research-article

Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems

Authors:

Jean-Luc Guermond,

Bojan PopovAuthors Info & Claims

SIAM Journal on Numerical Analysis, Volume 54, Issue 4

Pages 2466 - 2489

https://doi.org/10.1137/16M1074291

Published: 01 January 2016 Publication History

Abstract

We propose a numerical method for solving general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on nonuniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using strong stability preserving algorithms. This technique extends to continuous finite elements the work of [D. Hoff, Math. Comp., 33 (1979), pp. 1171--1193], [D. Hoff, Trans. Amer. Math. Soc., 289 (1985), pp. 591--610], and [H. Frid, Arch. Ration. Mech. Anal., 160 (2001), pp. 245--269].

References

[1]

M. Ainsworth, Pyramid algorithms for Bernstein--Bézier finite elements of high, nonuniform order in any dimension, SIAM J. Sci. Comput., 36 (2014), pp. A543--A569, http://dx.doi.org/10.1137/130914048 130914048.

[2]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), pp. 223--342.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl. 20, Oxford University Press, Oxford, UK, 2000.

[4]

A. J. Chorin, Random choice solution of hyperbolic systems, J. Comput. Phys., 22 (1976), pp. 517--533.

[5]

K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), pp. 373--392.

[6]

P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 171--200, http://dx.doi.org/10.1016/0021-9991(90)90233-Q

[7]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1--21, http://dx.doi.org/10.2307/2006218

[8]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences] 325, Springer-Verlag, Berlin, 2000, http://dx.doi.org/10.1007/3-540-29089-3_14 .

[9]

H. Frid, Maps of convex sets and invariant regions for finite-difference systems of conservation laws, Arch. Ration. Mech. Anal., 160 (2001), pp. 245--269, http://dx.doi.org/10.1007/s002050100166

[10]

J.-L. Guermond and M. Nazarov, A maximum-principle preserving ${C}^0$ finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Engrg., 272 (2013), pp. 198--213.

[11]

J.-L. Guermond and B. Popov, Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations, J. Comput. Phys., 321 (2016), pp. 908--926.

[12]

J.-L. Guermond and B. Popov, Error estimates of a first-order Lagrange finite element technique for nonlinear scalar conservation equations, SIAM J. Numer. Anal., 54 (2016), pp. 57--85, http://dx.doi.org/10.1137/140990863

[13]

D. Hoff, A finite difference scheme for a system of two conservation laws with artificial viscosity, Math. Comp., 33 (1979), pp. 1171--1193.

[14]

D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc., 289 (1985), pp. 591--610, http://dx.doi.org/10.2307/2000254

[15]

A. Jameson, Positive schemes and shock modelling for compressible flows, Internat. J. Numer. Methods Fluids, 20 (1995), pp. 743--776, http://dx.doi.org/10.1002/fld.1650200805

[16]

A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, 18 (2002), pp. 584--608, http://dx.doi.org/10.1002/num.10025

[17]

A. Kurganov, G. Petrova, and B. Popov, Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws, SIAM J. Sci. Comput., 29 (2007), pp. 2381--2401, http://dx.doi.org/10.1137/040614189

[18]

D. Kuzmin and S. Turek, Flux correction tools for finite elements, J. Comput. Phys., 175 (2002), pp. 525--558.

[19]

D. Kuzmin, R. Löhner, and S. Turek, eds., Flux-Corrected Transport. Principles, Algorithms, and Applications, Sci. Comput., Springer, Berlin, 2005.

[20]

M.-J. Lai and L. L. Schumaker, Spline Functions on Triangulations, Encyclopedia Math. Appl. 110, Cambridge University Press, Cambridge, UK, 2007, http://dx.doi.org/10.1017/CBO9780511721588 CBO9780511721588.

[21]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), pp. 537--566.

[22]

P.-L. Lions, B. Perthame, and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), pp. 599--638.

[23]

R. Liska and B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 25 (2003), pp. 995--1017, http://dx.doi.org/10.1137/S1064827502402120 S1064827502402120.

[24]

T. P. Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations, 18 (1975), pp. 218--234.

[25]

T. Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad., 44 (1968), pp. 642--646.

[26]

S. Osher, The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations, Proc. Amer. Math. Soc., 89 (1983), pp. 641--646, http://dx.doi.org/10.2307/2044598

[27]

B. Perthame and C.-W. Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), pp. 119--130, http://dx.doi.org/10.1007/s002110050187

[28]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Science] 258, Springer-Verlag, New York, Berlin, 1983.

[29]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 3rd ed., Springer-Verlag, Berlin, 2009.

[30]

R. Young, The $p$-system. I. The Riemann problem, in The Legacy of the Inverse Scattering Transform in Applied Mathematics (South Hadley, MA, 2001), Contemp. Math. 301, Amer. Math. Soc., Providence, RI, 2002, pp. 219--234, http://dx.doi.org/10.1090/conm/301/05166

Cited By

Milani RRenac FRuel J(2024)Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113254514:COnline publication date: 18-Oct-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113254
Dzanic T(2024)Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113010508:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113010
Dao TNazarov MTomas I(2024)A structure preserving numerical method for the ideal compressible MHD systemJournal of Computational Physics10.1016/j.jcp.2024.113009508:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113009
Show More Cited By

Recommendations

Invariant Domains Preserving Arbitrary Lagrangian Eulerian Approximation of Hyperbolic Systems with Continuous Finite Elements

A conservative invariant domain preserving arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. The method is explicit in time, works with continuous finite elements, and is first-order accurate in space. One original ...
Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy, and the local minimum ...
A domain decomposition Taylor Galerkin finite element approximation of a parabolic singularly perturbed differential equation

In this paper, we deal with a discrete Monotone Iterative Domain Decomposition (MIDD) method based on Schwarz alternating algorithm for solving parabolic singularly perturbed partial differential equations. A discrete iterative algorithm is proposed ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Numerical Analysis

SIAM Journal on Numerical Analysis Volume 54, Issue 4

DOI:10.1137/sjnaam.54.4

Issue’s Table of Contents

© 2016, Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2016

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

20
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 21 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Milani RRenac FRuel J(2024)Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113254514:COnline publication date: 18-Oct-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113254
Dzanic T(2024)Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113010508:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113010
Dao TNazarov MTomas I(2024)A structure preserving numerical method for the ideal compressible MHD systemJournal of Computational Physics10.1016/j.jcp.2024.113009508:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.113009
Ching EJohnson RKercher A(2024)Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part IJournal of Computational Physics10.1016/j.jcp.2024.112881505:COnline publication date: 15-May-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.112881
Ching EJohnson RKercher A(2024)Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part IIJournal of Computational Physics10.1016/j.jcp.2024.112878505:COnline publication date: 15-May-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.112878
Lin YChan J(2024)High order entropy stable discontinuous Galerkin spectral element methods through subcell limitingJournal of Computational Physics10.1016/j.jcp.2023.112677498:COnline publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1016/j.jcp.2023.112677
Ern AGuermond JWang Z(2024)Asymptotic and Invariant-Domain Preserving Schemes for Scalar Conservation Equations with Stiff Source Terms and Multiple Equilibrium PointsJournal of Scientific Computing10.1007/s10915-024-02628-9100:3Online publication date: 2-Aug-2024
https://dl.acm.org/doi/10.1007/s10915-024-02628-9
Guermond JMaier MPopov BSaavedra LTomas I(2024)First-Order Greedy Invariant-Domain Preserving Approximation for Hyperbolic Problems: Scalar Conservation Laws, and p-SystemJournal of Scientific Computing10.1007/s10915-024-02592-4100:2Online publication date: 27-Jun-2024
https://dl.acm.org/doi/10.1007/s10915-024-02592-4
Clayton BGuermond JMaier MPopov BTovar E(2023)Robust second-order approximation of the compressible Euler equations with an arbitrary equation of stateJournal of Computational Physics10.1016/j.jcp.2023.111926478:COnline publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1016/j.jcp.2023.111926
Lin YChan JTomas I(2023)A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equationsJournal of Computational Physics10.1016/j.jcp.2022.111850475:COnline publication date: 15-Feb-2023
https://dl.acm.org/doi/10.1016/j.jcp.2022.111850
Show More Cited By

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents