quaderni di matematica volume 24

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Dipartimento di Matematica Seconda Università di Napoli

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Classical Problems in Mechanics (R. Russo ed.) Recent Developments in Partial Differential Equations (V. A. Solonnikov ed.) Recent Progress in Function Spaces (G. Di Maio and Ľ. Holá eds.) Advances in Fluid Dynamics (P. Maremonti ed.) Methods of Discrete Mathematics (S. Löwe, F. Mazzocca, N. Melone and U. Ott eds.) Connections between Model Theory and Algebraic and Analytic Geometry (A. Macintyre ed.) Homage to Gaetano Fichera (A. Cialdea ed.) Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.) Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.) Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.) Model Theory and Applications (L. Belair, Z. Chatzidakis et al. eds.) Topics in Diagram Geometry (A. Pasini ed.) Complexity of Computations and Proofs (J. Krajíček ed.) Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi (D. Pallara ed.) Dispersive Nonlinear Problems in Mathematical Physics (P. D’Ancona and V. Georgev eds.) Kinetic Methods for Nonconservative and Reacting Systems (G. Toscani ed.) Set Theory: Recent Trends and Applications (A. Andretta ed.) Selection Principles and Covering Properties in Topology (Lj.D.R. Kočinac ed.) Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior (M. Šilhavý ed.) Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments (G. Casnati, F. Catanese, and R. Notari eds.) Theory and Applications of Proximity, Nearness and Uniformity (G. Di Maio and S. Naimpally eds.) On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments (A. Alvino, A. Mercaldo, F. Murat, and I. Peral eds.) Numerical Methods for Balance Laws (G. Puppo and G. Russo eds.)

Next issues Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini (F. Mazzocca, N. Melone and D. Olanda eds.)

quaderni di matematica errata-corrige

errata

corrige

vol. 22 Theory and Applications of Proxim-

vol. 22 Theory and Applications of Proxim-

ity, Nearness and Uniformity

ity, Nearness and Uniformity

(G. Di Maio and S. Naimpally eds.)

(G. Di Maio and S. Naimpally eds.)

2009

2008

vol. 23 On the notions of solution to nonlin-

vol. 23 On the notions of solution to nonlin-

ear elliptic problems: results and develop-

ear elliptic problems: results and develop-

ments

ments

(A. Alvino, A. Mercaldo, F. Murat, and I.

(A. Alvino, A. Mercaldo, F. Murat, and I.

Peral eds.)

Peral eds.)

2009

2008

Numerical Methods for Balance Laws

edited by

Gabriella Puppo and Giovanni Russo

Received September, 2009 c 2009 by Dipartimento di Matematica della Seconda Università di Napoli  Photocomposed copy prepared from a LATEX file. ISBN 978-88-548-3360-9

Authors’ addresses: Fausto Cavalli Department of Mathematics University of Brescia Via Valotti, 9 I-25133 Brescia, Italy email: [email protected]

Alina Chertock Department of Mathematics North Carolina State University Raleigh, NC 27695 USA email: [email protected]

Cory D. Hauck Computational Mathematics Group Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge, TN 37831 USA email: [email protected]

Christiane Helzel Fakultät für Mathematik Ruhr-Universität Bochum Universitätsstrasse 150 44780 Bochum, Germany email: [email protected]

Alexander Kurganov Mathematics Department Tulane University New Orleans, LA 70118 USA email: [email protected]

Randall J. LeVeque University of Washington Department of Applied Mathematics Box 352420 Seattle, WA 98195-2420, USA email: [email protected]

Robert B. Lowrie Los Alamos National Laboratory Computational Physics Group (CCS-2) Computer, Computational and Statistical Sciences Division MS D413, Los Alamos NM 87545 USA email: [email protected]

Ryan G. McClarren Department of Nuclear Engineering Texas A&M University College Station, TX 77843 USA email: [email protected]

Giovanni Naldi Department of Mathematics University of Milan via Saldini, 50 I-20133 Milano, Italy email: [email protected]

Sebastian Noelle Institute for Geometry and Practical Mathematics RWTH Aachen University Templergraben 55 52056 Aachen, Germany email: [email protected]

Carlos Parés Departamento de Análisis Matemático Universidad de Málaga 29071 Málaga, Spain email: [email protected]

Gabriella Puppo Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi, 24 I-10124 Torino, Italy email: [email protected]

Giovanni Russo Department of Mathematics and Computer Science University of Catania Viale Andrea Doria, 6 I-95125 Catania, Italy email: [email protected]

Matteo Semplice Department of Physics and Mathematics University of Insubria-Como Via Valleggio, 11 I-22100 Como, Italy email: [email protected]

Chi-Wang Shu Division of Applied Mathematics Brown University 182 George Street Providence, RI 02912, USA email: [email protected]

Yoshifumi Suzuki Department of Aerospace Engineering The University of Michigan 1320 Beal Ave., Ann Arbor MI 48109-2140, USA Present address: Desktop Aeronautics, Inc. 1900 Embarcadero Rd., Suite 101 Palo Alto, CA 94303-3310, USA email: [email protected]

Bram van Leer Department of Aerospace Engineering The University of Michigan 1320 Beal Ave., Ann Arbor MI 48109-2140, USA email: [email protected]

Yulong Xing Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge, TN 37831 and Department of Mathematics University of Tennessee Knoxville, TN 37996, USA email: [email protected]

Contents Preface High-order Well-balanced Schemes

1

Sebastian Noelle, Yulong Xing, and Chi-Wang Shu

Path-Conservative Numerical Methods for Nonconservative Hyperbolic Systems

67

Carlos Parés

Numerical Approximation of Stiff Reacting Flow

123

Christiane Helzel and Randall J. LeVeque

Relaxed Schemes Based on Diffusive Relaxation for HyperbolicParabolic Problems: Some New Developments

157

Fausto Cavalli, Giovanni Naldi, Gabriella Puppo, and Matteo Semplice

Methods for Diffusive Relaxation in the PN Equations

197

A Space-Time Discontinuous Galerkin Method for Extended Hydrodynamics

245

Cory D. Hauck, Robert B. Lowrie, and Ryan G. McClarren

Yoshifumi Suzuki and Bram van Leer

On Splitting-Based Numerical Methods for ConvectionDiffusion Equations Alina Chertock and Alexander Kurganov

303

Mathematics subject classification High-order Well-balanced Schemes: 35L65, 76M12, 65M12. Path-Conservative Numerical Methods for . . . : 65M06, 35L65, 76L05. Numerical Approximation of Stiff Reacting Flow: 35L65, 65M08, 80A32. Relaxed Schemes Based on Diffusive Relaxation for . . . : 65M20, 65M12, 35K65. Methods for Diffusive Relaxation in the PN Equations: 65M60, 35Q20, 82D75, 78M05. A Space-Time Discontinuous Galerkin Method for . . . : 65M08, 65M60, 65M22, 76P05. On Splitting-Based Numerical Methods for . . . : 65M06, 65M08, 65M25, 35K15.

quaderni di matematica, 24 ISBN 978-88-548-3360-9 DOI 10.4399/97888548336091 pag. XI–XV

Preface This book addresses recent developments in the numerical integration of balance laws, which can be defined as hyperbolic systems of equations with a source term. These equations arise in the modeling of several phenomena, spanning from physics to biology, from chemistry to mathematical finance. Source terms added to conservation laws may arise due to geometric effects, as occurs for instance in nozzle flow. Or they can be due to the presence of an external field, as is the case of gravity in shallow water. Another interesting case is the flow of a mixture of reacting gases. Here the exchanges of mass, momentum and energy due to the phenomena induced by the chemical reaction are modeled as gain and loss terms which alter the balance of conserved quantities, and appear in the equations as source terms. Similar exchanges of mass, momentum and energy also occur in multiphase flows, which are of interest in secondary oil recovery. Such systems have the structure of hyperbolic systems with relaxation, which is common to other mathematical models, such as hydrodinamic models for semiconductors, in which carrier dynamics is described by the balance equations of the carrier mass, momentum, and energy density, and the source term accounts mainly for the interaction with the lattice. Another class of phenomena modeled with balance laws occurs in convection diffusion phenomena, as Navier Stokes equations, or the porous media equation, in the cases in which convection terms are dominant. Similar equations arise in the so-called drift-diffusion model of carrier transport in semiconductor, or in the simulation of capillary growth under the action of a chemical stimulus, as in chemiotaxis. An interesting aspect of convection-diffusion problems is that sometimes they are obtained from relaxation models, under an appropriate scaling, for a small (finite) relaxation parameter. We also mention kinetic equations, such as the Boltzmann equation or simplified models, appearing in rarefied gas dynamics. In this case, one is interested in the evolution of a probability distribution in the single particle phase space.

These phenomena can also be written as balance laws: a linear convection coupled with a collision term, which accounts for the decay toward equilibrium of the distribution function. Scattering of particles through a medium can also be included in this class of phenomena. For kinetic problems, interesting balance laws derive as the departure from local equilibrium is rather small. In such a case a continuum description obtained through Euler or Navier-Stokes equations is not yet appropriate, and a full kinetic treatment is quite expensive. In this regime, macroscopic balance laws can be derived from Boltzmann equation, using Chapman-Enskog expansion or the moment closure method, as in the case of Burnett or Grad equations. In many cases, the inclusion of a source term does not increase significantly the complexity of the problem. If finite difference schemes are used, a source term can be discretized point-wise, adding its contribution to the numerical fluxes. On the other hand, if finite volume schemes are chosen, source terms require a simple quadrature, to evaluate the contribution of the source to the evolution of the cell averages of the solution. However in many important applications, such a naive approach simply does not work. The papers collected in this book address the main difficulties arising in the numerical integration of balance laws, when standard techniques fail. The technology to accurately integrate systems of conservation laws dates back from the second half of the 80’s to the 90’s. Current high resolution schemes employ ideas developed in those years. High order non linear reconstructions are available to guarantee accurate solutions, with sharp shock transitions, and vanishing spurious oscillations. The state of the art is by far not as mature when source terms must be taken care of. The main difficulties arise from the fact that convection and source terms may induce different scales in the phenomena been simulated. The most interesting problems arise either when the scales induced by the source and the hyperbolic terms are of the same order, or, by contrast, when the scales are very different so that one of the two terms prevails on the other one. We start from the case in which the source terms and the convective fluxes are of the same order. In this case, steady state solutions may exhist, in which flux and source term exactly balance. The most studied problem in this class

is given perhaps by the shallow water system of equations. Here non trivial steady states are found when the water is at rest, or in the most complex case of moving equilibria. The first paper in this collection, by Noelle, Xing and Shu, contains a review on recent developments in high order well balanced schemes. These schemes are characterized by a discretization of the flux and the source terms designed to enforce equilibria exactly at the discrete level. In this fashion, steady state solutions are exactly maintained, and it is possible to study flows close to equilibrium on relatively coarse grids. Similar techniques can be applied to simulate problems with non trivial equilibria, such as nozzle flows, chemiotaxis or multiphase flows. Much work has been dedicated to this fashinating problem, and one can find an extensive list of references in this paper. The second paper in this collection, by Pares, constructs well balanced schemes starting from a different perspective. Here the balance law is rewritten as a non conservative hyperbolic system of equations. The proper speed of discontinuities cannot be derived from Rankine-Hugoniot conditions, which assume a conservative flux, but it is computed choosing a suitable path connecting the jump in the solution in phase space. The correct path contains the physics of the problem and thus, in particular, preserves steady states. At the discrete level, steady states are exactly well balanced, provided the scheme preserves the chosen path (path conservative schemes). With the same techniques, steady states are exactly preserved also in complex non conservative flows, as, for instance, in two layers shallow water problems. A completely different class of problems arises when the source term induces a much faster time scale than the convective flux. In this case, the flow occurs at different length and time scales. In many cases, one is interested in the slower scale, but standard methods may become unstable or develop oscillations or converge to wrong solutions unless the faster scale is taken into account. A typical case occurs with detonation waves, which is the topic of the third paper in this series, by Helzel and LeVeque. It is well known that standard schemes for detonation problems converge to solutions which travel with a speed induced by the grid, unless the grid is fine enough. In this paper, a fractional step method designed to ensure correct propagation speeds on under-resolved grids

is analyzed, overcoming grid induced spurious effects. When the source term is characterized by a much faster time scale with respect to convection terms, the system of balance laws may relax on a lower order system of equations. This behavior is well known in kinetic models, when the system of macroscopic moment equations relaxes onto lower order systems, such as compressible gas dynamics, as the mean free path approaches zero. This concept can be exploited to generate systems of hyperbolic equations with stiff sources that relax on pre-assigned PDE’s. This is the idea behind relaxation schemes, which integrate non linear PDE’s via a linear hyperbolic system of equations with a stiff source, which decays on the original PDE as the relaxation parameter is drawn to zero. The fourth paper in the book, by Cavalli, Naldi, Puppo and Semplice contains recent developments on relaxation schemes for non linear, degenerate parabolic equations with convection terms. The scheme is based on a proper choice of the relaxation system, with an implicit-explicit time integration, so that the fast source term is computed implicitely, while the convection part is integrated explicitely in time. The following paper, by Hauck, Lowrie and McClarren also is dedicated to relaxation systems, but in this case, the relaxation system has a physical meaning: it describes neutron transport across a medium. The relaxation parameter is the mean free path between scattering events, and the challenge is to capture the correct diffusion limit, without resolving the small scales. Since in this case the relaxation parameter is not zero, difficulties arise in the treatment of the stiff terms. Similar problems are found in drift diffusion models for semiconductors or chemiotaxis. The following paper, by Suzuki and van Leer, discusses a Galerkin method for relaxation systems, applied to the higher oder hydrodynamical models for gas flow. Such a model have the structure of hyperbolic systems with relaxation. When the relaxation parameter vanishes, the system relaxes to the Euler equation of gas dynamics, while for small values of the relaxation parameter the equations should well approximate Navier-Stokes equations for compressible gas. The method uses Discontinuous Galerkin finite volume methods and particular care is given to study the structure of the error terms obtained with the discretization, since the goal is to be able to capture the diffusive behavior for small (but non zero) relaxation parameter.

The last paper in this series is concerned with convection diffusion problems, in the case in which convection is the dominant term and diffusion is non degenerate. Here the purpose is to avoid oscillations issued by the second order singular perturbation, without resolving the small scales. This goal is achieved with an operator splitting approach, integrating exactly or with FFT the diffusion term, and applying conservative high resolution schemes for the hyperbolic part. The list of topics in the collection is far from being complete. For example, numerical schemes for kinetic problems are not included, and the tools to deal with singular perturbation problems, possibly maintaining a uniform accuracy in the stiffness parameter, However, these topics are so complex that they would deserve a monograph on their own. The overview given by this book already contains the main problems and techniques arising in the numerical treatment of balance laws, and, on the other hand, these ideas are essential in the development of numerical methods for other more specialized fields. Giovanni Russo and Gabriella Puppo

quaderni di matematica, 24 ISBN 978-88-548-3360-9 DOI 10.4399/97888548336092 pag. 1–66

High-order Well-balanced Schemes Sebastian Noelle, Yulong Xing, and Chi-Wang Shu

Contents 1. Introduction (3). 2. Preliminaries: equilibria and the residual (4). 3. Schemes based on well-balanced finite difference operators (7). 4. Schemes based on well-balanced quadrature (22). 5. Numerical examples for the shallow water equations (38). 6. Conclusion (61).

High-order Well-balanced Schemes

3

1. Introduction In many applications we often encounter hyperbolic balance laws, which in one dimension are in the form (1.1)

𝑈𝑡 + 𝑓 (𝑈, 𝑥)𝑥 = 𝑠(𝑈, 𝑥)

where 𝑈 is the solution vector, 𝑓 (𝑈, 𝑥) is the flux and 𝑠(𝑈, 𝑥) is the source term. The source term may come from geometrical, reactive or other considerations. Examples of hyperbolic balance laws include the shallow water equation with a non-flat bottom topology, elastic wave equation [2], chemosensitive movement [15], nozzle flow [13], and two phase flow [23]. Comparing with the standard hyperbolic conservation laws, namely (1.1) with 𝑠(𝑈, 𝑥) = 0, the numerical approximation to the balance laws (1.1) is usually not too much more difficult: we simply need to put the point values (for finite difference schemes) or the cell averages (for finite volume schemes) of the source term 𝑠(𝑈, 𝑥) directly to the discretization of the spatial operator. There is, however, one noticeable exception. The balance law (1.1) often admits steady state solutions in which the source term 𝑠(𝑈, 𝑥) is exactly balanced by the flux gradient 𝑓 (𝑈, 𝑥)𝑥 . Such steady state solutions are usually non-trivial (they are usually not polynomials) and they often carry important physical meaning (for example, the still water or steady moving water solution of the shallow water equation, to be studied in more detail later in this paper). The objective of well-balanced schemes is to preserve exactly some of these steady state solutions. The most important advantage of well-balanced schemes is that they can accurately resolve small perturbations to such steady state solutions with relatively coarse meshes. In comparison, a non-well-balanced scheme will introduce truncation errors to the steady state solution, hence it cannot resolve small perturbations to such steady states unless the truncation error is already smaller than such perturbations, thus requiring a refined mesh. In Section 5 we will provide such examples. However, it is quite difficult to design well-balanced schemes which are high-order accurate and non-oscillatory in the presence of discontinuities in the solution. In this paper we use the shallow water equation as a prototype to survey a few recently developed well-balanced high-order finite difference, finite volume

4

S. Noelle, Y. Xing, and C.-W. Shu

and discontinuous Galerkin finite element methods. We attempt to explain the main ingredients in these algorithms which allow us to achieve the wellbalanced property without losing other nice properties of the original scheme, such as high-order accuracy and non-oscillatory performance in the presence of solution discontinuities. The paper is organized as follows. In Section 2 we first discuss a number of interesting equilibrium states. Then we introduce the residual which need to be well-balanced near stationary states. At this point the paper splits into two approaches: The first approach, see Section 3, applies to finite difference, finite volume and discontinuous Galerkin schemes. It treats equilibria for which the source term can be decomposed into sums of products of the form (3.4). The challenge is to construct finite difference operators which are high-order accurate and non-oscillatory for the conservative flux difference and the source term, and which coincide for both terms in the case of equilibrium solutions. The second approach, designed for general equilibria and finite volume schemes, is covered in Section 4. The key task is to find well-balanced quadratures for the integral of the residual, see equation (4.1). Subsection 4.2 presents a general framework to decompose this integral into suitable parts. Subsection 4.3 realizes this approach for moving water equilibria for shallow water flows. In Section 5 we present numerical results showing the accuracy and wellbalanced properties of both classes of schemes for a number of challenging flows. Section 6 contains some concluding remarks.

2. Preliminaries: equilibria and the residual In this section we introduce equilibrium variables which characterize stationary states, and discuss the residual which monitors the deviation of the system from stationary states. In particular, two forms of the residual are singled out which are the bases of the finite difference algorithms in Section 3 on one hand and the finite volume algorithms in Section 4 on the other hand.