Hyperbolic Systems of Conservation Laws. in One Space Dimension. II - Solutions to the Cauchy problem. Alberto Bressan

Size: px
Start display at page:

Download "Hyperbolic Systems of Conservation Laws. in One Space Dimension. II - Solutions to the Cauchy problem. Alberto Bressan"

Transcription

1 Hyperbolic Systems of Conservation Laws in One Space Dimension II - Solutions to the Cauchy problem Alberto Bressan Department of Mathematics, Penn State University 1

2 Global Solutions to the Cauchy Problem u t + f(u) = u(, ) = ū() Construct a sequence of approimate solutions u m Show that (a subsequence) converges: u m u in L 1 loc = u is a weak solution u u 1 2 u m To prevent oscillations, one needs an a-priori bound on the total variation (J. Glimm, 1965) 2

3 Building block: the Riemann Problem u t + f(u) = u(, ) = { u if < u + if > B. Riemann 186: 2 2 system of isentropic gas dynamics P. La 1957: n n systems (+ special assumptions) T. P. Liu 1975 n n systems (generic case) S. Bianchini 23 (vanishing viscosity limit for general hyperbolic systems, possibly non-conservative) t ω ω 2 1 ω = u ω = u + 3 invariant w.r.t. rescaling symmetry: u θ (t, ). = u(θt, θ) for all θ > 3

4 Riemann Problem for Linear Systems u t + Au = u(, ) = { u if < u + if > t / t = λ 2 / t = λ 1 ω 1 ω 2 / t = λ 3 ω = u ω = u + 3 u + u = n c j r j (sum of eigenvectors of A) j=1 intermediate states : ω i. = u + j i c j r j i-th jump: ω i ω i 1 = c i r i travels with speed λ i 4

5 Scalar Conservation Law u t + f(u) = u IR CASE 1: Linear flu. f(u) = λu. Jump travels with speed λ (contact discontinuity) f (u) = λ u + u(t) λ t u 5

6 CASE 2: the flu f is conve, so that u f (u) is increasing. u + > u = centered rarefaction wave f (u) u + u(t) f (u) t u u + < u = stable shock f (u) u + u λ t u(t) λ = f(u + ) f(u ) + u u 6

7 A class of nonlinear hyperbolic systems u t + f(u) = A(u) = Df(u) A(u)r i (u) = λ i (u)r i (u) Assumption (H) (P.La, 1957): Each i-th characteristic field is either genuinely nonlinear, so that λ i r i > for all u or linearly degenerate, so that λ i r i = for all u 7

8 genuinely nonlinear = characteristic speed λ i (u) is strictly increasing along integral curves of the eigenvectors r i linearly degenerate = characteristic speed λ i (u) is constant along integral curves of the eigenvectors r i u 2 r 1 r 2 u u 1 R 2 (σ)(u ) 8

9 Shock and Rarefaction curves u t + f(u) = A(u) = Df(u) i-rarefaction curve through u : σ R i (σ)(u ) = integral curve of the field of eigenvectors r i through u du dσ = r i(u), u() = u i-shock curve through u : σ S i (σ)(u ) = set of points u connected to u by an i-shock, so that u u is an i-eigenvector of the averaged matri A(u, u ) S i R i u r i (u ) 9

10 Elementary waves u t + f(u) = u(, ) = { u if < u + if > CASE 1 (Centered rarefaction wave). Let the i-th field be genuinely nonlinear. Assume u + = R i (σ)(u ) for some σ >, so that λ i (u + ) > λ i (u ). Then u(t, ) = { u if < tλ i (u ), R i (s)(u ) if = tλ i (s), s [, σ], u + if > tλ i (u + ), is a weak solution (piecewise smooth) of the Riemann problem. 1

11 Indeed, the previous construction achieves: u is constant along rays {/t = constant}, hence tu t + u = λ i (u(t, )) = /t u t, u are parallel to r i (u), hence they are i-eigenvectors of A(u) Therefore, for λ i (u ) < /t < λ i (u + ) one has = u t + t u = u t + λ i (u)u = u t + A(u)u 11

12 A centered rarefaction wave u 2 u u + u(t) t t = λ (u ) i t = λ (u + i ) u 1 u = u 12

13 CASE 2 (Shock or contact discontinuity). Assume that u + = S i (σ)(u ) for some i, σ. Let λ = λ i (u, u + ) be the shock speed. Then the function u(t, ) = { u if < λt, u + if > λt, is a weak solution to the Riemann problem. In the genuinely nonlinear case, this shock is admissible (La condition, Liu condition) iff σ <. t = t λ u = u u = u + 13

14 Solution to a 2 2 Riemann problem R 2 1 rarefaction t 2 shock u 2 ω 1 R 1 u = ω 1 u u = u u = u+ u + S 1 u 1 S 2 14

15 Solution of the general Riemann problem (P. La, 1957) u t + f(u) = u(, ) = { u if < u + if > Find states u = ω, ω 1,... ω n such that ω = u ω n = u + and every couple ω i 1, ω i are connected by an elementary wave (shock or rarefaction) either ω i = R i (σ i )(ω i 1 ) σ i or ω i = S i (σ i )(ω i 1 ) σ i < 15

16 General solution of the Riemann problem: concatenation of elementary waves t ω ω 2 1 ω = u ω = u

17 Assume: u + u small Implicit function theorem = eistence, uniqueness of the intermediate states ω, ω 1,..., ω n Ψ i (σ)(u) = { Ri (σ)(u) if σ S i (σ)(u) if σ < S i R i u r i (u ) (σ 1, σ 2,..., σ n ) Ψ n (σ n ) Ψ 2 (σ 2 ) Ψ 1 (σ 1 )(u ) Jacobian matri at the origin: J =. ( r 1 (u ) r 2 (u ) always has full rank ) r n (u ) 17

18 Global solution to the Cauchy problem u t + f(u) =, u(, ) = ū() Theorem (Glimm 1965). Assume: system is strictly hyperbolic each characteristic field is either linearly degenerate or genuinely nonlinear Then there eists a constant δ > such that, for every initial condition ū L 1 (IR; IR n ) with Tot.Var.(ū) δ, the Cauchy problem has an entropy admissible weak solution u = u(t, ) defined for all t. 18

19 Construction of a sequence of approimate solutions: by piecing together solutions of Riemann problems - on a fied grid in t- plane (Glimm scheme) t 2 t * * θ = 1/3 2 t * * * θ = 1/ at points where fronts interact (front tracking) t 4 t t 3 α β t 2 t 1 σ σ 19

20 Piecewise constant approimate solution to a Riemann problem replace centered rarefaction waves with piecewise constant rarefaction fans u(t) ω =ω i 1 i, ω i,1 ω i,2 ω i,3 R i ω =ω i i,4 t _ ω i 1 i,j ω i u t 2,1 1,1 ω 2 3,j ω 1 ω ω 3 2

21 Front Tracking Approimations t t 4 t 3 α β t 2 t 1 σ σ Approimate the initial data ū with a piecewise constant function. Construct a piecewise constant approimate solution to each Riemann problem at t = at each time t j where two fronts interact, construct a piecewise constant approimate solution to the new Riemann problem... NEED TO CHECK: - total variation remains small - number of wave fronts remains finite 21

22 Interaction Estimates GOAL: estimate the strengths of the waves in the solution of a Riemann problem, depending on the strengths of the two interacting waves σ, σ σ i σ j σ k u l σ u m σ" u r Incoming: a j-wave of strength σ and an i-wave of strength σ Outgoing: waves of strengths σ 1,..., σ n. Then σ i σ + σ j σ + σ k = O(1) σ σ k i,j 22

23 Incoming: two i-waves of strengths σ and σ Outgoing: waves of strengths σ 1,..., σ n. Then σ i σ σ + k i σ k = O(1) σ σ ( σ + σ ) σ k σ i σ σ" u l m u u r 23

24 Glimm Functionals Total strength of waves: V (t). = α σ α Wave interaction potential: Q(t) =. σ α σ β (α,β) A A =. couples of approaching wave fronts t σ σ β α τ σ α σ β σ γ 24

25 Changes in V, Q at time τ when the fronts σ α, σ β interact: V (τ) = O(1) σ α σ β Q(τ) = σ α σ β + O(1) V (τ ) σ α σ β Choosing a constant C large enough, the map t V (t) + C Q(t) is nonincreasing, as long as V remains small Total variation initially small = global BV bounds Tot.Var.{u(t, )} V (t) V () + C Q() Front tracking approimations can be constructed for all t 25

26 Keeping finite the number of wave fronts At each interaction point, the Accurate Riemann Solver yields a solution, possibly introducing several new fronts Number of fronts can become infinite in finite time accurate Riemann solver simplified Riemann solver Need: a Simplified Riemann Solver, producing only one non-physical front 26

27 A sequence of approimate solutions u t + f(u) = u(, ) = ū() let ε ν as ν (u ν ) ν 1 sequence of approimate front tracking solutions initial data satisfy u ν (, ) ū L 1 ε ν all shock fronts in u ν are entropy-admissible each rarefaction front in u ν has strength ε ν at each time t, the total strength of all non-physical fronts in u ν (t, ) is ε ν 27

28 t S R S S NP R t 2 t 1 α

29 Eistence of a convergent subsequence Tot.Var.{u ν (t, )} C u ν (t) u ν (s) L 1 (t s) [total strength of all wave fronts] [maimum speed] L (t s) Helly s compactness theorem = a subsequence converges u ν u in L 1 loc 28

30 Claim: u = lim ν u ν is a weak solution ( ) {φ t u + φ f(u)} ddt = φ Cc 1 ], [ IR Need to show: {φt lim u ν + φ f(u ν ) } ddt = ν 29

31 T { } φ t (t, )u ν (t, ) + φ (t, )f(u ν (t, )) R R S S ddt ( ) R Supp.φ Γ j n α NP S R = Rarefaction front S = Shock front NP = Non physical front Assume φ(t, ) = outside the strip [, T ] IR. Define u ν (t, α ). = u ν (t, α +) u ν (t, α ) f(u ν (t, α )) =. f(u ν (t, α +)) f(u ν (t, α )). Φ ν = (φ u ν, φ f(u ν )). Use the divergence theorem on each polygonal domain Γ j where u ν is constant: ( ) = div Φ ν (t, ) ddt = Φ ν n dσ j Γ j j Γ j 3

32 T R R S S R Supp.φ Γ j n α NP S R = Rarefaction front S = Shock front NP = Non physical front lim sup ν = Φ ν n dσ Γ j j lim sup ν lim sup ν α S R N P [ẋα (t) u ν (t, α ) f(u ν (t, α )) ] φ(t, α (t)) {O(1) ε ν σ α + O(1) α R α N P } σ α ma φ(t, ) t, 31

33 The Glimm scheme u t + f(u) = u(, ) = ū() Assume: all characteristic speeds satisfy λ i (u) [, 1] This is not restrictive. If λ i (u) [ M, M], simply change coordinates: y = + Mt, τ = 2Mt Choose: a grid in the t- plane with step size t = a sequence of numbers θ 1, θ 2, θ 3,... uniformly distributed over [, 1] #{j ; 1 j N, θ j [, λ] } lim N N θ 3 θ = λ for each λ [, 1]. θ 1 2 λ 1 32

34 Glimm approimations Grid points : j = j, t k = k t for each k, u(t k, ) is piecewise constant, with jumps at the points j. The Riemann problems are solved eactly, for t k t < t k+1 at time t k+1 the solution is again approimated by a piecewise constant function, by a sampling technique t * * * θ = 1/3 2 t * * * θ = 1/2 1 33

35 Eample: shock Glimm s scheme applied to a solution containing a single U(t, ) = { u + if > λt, u if < λt. t t u=u * * * * * * * * * * u=u + Fi T >, take = t = T/N (T ) = #{j ; 1 j N, θ j [, λ] } t = #{j ; 1 j N, θ j [, λ] } N T λt as N 34

36 Rate of convergence for Glimm approimations Random sampling at points determined by the equidistributed sequence (θ k ) k 1 #{j ; 1 j N, θ j [, λ] } lim N N = λ for each λ [, 1]. Need fast convergence to uniform distribution. Achieved by choosing: θ 1 =.1,..., θ 759 =.957,..., θ 3922 =.2293,... Error Estimate: u Glimm (T, ) u eact (T, ) lim L 1 = ln (A.Bressan & A.Marson, 1998) 35

Hyperbolic Systems of Conservation Laws. I - Basic Concepts

Hyperbolic Systems of Conservation Laws. I - Basic Concepts Hyperbolic Systems of Conservation Laws I - Basic Concepts Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 27 The

More information

The Scalar Conservation Law

The Scalar Conservation Law The Scalar Conservation Law t + f() = 0 = conserved qantity, f() =fl d dt Z b a (t, ) d = Z b a t (t, ) d = Z b a f (t, ) d = f (t, a) f (t, b) = [inflow at a] [otflow at b] f((a)) f((b)) a b Alberto Bressan

More information

Lecture Notes on Hyperbolic Conservation Laws

Lecture Notes on Hyperbolic Conservation Laws Lecture Notes on Hyperbolic Conservation Laws Alberto Bressan Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. bressan@math.psu.edu May 21, 2009 Abstract These notes provide

More information

Hyperbolic Systems of Conservation Laws

Hyperbolic Systems of Conservation Laws Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University http://www.math.psu.edu/bressan/

More information

Hyperbolic Systems of Conservation Laws. in One Space Dimension. I - Basic concepts. Alberto Bressan. Department of Mathematics, Penn State University

Hyperbolic Systems of Conservation Laws. in One Space Dimension. I - Basic concepts. Alberto Bressan. Department of Mathematics, Penn State University Hyperbolic Systems of Conservation Laws in One Space Dimension I - Basic concepts Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1 The Scalar Conservation

More information

On finite time BV blow-up for the p-system

On finite time BV blow-up for the p-system On finite time BV blow-up for the p-system Alberto Bressan ( ), Geng Chen ( ), and Qingtian Zhang ( ) (*) Department of Mathematics, Penn State University, (**) Department of Mathematics, University of

More information

Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions

Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions Tatsuo Iguchi & Philippe G. LeFloch Abstract For the Cauchy problem associated with a nonlinear, strictly hyperbolic

More information

Contractive Metrics for Nonsmooth Evolutions

Contractive Metrics for Nonsmooth Evolutions Contractive Metrics for Nonsmooth Evolutions Alberto Bressan Department of Mathematics, Penn State University, University Park, Pa 1682, USA bressan@mathpsuedu July 22, 212 Abstract Given an evolution

More information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

Instability of Finite Difference Schemes for Hyperbolic Conservation Laws

Instability of Finite Difference Schemes for Hyperbolic Conservation Laws Instability of Finite Difference Schemes for Hyperbolic Conservation Laws Alberto Bressan ( ), Paolo Baiti ( ) and Helge Kristian Jenssen ( ) ( ) Department of Mathematics, Penn State University, University

More information

On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws

On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws Stefano Bianchini and Alberto Bressan S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy. E-mail addresses: bianchin@mis.mpg.de,

More information

On the Front-Tracking Algorithm

On the Front-Tracking Algorithm JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 395404 998 ARTICLE NO. AY97575 On the Front-Tracking Algorithm Paolo Baiti S.I.S.S.A., Via Beirut 4, Trieste 3404, Italy and Helge Kristian Jenssen

More information

On the Cauchy Problems for Polymer Flooding with Gravitation

On the Cauchy Problems for Polymer Flooding with Gravitation On the Cauchy Problems for Polymer Flooding with Gravitation Wen Shen Mathematics Department, Penn State University Abstract We study two systems of conservation laws for polymer flooding in secondary

More information

On the Cauchy Problems for Polymer Flooding with Gravitation

On the Cauchy Problems for Polymer Flooding with Gravitation On the Cauchy Problems for Polymer Flooding with Gravitation Wen Shen Mathematics Department, Penn State University. Email: wxs27@psu.edu November 5, 2015 Abstract We study two systems of conservation

More information

The Riemann problem. The Riemann problem Rarefaction waves and shock waves

The Riemann problem. The Riemann problem Rarefaction waves and shock waves The Riemann problem Rarefaction waves and shock waves 1. An illuminating example A Heaviside function as initial datum Solving the Riemann problem for the Hopf equation consists in describing the solutions

More information

Global Existence of Large BV Solutions in a Model of Granular Flow

Global Existence of Large BV Solutions in a Model of Granular Flow This article was downloaded by: [Pennsylvania State University] On: 08 February 2012, At: 09:55 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered

More information

1. Introduction. Consider a strictly hyperbolic system of n conservation laws. 1 if i = j, 0 if i j.

1. Introduction. Consider a strictly hyperbolic system of n conservation laws. 1 if i = j, 0 if i j. SIAM J MATH ANAL Vol 36, No, pp 659 677 c 4 Society for Industrial and Applied Mathematics A SHARP DECAY ESTIMATE FOR POSITIVE NONLINEAR WAVES ALBERTO BRESSAN AND TONG YANG Abstract We consider a strictly

More information

Error bounds for a Deterministic Version of the Glimm Scheme

Error bounds for a Deterministic Version of the Glimm Scheme Error bounds for a Deterministic Version of the Glimm Scheme Alberto Bressan and Andrea Marson S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy. Abstract. Consider the hyperbolic system of conservation laws

More information

A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws

A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws Zhengfu Xu, Jinchao Xu and Chi-Wang Shu 0th April 010 Abstract In this note, we apply the h-adaptive streamline

More information

Global Riemann Solver and Front Tracking Approximation of Three-Component Gas Floods

Global Riemann Solver and Front Tracking Approximation of Three-Component Gas Floods Global Riemann Solver and Front Tracking Approximation of Three-Component Gas Floods Saeid Khorsandi (1), Wen Shen (2) and Russell T. Johns (3) (1) Department of Energy and Mineral Engineering, Penn State

More information

Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3

Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Tommaso Ruggeri Department of Mathematics and Research Center of Applied Mathematics University of Bologna January 21, 2017 ommaso

More information

Solutions in the sense of distributions. Solutions in the sense of distributions Definition, non uniqueness

Solutions in the sense of distributions. Solutions in the sense of distributions Definition, non uniqueness Solutions in the sense of distributions Definition, non uniqueness 1. Notion of distributions In order to build weak solutions to the Hopf equation, we need to define derivatives of non smooth functions,

More information

MATH 220: Problem Set 3 Solutions

MATH 220: Problem Set 3 Solutions MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, < 1, 1 +, 1 < < 0, ψ() = 1, 0 < < 1, 0, > 1, so that it verifies ψ 0, ψ() = 0 if 1 and ψ()d = 1. Consider (ψ j ) j 1 constructed as

More information

Intersection Models and Nash Equilibria for Traffic Flow on Networks

Intersection Models and Nash Equilibria for Traffic Flow on Networks Intersection Models and Nash Equilibria for Traffic Flow on Networks Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu (Los Angeles, November 2015) Alberto Bressan (Penn

More information

On the Dependence of Euler Equations on Physical Parameters

On the Dependence of Euler Equations on Physical Parameters On the Dependence of Euler Equations on Physical Parameters Cleopatra Christoforou Department of Mathematics, University of Houston Joint Work with: Gui-Qiang Chen, Northwestern University Yongqian Zhang,

More information

Hyperbolic Conservation Laws Past and Future

Hyperbolic Conservation Laws Past and Future Hyperbolic Conservation Laws Past and Future Barbara Lee Keyfitz Fields Institute and University of Houston bkeyfitz@fields.utoronto.ca Research supported by the US Department of Energy, National Science

More information

= H(W)z x, hu hu 2 + gh 2 /2., H(W) = gh

= H(W)z x, hu hu 2 + gh 2 /2., H(W) = gh THE EXACT RIEMANN SOUTIONS TO SHAOW WATER EQUATIONS EE HAN AND GERAD WARNECKE Abstract. We determine completely the eact Riemann solutions for the shallow water equations with a bottom step including the

More information

Scalar conservation laws with moving density constraints arising in traffic flow modeling

Scalar conservation laws with moving density constraints arising in traffic flow modeling Scalar conservation laws with moving density constraints arising in traffic flow modeling Maria Laura Delle Monache Email: maria-laura.delle monache@inria.fr. Joint work with Paola Goatin 14th International

More information

A Very Brief Introduction to Conservation Laws

A Very Brief Introduction to Conservation Laws A Very Brief Introduction to Wen Shen Department of Mathematics, Penn State University Summer REU Tutorial, May 2013 Summer REU Tutorial, May 2013 1 / The derivation of conservation laws A conservation

More information

H. L. Atkins* NASA Langley Research Center. Hampton, VA either limiters or added dissipation when applied to

H. L. Atkins* NASA Langley Research Center. Hampton, VA either limiters or added dissipation when applied to Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology H. L. Atkins* NASA Langley Research Center Hampton, A 68- Abstract The compact form of the discontinuous Galerkin method allows

More information

Traffic models on a network of roads

Traffic models on a network of roads Traic models on a network o roads Alberto Bressan Department o Mathematics, Penn State University bressan@math.psu.edu Center or Interdisciplinary Mathematics Alberto Bressan (Penn State) Traic low on

More information

Coupling conditions for transport problems on networks governed by conservation laws

Coupling conditions for transport problems on networks governed by conservation laws Coupling conditions for transport problems on networks governed by conservation laws Michael Herty IPAM, LA, April 2009 (RWTH 2009) Transport Eq s on Networks 1 / 41 Outline of the Talk Scope: Boundary

More information

Notes: Outline. Shock formation. Notes: Notes: Shocks in traffic flow

Notes: Outline. Shock formation. Notes: Notes: Shocks in traffic flow Outline Scalar nonlinear conservation laws Traffic flow Shocks and rarefaction waves Burgers equation Rankine-Hugoniot conditions Importance of conservation form Weak solutions Reading: Chapter, 2 R.J.

More information

Nonclassical Shocks and the Cauchy Problem: General Conservation Laws

Nonclassical Shocks and the Cauchy Problem: General Conservation Laws Contemporary Mathematics Contemporary Mathematics Volume 238, 1999 Volume 00, 1997 B 0-8218-1196-7-03536-2 Nonclassical Shocks and the Cauchy Problem: General Conservation Laws Paolo Baiti, Philippe G.

More information

c 2007 Society for Industrial and Applied Mathematics

c 2007 Society for Industrial and Applied Mathematics SIAM J MATH ANAL Vol 38, No 5, pp 474 488 c 007 Society for Industrial and Applied Mathematics OPTIMAL TRACING OF VISCOUS SHOCKS IN SOLUTIONS OF VISCOUS CONSERVATION LAWS WEN SHEN AND MEE REA PARK Abstract

More information

Math Partial Differential Equations 1

Math Partial Differential Equations 1 Math 9 - Partial Differential Equations Homework 5 and Answers. The one-dimensional shallow water equations are h t + (hv) x, v t + ( v + h) x, or equivalently for classical solutions, h t + (hv) x, (hv)

More information

M. HERTY, CH. JÖRRES, AND B. PICCOLI

M. HERTY, CH. JÖRRES, AND B. PICCOLI EXISTENCE OF SOLUTION TO SUPPLY CHAIN MODELS BASED ON PARTIAL DIFFERENTIAL EQUATION WITH DISCONTINUOUS FLUX FUNCTION M. HERTY, CH. JÖRRES, AND B. PICCOLI Abstract. We consider a recently [2] proposed model

More information

Numerical Methods for Hyperbolic Conservation Laws Lecture 4

Numerical Methods for Hyperbolic Conservation Laws Lecture 4 Numerical Methods for Hyperbolic Conservation Laws Lecture 4 Wen Shen Department of Mathematics, Penn State University Email: wxs7@psu.edu Oxford, Spring, 018 Lecture Notes online: http://personal.psu.edu/wxs7/notesnumcons/

More information

Applications of the compensated compactness method on hyperbolic conservation systems

Applications of the compensated compactness method on hyperbolic conservation systems Applications of the compensated compactness method on hyperbolic conservation systems Yunguang Lu Department of Mathematics National University of Colombia e-mail:ylu@unal.edu.co ALAMMI 2009 In this talk,

More information

Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2

Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2 Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation

More information

Nonlinear stability of compressible vortex sheets in two space dimensions

Nonlinear stability of compressible vortex sheets in two space dimensions of compressible vortex sheets in two space dimensions J.-F. Coulombel (Lille) P. Secchi (Brescia) CNRS, and Team SIMPAF of INRIA Futurs Evolution Equations 2006, Mons, August 29th Plan 1 2 3 Related problems

More information

0.3.4 Burgers Equation and Nonlinear Wave

0.3.4 Burgers Equation and Nonlinear Wave 16 CONTENTS Solution to step (discontinuity) initial condition u(x, 0) = ul if X < 0 u r if X > 0, (80) u(x, t) = u L + (u L u R ) ( 1 1 π X 4νt e Y 2 dy ) (81) 0.3.4 Burgers Equation and Nonlinear Wave

More information

Optima and Equilibria for Traffic Flow on Networks with Backward Propagating Queues

Optima and Equilibria for Traffic Flow on Networks with Backward Propagating Queues Optima and Equilibria for Traffic Flow on Networks with Backward Propagating Queues Alberto Bressan and Khai T Nguyen Department of Mathematics, Penn State University University Park, PA 16802, USA e-mails:

More information

A maximum principle for optimally controlled systems of conservation laws

A maximum principle for optimally controlled systems of conservation laws RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ALBERTO BRESSAN ANDREA MARSON A maximum principle for optimally controlled systems of conservation laws Rendiconti del Seminario Matematico

More information

Annalisa Ambroso 1, Christophe Chalons 2, Frédéric Coquel 3 and Thomas. Introduction

Annalisa Ambroso 1, Christophe Chalons 2, Frédéric Coquel 3 and Thomas. Introduction Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher RELAXATION AND NUMERICAL APPROXIMATION OF A TWO-FLUID TWO-PRESSURE DIPHASIC MODEL

More information

NUMERICAL CONSERVATION PROPERTIES OF H(DIV )-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION

NUMERICAL CONSERVATION PROPERTIES OF H(DIV )-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION SIAM JOURNAL ON SCIENTIFIC COMPUTING, SUBMITTED JUNE 23, ACCEPTED JUNE 24 NUMERICAL CONSERVATION PROPERTIES OF H(DIV )-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION H. DE STERCK,

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear

More information

NONCLASSICAL SHOCK WAVES OF CONSERVATION LAWS: FLUX FUNCTION HAVING TWO INFLECTION POINTS

NONCLASSICAL SHOCK WAVES OF CONSERVATION LAWS: FLUX FUNCTION HAVING TWO INFLECTION POINTS Electronic Journal of Differential Equations, Vol. 2006(2006), No. 149, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) NONCLASSICAL

More information

Shooting methods for numerical solutions of control problems constrained. by linear and nonlinear hyperbolic partial differential equations

Shooting methods for numerical solutions of control problems constrained. by linear and nonlinear hyperbolic partial differential equations Shooting methods for numerical solutions of control problems constrained by linear and nonlinear hyperbolic partial differential equations by Sung-Dae Yang A dissertation submitted to the graduate faculty

More information

Conservation laws and some applications to traffic flows

Conservation laws and some applications to traffic flows Conservation laws and some applications to traffic flows Khai T. Nguyen Department of Mathematics, Penn State University ktn2@psu.edu 46th Annual John H. Barrett Memorial Lectures May 16 18, 2016 Khai

More information

Alberto Bressan. Department of Mathematics, Penn State University

Alberto Bressan. Department of Mathematics, Penn State University Non-cooperative Differential Games A Homotopy Approach Alberto Bressan Department of Mathematics, Penn State University 1 Differential Games d dt x(t) = G(x(t), u 1(t), u 2 (t)), x(0) = y, u i (t) U i

More information

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS HASEENA AHMED AND HAILIANG LIU Abstract. High resolution finite difference methods

More information

Balance laws with integrable unbounded sources

Balance laws with integrable unbounded sources Università di Milano Bicocca Quaderni di Matematica Balance laws with integrable unbounded sources Graziano Guerra, Francesca Marcellini and Veronika Schleper Quaderno n. 8/28 arxiv:89.2664v Stampato nel

More information

Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws

Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws 1 Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws Sébastien Blandin, Xavier Litrico, Maria Laura Delle Monache, Benedetto Piccoli and Alexandre Bayen Abstract

More information

Numerical explorations of a forward-backward diffusion equation

Numerical explorations of a forward-backward diffusion equation Numerical explorations of a forward-backward diffusion equation Newton Institute KIT Programme Pauline Lafitte 1 C. Mascia 2 1 SIMPAF - INRIA & U. Lille 1, France 2 Univ. La Sapienza, Roma, Italy September

More information

Functions with orthogonal Hessian

Functions with orthogonal Hessian Functions with orthogonal Hessian B. Dacorogna P. Marcellini E. Paolini Abstract A Dirichlet problem for orthogonal Hessians in two dimensions is eplicitly solved, by characterizing all piecewise C 2 functions

More information

arxiv: v2 [math.ap] 1 Jul 2011

arxiv: v2 [math.ap] 1 Jul 2011 A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime arxiv:1105.3074v2 [math.ap] 1 Jul 2011 Abstract Philippe G. efloch 1 and Mai Duc Thanh 2 1 aboratoire

More information

2 Introduction T(,t) = = L Figure..: Geometry of the prismatical rod of Eample... T = u(,t) = L T Figure..2: Geometry of the taut string of Eample..2.

2 Introduction T(,t) = = L Figure..: Geometry of the prismatical rod of Eample... T = u(,t) = L T Figure..2: Geometry of the taut string of Eample..2. Chapter Introduction. Problems Leading to Partial Dierential Equations Many problems in science and engineering are modeled and analyzed using systems of partial dierential equations. Realistic models

More information

Stability of Patterns

Stability of Patterns Stability of Patterns Steve Schecter Mathematics Department North Carolina State University 1 2 Overview Motivation Systems of viscous conservation laws and their variants are partial differential equations

More information

SHOCK WAVES FOR RADIATIVE HYPERBOLIC ELLIPTIC SYSTEMS

SHOCK WAVES FOR RADIATIVE HYPERBOLIC ELLIPTIC SYSTEMS SHOCK WAVES FOR RADIATIVE HYPERBOLIC ELLIPTIC SYSTEMS CORRADO LATTANZIO, CORRADO MASCIA, AND DENIS SERRE Abstract. The present paper deals with the following hyperbolic elliptic coupled system, modelling

More information

Singularity formation for compressible Euler equations

Singularity formation for compressible Euler equations Singularity formation for compressible Euler equations Geng Chen Ronghua Pan Shengguo Zhu Abstract In this paper, for the p-system and full compressible Euler equations in one space dimension, we provide

More information

SBV REGULARITY RESULTS FOR SOLUTIONS TO 1D CONSERVATION LAWS

SBV REGULARITY RESULTS FOR SOLUTIONS TO 1D CONSERVATION LAWS SBV REGULARITY RESULTS FOR SOLUTIONS TO 1D CONSERVATION LAWS LAURA CARAVENNA Abstract. A well-posedness theory has been established for entropy solutions to strictly hyperbolic systems of conservation

More information

Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form

Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form Abdelaziz Beljadid, Philippe Lefloch, Siddhartha Mishra, Carlos Parés To cite this version: Abdelaziz Beljadid, Philippe

More information

CapSel Roe Roe solver.

CapSel Roe Roe solver. CapSel Roe - 01 Roe solver keppens@rijnh.nl modern high resolution, shock-capturing schemes for Euler capitalize on known solution of the Riemann problem originally developed by Godunov always use conservative

More information

Math 660-Lecture 23: Gudonov s method and some theories for FVM schemes

Math 660-Lecture 23: Gudonov s method and some theories for FVM schemes Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,

More information

DIMINISHING FUNCTIONALS FOR NONCLASSICAL ENTROPY SOLUTIONS SELECTED BY KINETIC RELATIONS

DIMINISHING FUNCTIONALS FOR NONCLASSICAL ENTROPY SOLUTIONS SELECTED BY KINETIC RELATIONS DIMINISHING FUNCTIONALS FOR NONCLASSICAL ENTROPY SOLUTIONS SELECTED BY KINETIC RELATIONS MARC LAFOREST AND PHILIPPE G. LEFLOCH Abstract. We consider nonclassical entropy solutions to scalar conservation

More information

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow D.W. Schwendeman, C.W. Wahle 2, and A.K. Kapila Department of Mathematical Sciences, Rensselaer Polytechnic

More information

Effects of a Saturating Dissipation in Burgers-Type Equations

Effects of a Saturating Dissipation in Burgers-Type Equations Effects of a Saturating Dissipation in Burgers-Type Equations A. KURGANOV AND P. ROSENAU Tel-Aviv University Abstract We propose and study a new variant of the Burgers equation with dissipation flues that

More information

SOME PRELIMINARY NOTES ON HYPERBOLIC CONSERVATION LAWS

SOME PRELIMINARY NOTES ON HYPERBOLIC CONSERVATION LAWS SOME PRELIMINARY NOTES ON HYPERBOLIC CONSERVATION LAWS KAYYUNNAPARA T. JOSEPH CENTRE FOR APPLICABLE MATHEMATICS TATA INSTITUTE OF FUNDAMENTAL RESEARCH BANGALORE 5665 1. Introduction Many of the physical

More information

Optimality Conditions for Solutions to Hyperbolic Balance Laws

Optimality Conditions for Solutions to Hyperbolic Balance Laws Optimality Conditions for Solutions to Hyperbolic Balance Laws Alberto Bressan and Wen Shen Department of Mathematics, Penn State University University Park, Pa. 1682, U.S.A. e-mails: bressan@math.psu.edu,

More information

Numerische Mathematik

Numerische Mathematik Numer. Math. (2007 106:369 425 DOI 10.1007/s00211-007-0069-y Numerische Mathematik Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem Matania Ben-Artzi Jiequan Li Received:

More information

Entropy-stable discontinuous Galerkin nite element method with streamline diusion and shock-capturing

Entropy-stable discontinuous Galerkin nite element method with streamline diusion and shock-capturing Entropy-stable discontinuous Galerkin nite element method with streamline diusion and shock-capturing Siddhartha Mishra Seminar for Applied Mathematics ETH Zurich Goal Find a numerical scheme for conservation

More information

E = where γ > 1 is a constant spesific to the gas. For air, γ 1.4. Solving for p, we get. 2 ρv2 + (γ 1)E t

E = where γ > 1 is a constant spesific to the gas. For air, γ 1.4. Solving for p, we get. 2 ρv2 + (γ 1)E t . The Euler equations The Euler equations are often used as a simplification of the Navier-Stokes equations as a model of the flow of a gas. In one space dimension these represent the conservation of mass,

More information

Computation of entropic measure-valued solutions for Euler equations

Computation of entropic measure-valued solutions for Euler equations Computation of entropic measure-valued solutions for Euler equations Eitan Tadmor Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics, Institute for Physical

More information

Conservation law equations : problem set

Conservation law equations : problem set Conservation law equations : problem set Luis Silvestre For Isaac Neal and Elia Portnoy in the 2018 summer bootcamp 1 Method of characteristics For the problems in this section, assume that the solutions

More information

A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway

A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway Manuscript A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway Peng Zhang, S.C. Wong and Chi-Wang Shu. Department of Civil Engineering,

More information

AMath 574 February 11, 2011

AMath 574 February 11, 2011 AMath 574 February 11, 2011 Today: Entropy conditions and functions Lax-Wendroff theorem Wednesday February 23: Nonlinear systems Reading: Chapter 13 R.J. LeVeque, University of Washington AMath 574, February

More information

c 2012 Society for Industrial and Applied Mathematics

c 2012 Society for Industrial and Applied Mathematics SIAM J. NUMER. ANAL. Vol. 5, No., pp. 544 573 c Society for Industrial and Applied Mathematics ARBITRARILY HIGH-ORDER ACCURATE ENTROPY STABLE ESSENTIALLY NONOSCILLATORY SCHEMES FOR SYSTEMS OF CONSERVATION

More information

Info. No lecture on Thursday in a week (March 17) PSet back tonight

Info. No lecture on Thursday in a week (March 17) PSet back tonight Lecture 0 8.086 Info No lecture on Thursday in a week (March 7) PSet back tonight Nonlinear transport & conservation laws What if transport becomes nonlinear? Remember: Nonlinear transport A first attempt

More information

On Asymptotic Variational Wave Equations

On Asymptotic Variational Wave Equations On Asymptotic Variational Wave Equations Alberto Bressan 1, Ping Zhang 2, and Yuxi Zheng 1 1 Department of Mathematics, Penn State University, PA 1682. E-mail: bressan@math.psu.edu; yzheng@math.psu.edu

More information

The Slow Erosion Limit in a Model of Granular Flow

The Slow Erosion Limit in a Model of Granular Flow Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor Debora Amadori Wen Shen The Slow Erosion Limit in a Model of Granular Flow Abstract We study a 2 2 system of balance

More information

WEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II

WEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC

More information

Radon measure solutions for scalar. conservation laws. A. Terracina. A.Terracina La Sapienza, Università di Roma 06/09/2017

Radon measure solutions for scalar. conservation laws. A. Terracina. A.Terracina La Sapienza, Università di Roma 06/09/2017 Radon measure A.Terracina La Sapienza, Università di Roma 06/09/2017 Collaboration Michiel Bertsch Flavia Smarrazzo Alberto Tesei Introduction Consider the following Cauchy problem { ut + ϕ(u) x = 0 in

More information

Chapter 9. Derivatives. Josef Leydold Mathematical Methods WS 2018/19 9 Derivatives 1 / 51. f x. (x 0, f (x 0 ))

Chapter 9. Derivatives. Josef Leydold Mathematical Methods WS 2018/19 9 Derivatives 1 / 51. f x. (x 0, f (x 0 )) Chapter 9 Derivatives Josef Leydold Mathematical Methods WS 208/9 9 Derivatives / 5 Difference Quotient Let f : R R be some function. The the ratio f = f ( 0 + ) f ( 0 ) = f ( 0) 0 is called difference

More information

Nonlinear Regularizing Effects for Hyperbolic Conservation Laws

Nonlinear Regularizing Effects for Hyperbolic Conservation Laws Nonlinear Regularizing Effects for Hyperbolic Conservation Laws Ecole Polytechnique Centre de Mathématiques Laurent Schwartz Collège de France, séminaire EDP, 4 mai 2012 Motivation Consider Cauchy problem

More information

Équation de Burgers avec particule ponctuelle

Équation de Burgers avec particule ponctuelle Équation de Burgers avec particule ponctuelle Nicolas Seguin Laboratoire J.-L. Lions, UPMC Paris 6, France 7 juin 2010 En collaboration avec B. Andreianov, F. Lagoutière et T. Takahashi Nicolas Seguin

More information

A FRONT-TRACKING METHOD FOR HYPERBOLIC THREE-PHASE MODELS

A FRONT-TRACKING METHOD FOR HYPERBOLIC THREE-PHASE MODELS A FRONT-TRACKING METHOD FOR HYPERBOLIC THREE-PHASE MODELS Ruben Juanes 1 and Knut-Andreas Lie 2 1 Stanford University, Dept. Petroleum Engineering, USA 2 SINTEF IKT, Dept., Norway ECMOR IX, August 30 September

More information

Introduction to hyperbolic PDEs and Shallow Water Equations

Introduction to hyperbolic PDEs and Shallow Water Equations Introduction to hyperbolic PDEs and Shallow Water Equations Technische Universität München Fundamentals of Wave Simulation - Solving Hyperbolic Systems of PDEs Supervisor Leonhard Rannabauer 8/12/2017

More information

L 1 Stability for scalar balance laws. Control of the continuity equation with a non-local flow.

L 1 Stability for scalar balance laws. Control of the continuity equation with a non-local flow. L 1 Stability for scalar balance laws. Control of the continuity equation with a non-local flow. Magali Mercier Institut Camille Jordan, Lyon Beijing, 16th June 2010 Pedestrian traffic We consider tu +

More information

1. Introduction. We consider finite element (FE) methods of least-squares (LS) type [7] for the scalar nonlinear hyperbolic conservation law

1. Introduction. We consider finite element (FE) methods of least-squares (LS) type [7] for the scalar nonlinear hyperbolic conservation law SIAM J. SCI. COMPUT. Vol. 26, No. 5, pp. 573 597 c 25 Society for Industrial and Applied Mathematics NUMERICAL CONSERVATION PROPERTIES OF H(div)-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE

More information

STRUCTURAL STABILITY OF SOLUTIONS TO THE RIEMANN PROBLEM FOR A NON-STRICTLY HYPERBOLIC SYSTEM WITH FLUX APPROXIMATION

STRUCTURAL STABILITY OF SOLUTIONS TO THE RIEMANN PROBLEM FOR A NON-STRICTLY HYPERBOLIC SYSTEM WITH FLUX APPROXIMATION Electronic Journal of Differential Equations, Vol. 216 (216, No. 126, pp. 1 16. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STRUCTURAL STABILITY OF SOLUTIONS TO THE RIEMANN

More information

Stable, Oscillatory Viscous Profiles of Weak, non-lax Shocks in Systems of Stiff Balance Laws

Stable, Oscillatory Viscous Profiles of Weak, non-lax Shocks in Systems of Stiff Balance Laws Stable, Oscillatory Viscous Profiles of Weak, non-lax Shocks in Systems of Stiff Balance Laws Stefan Liebscher liebsch@math.fu-berlin.de http://www.math.fu-berlin.de/ Dynamik/ Dissertation eingereicht

More information

arxiv: v1 [math.ap] 22 Sep 2017

arxiv: v1 [math.ap] 22 Sep 2017 Global Riemann Solvers for Several 3 3 Systems of Conservation Laws with Degeneracies arxiv:1709.07675v1 [math.ap] 22 Sep 2017 Wen Shen Mathematics Department, Pennsylvania State University, University

More information

Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations

Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations Steve Bryson, Aleander Kurganov, Doron Levy and Guergana Petrova Abstract We introduce a new family of Godunov-type

More information

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,

More information

1 Basic Second-Order PDEs

1 Basic Second-Order PDEs Partial Differential Equations A. Visintin a.a. 2011-12 These pages are in progress. They contain: an abstract of the classes; notes on some (few) specific issues. These notes are far from providing a

More information

Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions

Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 2, May 2016, pp. 199 206. c Indian Academy of Sciences Ray equations of a weak in a hyperbolic system of conservation laws in multi-dimensions PHOOLAN

More information

Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)

Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1) Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL

More information

Entropy-based moment closure for kinetic equations: Riemann problem and invariant regions

Entropy-based moment closure for kinetic equations: Riemann problem and invariant regions Entropy-based moment closure for kinetic equations: Riemann problem and invariant regions Jean-François Coulombel and Thierry Goudon CNRS & Université Lille, Laboratoire Paul Painlevé, UMR CNRS 854 Cité

More information

c1992 Society for Industrial and Applied Mathematics THE CONVERGENCE RATE OF APPROXIMATE SOLUTIONS FOR NONLINEAR SCALAR CONSERVATION LAWS

c1992 Society for Industrial and Applied Mathematics THE CONVERGENCE RATE OF APPROXIMATE SOLUTIONS FOR NONLINEAR SCALAR CONSERVATION LAWS SIAM J. NUMER. ANAL. Vol. 9, No.6, pp. 505-59, December 99 c99 Society for Industrial and Applied Mathematics 00 THE CONVERGENCE RATE OF APPROXIMATE SOLUTIONS FOR NONLINEAR SCALAR CONSERVATION LAWS HAIM

More information