element method applied to less traditional fields, such as aerodynamics, seismology, climate modelling, and magneto-hydrodynamics and they included some elements of modelling such applications in this revised edition. Another objective for revision has been to provide more details on implementing various aspects of the method. In order to meet this end, the authors have put some emphasis on implementation and technical issues with exercises in the founding Chapters 2–5 to aid in implementing basic spectral element solvers, which can be used as building blocks for more complex application codes.

Two distinct margin identifiers have been introduced in order to highlight formulation and implementation details. The book has been increased in material by almost 50%.

The first chapter presents reduced models of compressible and incompressible Navier-Stokes equations which are used in various discretizations discussed in the following chapters. The convergence philosophy of spectral and finite element methods, the combination of which provides a dual path of convergence, is also introduced.

In Chapter 2 fundamental concepts behind design and implementation of the spectral/\(hp\) element method for one-dimensional linear elliptic problems are illustrated. The basic mechanics for this formulation is helpful to illustrate useful techniques for a variety of different types of mathematical problems, such as hyperbolic and parabolic equations, as well as different types of formulations such as the discontinuous Galerkin formulation discussed in Chapter 6, and provide basis for understanding the multidimensional formulation discussed in Chapter 3 and 4.

General framework of different formulations in the context of the method of weighted residuals is introduced in Section 2.3 where the authors discuss \(h\)-type elemental decomposition from a global expansion and then nodal \(p\)-type polynomial expansion within each elemental region. In Section 2.4 principal elemental operations of numerical integration and differentiation are detailed. In this new edition in Section 2.2.1.2 is discussed Galerkin implementation of Neumann boundary conditions through the weak form of Galerkin problem, and the enforcement of Dirichlet boundary conditions through homogenization of the solution in Subsection 2.2.1.3. Also new sections on nodal \(p\)-type expansion and on integration errors and polynomial aliasing have been added. In Section 2.4.1.2 the effect of under-integration of \(n\times n\)-linear products of the polynomial solution is discussed, which is important when considering the nonlinear advection terms of Navier-Stokes equations. Finally, in Section 2.5 the basic formulations and error estimation results associated with one-dimensional spectral/\(hp\) element methods are outlined.

In Chapter 3 the authors consider the extension of the one-dimensional formulation to two and three dimensions by the development of expansion basis in standard regions such as triangles or rectangles in two dimensions, and tetrahedrons, prisms, pyramids, and hexahedrons in three dimensions. The construction of these bases uses a unified approach which permits the development of computationally efficient expansions. In this new revision the modal basis is formulated as solutions to a generalized Sturm-Liouville problem. Optimal nodal points, the so-called Fekete points, as well as electrostatic points on a simplex are presented, and related approximation results are included. In Section 3.1 a comprehensive discussion of the tensorial extension for all hybrid regions is given, and also underlying concepts, which will be helpful when constructing a tensorial basis for the unstructured region, are introduced. The most commonly used spectral/\(hp\) element bases are those which can be expanded into a globally \(C^0\) continuous expansion. This expansion makes use of the collapsed co-ordinate system. Finally, two non-tensorial nodal sets of points in a triangular region, compatible with the nodal quadrilateral expansion, are introduced and defined in Sections 3.3.3 and 3.3.4 as well as in Appendix D.

Chapter 4 has been reconstructed with extra emphasis on implementation aspects. Herein it is explained how two- and three-dimensional expansions developed in Chapter 3 can be extended to a tesselation of multiple domains. These extensions are decomposed into three sections: local operations such as integration and differentiation; global operations such as the construction of global matrix systems; and pre- and post-processing aspects of general multidimensional solver such as boundary condition representation, curvilinear mesh generation, and consistent particle tracking in high-order elements. Herein is introduced a matrix formulation to help illustrate the algebraic systems which need to be constructed when computationally implementing the spectral/\(hp\) method. Formulations of both Galerkin and collocation projections are considered in this manner. Matrix construction is convenient to clarify many of the numerical operations; however, when using tensorial-based operations, it is computationally more efficient to use the sum-factorization technique. Many of the local operations are also relevant to non-tensorial expansion bases for simplexes. When using non-tensorial basis, the sum-factorization technique cannot be used, and so matrix operators are applied. In the revised version Chapter 5 now considers diffusion equation; an implicit discretisation leads to the Helmholtz equation. This chapter discusses both the temporal discretization and eigenspectra of second-order operators that dictate timestep restrictions. Appropriate preconditioning techniques for inversion of the stiffness matrix are dealt with. The first four sections of this chapter consider solutions which are sufficiently smooth. However, elliptic problems, which can be seen as the steady state of the parabolic diffusion problem, may contain non-smooth solutions, due to singularities. Convergence of the spectral/\(hp\) method for domains with corner singularities is analysed herein, and possible ways of restoring high accuracy are suggested.

In Chapter 6 the focus is on scalar advection equation and development of Galerkin discretization using the techniques described in Chapter 4. Extended presentation of the discontinuous Galerkin formulation is also included. Eigenspectra of the advection operators in both two and three dimensions which are relevant for explicit time stepping are also reviewed. A further new addition is the discussion on two forms of a semi-Lagrangian method for advection (strong and auxiliary forms) that could potentially prove to be very effective in enhancing the speed and accuracy of spectral/\(hp\) element methods in advection-dominated problem. In Section 6.5 discontinuous solutions are considered and issues of monotonicity are addressed. It is demonstrated how filtering, artificial viscosity, super-collocation and upwind nodal distribution can be used to control high-frequency oscillations. The key issue here is how to maintain monotonicity and thus stability without sacrificing the exponential convergence of the method. These are very important issues for shock wave dynamics and are revisited in Chapter 10.

The formulation presented so far for multidimensional space (Chapter 4 and 5) deals with conformity of elements where vertices of adjoining elements coincide, and correspondingly a \(C^0\)-functional condition is satisfied at the elemental interfaces. In Chapter 7, as in Chapter 5, the authors consider again second-order spatial operators, but allow for non-conforming discretizations. That is, it is no longer required that the vertices of adjoining elements coincide. Instead, a framework is developed that allows for arbitrary connections between elements.

In the first part of the chapter two formulations are introduced that employ geometrically non-conforming elements but which maintain \(C^0\) continuity of the global polynomial expansion: referred to as the iterative patching and constrained approximation. In the second part the mortar element and the discontinuous Galerkin method for second-order elliptic and parabolic problems are introduced. In these cases the \(C^0\) continuity is no longer imposed, and new weak forms of the problem are developed. Some interesting possibilities exist with the discontinuous Galerkin method for second-order elliptic and parabolic problems, and particular attention is paid to this formulation.

In Chapter 8 the authors discuss algorithms for incompressible flows appropriate for direct numerical (DNS) and large eddy simulations (LES) in complex-geometry domains. They are primarily interested in those formulations which are extendable to three dimensions, and therefore they focus on primitive variables (velocity-pressure), velocity-vorticity, and the gauge formulations. Both coupled, splitting and least-square formulations for primitive variables are discussed. In this revision both Uzawa coupled algorithm and a new substructured solver are considered. The discussion on primitive variables time-splitting has been rewritten to include recent theoretical advances in pressure-connection and velocity-correction schemes as well as the rotational formulations of the pressure boundary condition. The discretization of the nonlinear terms in the Navier-Stokes formulation both in space as well as time using semi-Lagrangian formulation has been discussed. References also have been given for other formulations for general discretizations, i.e. a stream function-vorticity formulation of the unstructured spectral/\(hp\) element method.

Chapter 9 presents several examples of incompressible flow simulations. First, a series of relatively simple laminar flows which have exact solutoins and thus can be considered as benchmark problems are taken up. These solutions are useful for verification purposes, i.e. to ensure that the spectral/\(hp\) element discretisation is applied and the associated software is implemented correctly. Biglobal instability of small amplitude disturbances superimposed on two-dimensional flow states is also considered. This type of analysis can aid in understanding the onset of three-dimensional transition to turbulence. Further, the authors discuss numerical issues in DNS, including diagnostics for under-resolution. Specifically, they include a case study of turbulent channel flow which has been extensively simulated with global spectral methods. Subsequently, the main issues in LES with focus on high-order discretization, are summarised. The results here are compared with available experimental data and serve to validate the LES models. The chapter is concluded by considering computing mode that involves resolution steering. To meet this end, two adaptive procedures based on error estimates as well as other heuristics, and a parallel paradigm suitable for dynamic \(p\)-refinement, are presented.

In Chapter 10, the important issues of conservativity and monotonicity are revisited, thus extending the material presented earlier in Section 6.5. The emphasis in this chapter is primarily on systems of hyperbolic conservation laws, which are solved in conservative form employing characteristic decomposition. This is an area in which high-order methods have had little success in the past. The principal issue is how to effectively use the high-order expansions of spectral/\(hp\) methods while honouring the inherent monotonicity and conservation properties of analytic system. Different ways of dealing with these fundamental issues for both the compressible Euler and Navier-Stokes equations are considered. The issue of boundary conditions is crucial in wave propagation phenomena and is often misunderstood for compressible flows and magneto-hydrodynamics. To this end the authors discuss this issue in many different formulations that are presented, including a flux-corrected transport approach, a discontinuous Galerkin method and penalty formulation.

Spectral and multi-domain spectral methods are particularly effective in modelling propagation phenomena governed by hyperbolic conservation laws. For example, in atmospheric and ocean and seismic modelling, resolution of high frequencies and high wave is crucial, especially for long term forecasting.

In Section 10.5 Navier-Stokes equations are considered and appropriate solution algorithms are formulated. Thereafter shock-fitting techniques appropriate for high Mach number flows are discussed in Section 10.6. The last section discusses modelling of plasma flows and ocean modelling. Spectral methods have inherently small numerical dispersion and superb resolution properties, requiring typically four to six points (or modes) per wavelength. In addition, spectral elements provide several computational advantages, including nearest-neighbour communications and cache-blocked computations. This has been recognized by application scientists, and new scalable codes for large-scale simulations of atmospheric general circulation models and general seismology have been designed based on spectral element methods (see references 273, 308).

Appendices have been included which provide details on Jacobi and Askey polynomials as well as numerical integration and differentiation which are essential building blocks of spectral/\(hp\) element techniques. A full description of commonly used expansion bases is provided, which now includes nodal points for non-tensorial electrostatic problems, and Fekete point distribution in simplex domains, The final appendix also details Riemann solvers commonly used in solution of Euler equations.

The book incorporates latest developments in the subject, is outstanding and of great interest to students, academics and practitioners in computational fluid mechanics, applied and numerical mathematics aerospace and mechanical engineering, magneto-hydrodynamics and climate/ocean modelling.

The book is the revised version of the first edition published in 1997 [see for the review Zbl 0954.76001]. The motivation for the revision are new developments which are primarily in the discontinuous Galerkin methods, in non-tensorial nodal spectral element methods in complex domains, and in stabilization and filtering techniques. From practical print of view, high-order solutions in complex geometries require high-order meshes and high-order post-processing, a subject that is often neglected in everyday ‘production’ computing and simulation. Such subjects are now addressed in detail in this new edition of the book. The authors had also seen the spectral/\(hp\) element method applied to less traditional fields, such as aerodynamics, seismology, climate modelling, and magneto-hydrodynamics and they included some elements of modelling such applications in this revised edition. Another objective for revision has been to provide more details on implementing various aspects of the method. In order to meet this end, the authors have put some emphasis on implementation and technical issues with exercises in the founding Chapters 2–5 to aid in implementing basic spectral element solvers, which can be used as building blocks for more complex application codes.

Two distinct margin identifiers have been introduced in order to highlight formulation and implementation details. The book has been increased in material by almost 50%.

The first chapter presents reduced models of compressible and incompressible Navier-Stokes equations which are used in various discretizations discussed in the following chapters. The convergence philosophy of spectral and finite element methods, the combination of which provides a dual path of convergence, is also introduced.

In Chapter 2 fundamental concepts behind design and implementation of the spectral/\(hp\) element method for one-dimensional linear elliptic problems are illustrated. The basic mechanics for this formulation is helpful to illustrate useful techniques for a variety of different types of mathematical problems, such as hyperbolic and parabolic equations, as well as different types of formulations such as the discontinuous Galerkin formulation discussed in Chapter 6, and provide basis for understanding the multidimensional formulation discussed in Chapter 3 and 4.

General framework of different formulations in the context of the method of weighted residuals is introduced in Section 2.3 where the authors discuss \(h\)-type elemental decomposition from a global expansion and then nodal \(p\)-type polynomial expansion within each elemental region. In Section 2.4 principal elemental operations of numerical integration and differentiation are detailed. In this new edition in Section 2.2.1.2 is discussed Galerkin implementation of Neumann boundary conditions through the weak form of Galerkin problem, and the enforcement of Dirichlet boundary conditions through homogenization of the solution in Subsection 2.2.1.3. Also new sections on nodal \(p\)-type expansion and on integration errors and polynomial aliasing have been added. In Section 2.4.1.2 the effect of under-integration of \(n\times n\)-linear products of the polynomial solution is discussed, which is important when considering the nonlinear advection terms of Navier-Stokes equations. Finally, in Section 2.5 the basic formulations and error estimation results associated with one-dimensional spectral/\(hp\) element methods are outlined.

In Chapter 3 the authors consider the extension of the one-dimensional formulation to two and three dimensions by the development of expansion basis in standard regions such as triangles or rectangles in two dimensions, and tetrahedrons, prisms, pyramids, and hexahedrons in three dimensions. The construction of these bases uses a unified approach which permits the development of computationally efficient expansions. In this new revision the modal basis is formulated as solutions to a generalized Sturm-Liouville problem. Optimal nodal points, the so-called Fekete points, as well as electrostatic points on a simplex are presented, and related approximation results are included. In Section 3.1 a comprehensive discussion of the tensorial extension for all hybrid regions is given, and also underlying concepts, which will be helpful when constructing a tensorial basis for the unstructured region, are introduced. The most commonly used spectral/\(hp\) element bases are those which can be expanded into a globally \(C^0\) continuous expansion. This expansion makes use of the collapsed co-ordinate system. Finally, two non-tensorial nodal sets of points in a triangular region, compatible with the nodal quadrilateral expansion, are introduced and defined in Sections 3.3.3 and 3.3.4 as well as in Appendix D.

Chapter 4 has been reconstructed with extra emphasis on implementation aspects. Herein it is explained how two- and three-dimensional expansions developed in Chapter 3 can be extended to a tesselation of multiple domains. These extensions are decomposed into three sections: local operations such as integration and differentiation; global operations such as the construction of global matrix systems; and pre- and post-processing aspects of general multidimensional solver such as boundary condition representation, curvilinear mesh generation, and consistent particle tracking in high-order elements. Herein is introduced a matrix formulation to help illustrate the algebraic systems which need to be constructed when computationally implementing the spectral/\(hp\) method. Formulations of both Galerkin and collocation projections are considered in this manner. Matrix construction is convenient to clarify many of the numerical operations; however, when using tensorial-based operations, it is computationally more efficient to use the sum-factorization technique. Many of the local operations are also relevant to non-tensorial expansion bases for simplexes. When using non-tensorial basis, the sum-factorization technique cannot be used, and so matrix operators are applied. In the revised version Chapter 5 now considers diffusion equation; an implicit discretisation leads to the Helmholtz equation. This chapter discusses both the temporal discretization and eigenspectra of second-order operators that dictate timestep restrictions. Appropriate preconditioning techniques for inversion of the stiffness matrix are dealt with. The first four sections of this chapter consider solutions which are sufficiently smooth. However, elliptic problems, which can be seen as the steady state of the parabolic diffusion problem, may contain non-smooth solutions, due to singularities. Convergence of the spectral/\(hp\) method for domains with corner singularities is analysed herein, and possible ways of restoring high accuracy are suggested.

In Chapter 6 the focus is on scalar advection equation and development of Galerkin discretization using the techniques described in Chapter 4. Extended presentation of the discontinuous Galerkin formulation is also included. Eigenspectra of the advection operators in both two and three dimensions which are relevant for explicit time stepping are also reviewed. A further new addition is the discussion on two forms of a semi-Lagrangian method for advection (strong and auxiliary forms) that could potentially prove to be very effective in enhancing the speed and accuracy of spectral/\(hp\) element methods in advection-dominated problem. In Section 6.5 discontinuous solutions are considered and issues of monotonicity are addressed. It is demonstrated how filtering, artificial viscosity, super-collocation and upwind nodal distribution can be used to control high-frequency oscillations. The key issue here is how to maintain monotonicity and thus stability without sacrificing the exponential convergence of the method. These are very important issues for shock wave dynamics and are revisited in Chapter 10.

The formulation presented so far for multidimensional space (Chapter 4 and 5) deals with conformity of elements where vertices of adjoining elements coincide, and correspondingly a \(C^0\)-functional condition is satisfied at the elemental interfaces. In Chapter 7, as in Chapter 5, the authors consider again second-order spatial operators, but allow for non-conforming discretizations. That is, it is no longer required that the vertices of adjoining elements coincide. Instead, a framework is developed that allows for arbitrary connections between elements.

In the first part of the chapter two formulations are introduced that employ geometrically non-conforming elements but which maintain \(C^0\) continuity of the global polynomial expansion: referred to as the iterative patching and constrained approximation. In the second part the mortar element and the discontinuous Galerkin method for second-order elliptic and parabolic problems are introduced. In these cases the \(C^0\) continuity is no longer imposed, and new weak forms of the problem are developed. Some interesting possibilities exist with the discontinuous Galerkin method for second-order elliptic and parabolic problems, and particular attention is paid to this formulation.

In Chapter 8 the authors discuss algorithms for incompressible flows appropriate for direct numerical (DNS) and large eddy simulations (LES) in complex-geometry domains. They are primarily interested in those formulations which are extendable to three dimensions, and therefore they focus on primitive variables (velocity-pressure), velocity-vorticity, and the gauge formulations. Both coupled, splitting and least-square formulations for primitive variables are discussed. In this revision both Uzawa coupled algorithm and a new substructured solver are considered. The discussion on primitive variables time-splitting has been rewritten to include recent theoretical advances in pressure-connection and velocity-correction schemes as well as the rotational formulations of the pressure boundary condition. The discretization of the nonlinear terms in the Navier-Stokes formulation both in space as well as time using semi-Lagrangian formulation has been discussed. References also have been given for other formulations for general discretizations, i.e. a stream function-vorticity formulation of the unstructured spectral/\(hp\) element method.

Chapter 9 presents several examples of incompressible flow simulations. First, a series of relatively simple laminar flows which have exact solutoins and thus can be considered as benchmark problems are taken up. These solutions are useful for verification purposes, i.e. to ensure that the spectral/\(hp\) element discretisation is applied and the associated software is implemented correctly. Biglobal instability of small amplitude disturbances superimposed on two-dimensional flow states is also considered. This type of analysis can aid in understanding the onset of three-dimensional transition to turbulence. Further, the authors discuss numerical issues in DNS, including diagnostics for under-resolution. Specifically, they include a case study of turbulent channel flow which has been extensively simulated with global spectral methods. Subsequently, the main issues in LES with focus on high-order discretization, are summarised. The results here are compared with available experimental data and serve to validate the LES models. The chapter is concluded by considering computing mode that involves resolution steering. To meet this end, two adaptive procedures based on error estimates as well as other heuristics, and a parallel paradigm suitable for dynamic \(p\)-refinement, are presented.

In Chapter 10, the important issues of conservativity and monotonicity are revisited, thus extending the material presented earlier in Section 6.5. The emphasis in this chapter is primarily on systems of hyperbolic conservation laws, which are solved in conservative form employing characteristic decomposition. This is an area in which high-order methods have had little success in the past. The principal issue is how to effectively use the high-order expansions of spectral/\(hp\) methods while honouring the inherent monotonicity and conservation properties of analytic system. Different ways of dealing with these fundamental issues for both the compressible Euler and Navier-Stokes equations are considered. The issue of boundary conditions is crucial in wave propagation phenomena and is often misunderstood for compressible flows and magneto-hydrodynamics. To this end the authors discuss this issue in many different formulations that are presented, including a flux-corrected transport approach, a discontinuous Galerkin method and penalty formulation.

Spectral and multi-domain spectral methods are particularly effective in modelling propagation phenomena governed by hyperbolic conservation laws. For example, in atmospheric and ocean and seismic modelling, resolution of high frequencies and high wave is crucial, especially for long term forecasting.

In Section 10.5 Navier-Stokes equations are considered and appropriate solution algorithms are formulated. Thereafter shock-fitting techniques appropriate for high Mach number flows are discussed in Section 10.6. The last section discusses modelling of plasma flows and ocean modelling. Spectral methods have inherently small numerical dispersion and superb resolution properties, requiring typically four to six points (or modes) per wavelength. In addition, spectral elements provide several computational advantages, including nearest-neighbour communications and cache-blocked computations. This has been recognized by application scientists, and new scalable codes for large-scale simulations of atmospheric general circulation models and general seismology have been designed based on spectral element methods (see references 273, 308).

Appendices have been included which provide details on Jacobi and Askey polynomials as well as numerical integration and differentiation which are essential building blocks of spectral/\(hp\) element techniques. A full description of commonly used expansion bases is provided, which now includes nodal points for non-tensorial electrostatic problems, and Fekete point distribution in simplex domains, The final appendix also details Riemann solvers commonly used in solution of Euler equations.

The book incorporates latest developments in the subject, is outstanding and of great interest to students, academics and practitioners in computational fluid mechanics, applied and numerical mathematics aerospace and mechanical engineering, magneto-hydrodynamics and climate/ocean modelling.

Reviewer: Hari Krishan Verma (Ludhiana)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76M10 | Finite element methods applied to problems in fluid mechanics |

86-08 | Computational methods for problems pertaining to geophysics |

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |