Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models
|
|
- Laurence Harrington
- 6 years ago
- Views:
Transcription
1 B. Després+ X. Blanc LJLL-Paris VI+CEA Thanks to same collegues as before plus C. Buet, H. Egly and R. Sentis methods for FCI Part IV Multi-temperature fluid s s B. Després+ X. Blanc LJLL-Paris VI+CEA Thanks to same collegues as before plus C. Buet, H. Egly and R. Sentis methods for FCI Part IV Multi-temperature fluid s s p. / 7
2 FCI scenario s During the implosion, pure hydrodynamics is a very strong hypothesis. It is much more relevant to consider multi-temperature s. methods for FCI Part IV Multi-temperature fluid s s p. / 7
3 Plan s One temperature for ions T i and one temperature for electrons T e : T i T e Basic considerations One temperature for the matter, and one temperature for radiatiopn T r for the stability of the ablation front methods for FCI Part IV Multi-temperature fluid s s p. 3 / 7
4 The simplified T i T e s Starting point is (page 9 and page 3 of the notes) D t ρ + ρ u = 0, ρd t u + p = F r, ρd t ε e + p e.u (χ e T e ) + W ei = Q r + S, ρd t ε i + p i u + (χ i T i ) W ei = 0, The unknowns of this system are the density ρ(x, t) R + of the plasma, its velocity u(x, t) R 3 and its pressure p(x, t) R. We also have electronic and ionic values : pressures p e (x, t), p i (x, t) R (with p = p e + p i ), energies ε e (x, t), ε i (x, t) R (with E = E e + E i ), and temperatures T e (x, t), T i (x, t). Here, The terms F r and Q r are the radiative sources, and S is an additional source term ling the laser energy drop. This set of equations is closed by an adapted equation of state (ε e, p e, ε i, p i ) = F(ρ, T e, T i ). We will assume that the fluid is described by a perfect gas EOS p i = (γ i )ρc vi T i = (γ i )ρε i, ε i = C vi T i and the electronic part is described by a perfect gas EOS p e = (γ e )ρc ve T e = (γ e )ρε e, ε e = C ve T e. Since electrons are monoatomic γ e = 5 3. methods for FCI Part IV Multi-temperature fluid s s p. 4 / 7
5 Hydrodynamics of the T i T e s The hydrodynamic part is D t ρ + ρ u = 0, ρd t u + p = 0, ρd t ε e + p e.u = 0, ρd t ε i + p i u = 0, This system is non conservative. For discontinuous functions a and b the product a x b is not defined. What is a shock in such a system? We need to transform it into a conservative system of conservation laws. Universal principles : mass is preserved, momentum is preserved ad total energy is preserved. We get after convenient manipulations the correct conservative formulation (in D) t ρ + x (ρu) = 0, t (ρu) + x ρu + p = 0, t (ρe) + x (ρue + pu) = 0, with p = p i + p e and e = ε i + ε e + u. Question : Is there a fourth conservation laws? methods for FCI Part IV Multi-temperature fluid s s p. 5 / 7
6 Conservation of the electronic entropy s Convenient manipulations show that smooth solutions satisfy t (ρ(αs i + βs e )) + x (ρu(αs i + βs e )) = 0, α, β. For discontinuous solutions, these relations are not equivalent. The correct choice is α = 0 and β =. This is called the Born-Oppenheimer hypothesis. It is related to the fact that me is small. m i Zeldovith-Raizer, Cordier (PhD thesis 96), Degond-Luquin, Massot,... Finally t ρ + x (ρu) = 0, t (ρu) + x ρu + p = 0, t (ρs e ) + x (ρus e ) = 0, t (ρe) + x (ρue + pu) = 0. The mathematical entropy law writes t (ρs i ) + x (ρus i ) 0. methods for FCI Part IV Multi-temperature fluid s s p. 6 / 7
7 Shock relations As a consequence the ionic entropy increases at shocks while the electronic entropy is constant as shocks s Exercise : prove it. Solution < : S + i > S i, S + e = S e. σ (ρ R ρ L ) + (ρ R u R ρ L u L ) = 0, σ `ρ R S e,r ρ L S e,l + `ρr u R S e,r ρ L u L S e,l = 0, σ `ρ R S i,r ρ L S i,l + `ρr u R S i,r ρ L u L S i,l > 0. So ρ R (u R σ) = ρ L (u L σ). This is the constant mass flux D = ρ R (u R σ) = ρ L (u L σ). Therefore DS e,r = DS e,l and DS i,r > DS i,l. Assume a shock and the mass flux is positive D > 0. Then S e,r = S e,l and S i,r > S i,l. CQFD This behavior is absolutely fundamental : it explains that ions and electrons behave differently at shocks. In summary physical considerations show that is the correct eulerian system of conservation laws to analyze for the two temperature T i T e. methods for FCI Part IV Multi-temperature fluid s s p. 7 / 7
8 Lagrangian T i T e s In Lagrange variable in dimension one, one gets t τ mu = 0, t u + mp = 0, p = p i + p e, t S e = 0, t e + m(pu) = 0. Set ρ c i = p i τ S i, ρ c e = p i τ S e and ρ c = p i τ S i pe τ S e The sound speed of the lagrangian system is ρc where c = c i + c e. The natural Lagrangian scheme is now M j t (τ j L τj n ) u j+ + u j = 0, M j t (ul j uj n ) + p j+ p j = 0, (S e ) L j (S e ) n j = 0, M j t (el j ej n ) + p j+ u j+ p j u j = 0, with the solver u j+ = (un j + uj+ n ) + ρc (pn j pj+ n ) p j+ = (pn j + pj+ n ) + ρc (un j uj+ n ), h i (ρc) j+ = (ρc) n j + (ρc) n j+. A pure Lagrangian scheme is such that f n+ = f L for all f. methods for FCI Part IV Multi-temperature fluid s s p. / 7
9 With source terms s A simplified eulerian T i T e system with source terms writes t ρ + x ρu = 0 t ρu + x (ρu + p i + p e ) = 0 t ρε i + x ρuε i + p i x u = τ ei (T e T i ) t ρε e + x ρuε e + p e x u = τ ei (T i T e ) + x (K e x T e ). The relaxation time is τ ei. The electronic diffusion coefficient is K e. We assume that ε i = C vi T i et ε e = C ve T e. The rigorous way to write this is t ρ + x ρu = 0 t ρu + x (ρu + p i + p e ) = 0 t ρs e + x ρus e = τ ei Te (T i T e ) + Te x (Ke x Te ) t ρe + x (ρue + p i u + p e u) = x (K e x T e ),. where the unknowns are the density ρ, the momentum ρu, the electronic entropy ρs e and the total energy ρe. methods for FCI Part IV Multi-temperature fluid s s p. 9 / 7
10 solution base on a splitting strategy s First stage : solve the hydro. Second stage solve the remaining part We use the linear law t ρ = 0 t ρu = 0 t ρε i = τ ei (T e T i ) t ρε e = τ ei (T i T e ) + x (K e x T e ). ε i = C vi T i and ε e = C ve T e. The numerical solution of the system can be computed with an implicit linear solver in case the gas is described by perfect gas equations of state. ρ n+ = ρ L, u n+ = u L, ρ L (T i ) n+ (T j i ) L j C vi = t τ ((T e ) n+ ei j ρ L (Te ) n+ j (Te ) L j C ve = t τ ((T i ) n+ ei j K e,i+ (Te ) n+ j+ + (T i ) n+ ), j (T e ) n+ j (Te )n+ j ) K e,i (Te ) n+ j (Te ) n+ j x. methods for FCI Part IV Multi-temperature fluid s s p. 0 / 7
11 An example relevant for ICF in direct drive o The Piston velocity is W p > 0. The domain is Ω(t) = nx L = 0 < x < x R (t) = xr 0 twp. t s x L x(t) R x (0) R Radiation push The ionic temperature is the discontinuous curve. The electronic temperature is the continuous curve. The initial temperature is the blue curve (5 000 K). 9e+06 e+06 7e+06 6e+06 Kelvin 5e+06 4e+06 3e+06 e cm The ionic part of the gas is violently heated by the shock. The electronic temperature is continuous everywhere. The temperature relaxation is visible behind the shock. In front of the shock a prehating phenomenon is visible. This calculation shows the great importance of shocks for ICF flows in the context of direct drive. methods for FCI Part IV Multi-temperature fluid s s p. / 7
12 The Rankine-Hugoniot relation for the T i T e s Start from The Rankine-Hugoniot relations are t ρ + x ρu = 0 t ρu + x (ρu + p i + p e ) = 0 t ρs e + x ρus e = τ ei Te (T i T e ) + Te x (Ke x Te ) t ρe + x (ρue + p i u + p e u) = x (K e x T e ), σ[ρ] + [ρu] = 0, σ[ρu] + [ρu + p i + p e ] = 0, [T e ] = 0, σ[ρs e ] + [ρus e ] = Te [Ke x Te ], σ[ρe] + [ρue + p i u + p e u] = [K e x T e ].. Notice that the continuity of T e is provided by diffusion operator. Problem Prove that the continuity of S e is recovered in the limit K e 0 +. All numerical results support the conjecture. Works by Lefloch, Coquel and coworks (Chalon, Berthon,...) on a similar problem. methods for FCI Part IV Multi-temperature fluid s s p. / 7
13 Discretization of boundary conditions Assume an additional no-heat flux physical conditions. The boundary conditions are x = 0 = x L, u 0,t = 0, x T e = 0 s and x = x R (t) = x R (0) tw p, u xr (t),t = Wp, x Te = 0 One has boundary conditions and 4 equations #(bc ) =, #(eq ) = 4. Question : how can we do? There is no problem for the second stage of the algorithm t ρ = 0 t ρu = 0 t ρε i = τ ei (T e T i ) t ρε e = τ ei (T i T e ) + x (K e x T e ). plus homogeneous Neumann conditions x T e = 0 at boundaries. So the real problem is for the Lagrangian stage of the algorithme. We are left with #(bc ) =, #(eq ) = 4. But it works : this is the miracle of Lagrangian scheme+splitting for this problem!! methods for FCI Part IV Multi-temperature fluid s s p. 3 / 7
14 Solution Structure of the discrete problem s M j t (τ j L τj n ) u j+ + u j = 0, M j t (ul j uj n ) + p j+ p j = 0, (S e ) L j (S e ) n j = 0, M j t (el j ej n ) + p j+ u j+ p j u j = 0, #(bc ) =, #(what is needed ) =. Recall the rule : the equations of the discrete Riemann solver are more important than its solution j pb p L + (ρc) L (u B u L ) = 0, u P = W p. j pb = p L + (ρc) L (u B u L ), u P = W p. So we use in the last j = J max < : p j+ = p j + (ρc) j ( W p u j ), u j+ = W p. that we plug into the scheme. Finally : boundary condition is enough for the hydro!! methods for FCI Part IV Multi-temperature fluid s s p. 4 / 7
15 A simple grey non equilibrium s page 33 of the notes, one group Z E r = E ν dν 0 plus hydrodynamics. A rigorous justification is possible, but with an almost physical scaling. Consider the simplified for +matter (ρ) +.(ρu) = 0, t (ρu) +.(ρu u) + (p + pr ) = 0, t (ρe + Er ) +.((ρe + Er )u + (p + pr )u) =.( t 3σt T r 4 ), Er +.(uer ) + pr.u =.( t 3σt T r 4 ) + σa(t 4 Tr 4 ), with the grey hypothesis σ t = σ a + σ s and p r = Er. Fundamental is 3 E r = at 4 r, a = π5 k 4 See Buet+D. for a rigorous justification. 5c 3 h 3 = Stefan-Boltzmann constant. Trick : Define ε r E r = ρε r. The equation rewrites ρεr +.(uρεr ) + pr.u =.( T 4 r t 3σ ) + σa(t 4 T 4 r ) t and the radiative pressure rewrites p r = (γ r )ρε r with γ r = 4 3. methods for FCI Part IV Multi-temperature fluid s s p. 5 / 7
16 Hydrodynamic analysis The first task is to discretize the non equilibrium asymptotic set of equations. Setting σ a = 0 and σ s = + then we obtain the simplified hyperbolic set of equations in D t (ρ) + x (ρv) = 0, t (ρv) + x (ρv + p + p r ) = 0, p r = Er 3, t (ρe + Er ) + x (ρev + pv + pr v) = 0, Er = T r 4, t Sr + x (Sr v) = 0, Sr = T r 3. s The solver is compatible with the D Rankine-Hugoniot relations σ[ρ] + [ρv] = 0, σ[ρv] + [ρv + p + p r ] = 0, σ[ρe + E r ] + [ρve + pv + p r v] = 0, σ[s r ] + [vs r ] = 0. All this is compatible with the fact that S r = T 3 r is the number of photons if the is Planckian. methods for FCI Part IV Multi-temperature fluid s s p. 6 / 7
17 Explanation s The group of photons in direction Ω S and with frequency ν > 0 has intensity I (ν, Ω) The energy of is E r = R ν,ω IdνdΩ. v photon = cω, e ν = hν, I (ν, Ω) = n photons e ν. The number of photons is N r = R ν,ω hν I dνdω. The entropy of is S r = k R c 3 ν,ω ν (n log n (n + ) log(n + )) dνdω, n = I ν 3. Asumme a Planckian distribution with radiative temperature T r I = h 3 c ν. hν e ktr Then dimension analysis shows that : E r = αt 4 r, Nr = βt 3 r and S r = γt 3 r. Therefore S r is the number of photons. Read the facinating book by Steven Weinberg : The first three minutes (of universe). methods for FCI Part IV Multi-temperature fluid s s p. 7 / 7
18 Some numerical results The first test problem is a radiative Riemann problem. On the left we plot density, velocity and total pressure versus the position x at t = 0. with 000 cells : CFL= Density Velocity Total pressure 4 Sr/rho s On the right. Radiative entropy s r = T r 3 ρ = Sr versus x at t = 0. with 000 cells : CFL=0.5. One notices ρ the exact preservation of s r across the shock and in the rarefaction fan methods for FCI Part IV Multi-temperature fluid s s p. / 7
19 comparison with a moment Full system. The D solver is implicit. s Tm diffusion Tr diffusion Tm M Tr M cells, T = Right :T and T r for diffusion and the variable Eddington factor. Left : E r, F r and f = Fr Er. One notices that the non-equilibirum diffusion overpredicts the propagation of. It justifies (numerically) higher order s. methods for FCI Part IV Multi-temperature fluid s s p. 9 / 7
20 s : starting point s Chapter 4 of the notes. With H. Egly and R. Sentis. Use a cold coupled with one group for + hydrodynamics + T = T r and ρe + at 4 = ρc v T + at 4 ρc v T. The starting point of our analysis is the compressible Euler with non linear heat flux < : t ρ +.(ρu) = 0 t ρu +.(ρu u) + p = 0 t (ρe) + (ρue + pu κnt n T ) = 0. The Spitzer non linear coefficient is n [ 5, 7 ]. The heat flux boundary condition on the exterior boundary Γ r is non linear κnt n nt Γe = b given. methods for FCI Part IV Multi-temperature fluid s s p. 0 / 7
21 D cut of an ablation front The isobar regime is p = (γ )ρc v T C. left=cold ρ c right=hot T h s T c Ablation front velocity Fluid velocity ρ h x=0 Two other important ingredients are x=x (t) f x r ε =! n Tc, and u(t = 0, x) = u c (t), x the cold region. T h methods for FCI Part IV Multi-temperature fluid s s p. / 7
22 Quasi-isobar s Let us define the acceleration g = u c (t) and reset the velocity u u uc. After rescaling one gets where D t ρ + ρ.u = 0 ρ D t u + M p = ρ g F r Dt p + γ γ γ p u (HnT n T ) = 0. The Mach number M = ρ u p. The Froude number F r = u g l = u t g is a measure of the acceleration of the particles coming from the left boundary Γ l in the cold region. H = κn(t ) n+ p u = κn(t ) n+ ρ u 3 measures the velocity of the particles in the hot region. methods for FCI Part IV Multi-temperature fluid s s p. / 7
23 Slow Mach number expansion s We perform an asymptotic expansion of all variables with respect to the square of the Mach number, ρ = ρ (0) + M ρ () +,... One gets the quasi-isobar t ρ +.(ρu) = 0 t (ρu) +.(ρu u) + p = ρg (u nt n T ) = 0, ρt =. or also > < t T + u vort. T T T n = 0, t u + u. u + T p = g, u = nt n T + u vort = u therm + u vort,.u vort = 0. methods for FCI Part IV Multi-temperature fluid s s p. 3 / 7
24 : u vort = 0 Conjecture : For well prepared data, the solution of t T T T n = 0 is approximated by max(ε, θ) n where θ T n is a solution of the θ = 0, x Ω(t), nθ = b, x Γ equation : h, θ = 0, x Γ f (t), t Γ f (t) = θ γ(t), x Γ f (t). s A numerical example is Proof in D! Set Θ = T n. Then t Θ n + xx Θ = 0. Progressive waves are defined by Θ = Θ(x + vt). The generating Kull s function is the progressive wave with v = and Θ( ) = K (x) = K n «n + n (x), normalisation K(0) = e. n + x x0 +vt Set T = εk n. Then (T n )( ) = ε n, (T n ) (+ ) = v and ε n ε t T T xx T n = 0. methods for FCI Part IV Multi-temperature fluid s s p. 4 / 7
25 The classical problem (9) s < : p = 0, p = 0, nx = p, x Ω in (t) = blue region, x Ω in (t), x Ω in (t). The Web page of Howison (Ociam, Oxford) for some historical references about (and also with fresh science) : Mr (inv. of the variable pitch propeller) worked on the propeller of the cruisers of her gracious majesty methods for FCI Part IV Multi-temperature fluid s s p. 5 / 7
26 Ablative in D We use a Finite Element Method for the Poisson equation, and markers for the front. The markers move accordingly to the. Solution numerique Solution analytique 0.4 solution numerique solution analytique t=0 t=0.06 R(t) s temps On the left Γ int (t). On the middle t r(t). On the right : The smoothing effect of the equation for convergent front is visible ; In this regime, ablation fronts are stable. The full ablative : u vort 0 The writes Θ = 0, x Ω Ω in, nθ = v, x Ω = Γ ext, Θ = 0, x Ω in = Γ in (t), x (t) = Θ + u vort, x(t) Γ in (t), t V +. (u therm V) = S, x Ω Ω in, ϕ = T V, x Ω Ω in, u vort = ϕ, x Ω Ω in. The vorticity source is S. The thermic velocity if u therm = n+ n T n+ = Θ n Θ. The vorticity is V = ω T. methods for FCI Part IV Multi-temperature fluid s s p. 6 / 7
27 More results Passive vorticity : S 0 but u vort = 0 The initial data is a front Γ int discretize with 00 markers and with a mode 9. We plot the vorticity. s Active vorticity : S 0 and u vort In this regime, ablation fronts may be unstable. Open problem : design a two-temperature. methods for FCI Part IV Multi-temperature fluid s s p. 7 / 7
All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients
All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients Christophe Chalons LMV, Université de Versailles Saint-Quentin-en-Yvelines
More informationAll-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach regimes
All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach regimes Christophe Chalons LMV, Université de Versailles Saint-Quentin-en-Yvelines
More informationNumerical methods for inertial confinement fusion
Numerical methods for inertial confinement fusion X. Blanc 1, B. Després 2 1 CEA, DAM, DIF, Bruyères-le-Chatel, F-91297 Arpajon, France. 2 Université Pierre et Marie Curie-Paris6, UMR 7598, laboratoire
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More informationAnswers to Problem Set Number 04 for MIT (Spring 2008)
Answers to Problem Set Number 04 for 18.311 MIT (Spring 008) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139). March 17, 008. Course TA: Timothy Nguyen, MIT, Dept. of Mathematics, Cambridge,
More informationAngular momentum preserving CFD on general grids
B. Després LJLL-Paris VI Thanks CEA and ANR Chrome Angular momentum preserving CFD on general grids collaboration Emmanuel Labourasse (CEA) B. Després LJLL-Paris VI Thanks CEA and ANR Chrome collaboration
More informationModel adaptation in hierarchies of hyperbolic systems
Model adaptation in hierarchies of hyperbolic systems Nicolas Seguin Laboratoire J.-L. Lions, UPMC Paris 6, France February 15th, 2012 DFG-CNRS Workshop Nicolas Seguin (LJLL, UPMC) 1 / 29 Outline of the
More informationInvestigation of an implicit solver for the simulation of bubble oscillations using Basilisk
Investigation of an implicit solver for the simulation of bubble oscillations using Basilisk D. Fuster, and S. Popinet Sorbonne Universités, UPMC Univ Paris 6, CNRS, UMR 79 Institut Jean Le Rond d Alembert,
More informationNumerical resolution of a two-component compressible fluid model with interfaces
Numerical resolution of a two-component compressible fluid model with interfaces Bruno Després and Frédéric Lagoutière February, 25 Abstract We study a totally conservative algorithm for moving interfaces
More informationOn a class of numerical schemes. for compressible flows
On a class of numerical schemes for compressible flows R. Herbin, with T. Gallouët, J.-C. Latché L. Gastaldo, D. Grapsas, W. Kheriji, T.T. N Guyen, N. Therme, C. Zaza. Aix-Marseille Université I.R.S.N.
More informationA 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 informationLaser-plasma interactions
Chapter 2 Laser-plasma interactions This chapter reviews a variety of processes which may take place during the interaction of a laser pulse with a plasma. The discussion focuses on the features that are
More informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More informationUn schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme
Un schéma volumes finis well-balanced pour un modèle hyperbolique de chimiotactisme Christophe Berthon, Anaïs Crestetto et Françoise Foucher LMJL, Université de Nantes ANR GEONUM Séminaire de l équipe
More information0.2. CONSERVATION LAW FOR FLUID 9
0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is
More informationNon-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 informationA Godunov-type method for the diphasic Baer-Nunziato model
A Godunov-type method for the diphasic Baer-Nunziato model C. Chalons Joint work with : A. Ambroso, P.-A. Raviart with benefit from discussions with F. Coquel (CNRS, Ecole Polytechnique/CMAP) N. Seguin
More informationComputational Astrophysics 7 Hydrodynamics with source terms
Computational Astrophysics 7 Hydrodynamics with source terms Oscar Agertz Outline - Optically thin radiative hydrodynamics - Relaxation towards the diffusion limit - Hydrodynamics with gravity source term
More informationLecture 5.7 Compressible Euler Equations
Lecture 5.7 Compressible Euler Equations Nomenclature Density u, v, w Velocity components p E t H u, v, w e S=c v ln p - c M Pressure Total energy/unit volume Total enthalpy Conserved variables Internal
More informationOn a simple model of isothermal phase transition
On a simple model of isothermal phase transition Nicolas Seguin Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Paris 6 France Micro-Macro Modelling and Simulation of Liquid-Vapour Flows
More informationNumerical 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 informationTHE numerical simulation of the creation and evolution
Proceedings of the World Congress on Engineering Vol III WCE, July 4-6,, London, U.K. Numerical Simulation of Compressible Two-phase Flows Using an Eulerian Type Reduced Model A. Ballil, Member, IAENG,
More informationFluid Dynamics. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/14
Fluid Dynamics p.1/14 Fluid Dynamics Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/14 The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic). Use
More information0.3.4 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 informationIV. Compressible flow of inviscid fluids
IV. Compressible flow of inviscid fluids Governing equations for n = 0, r const: + (u )=0 t u + ( u ) u= p t De e = + ( u ) e= p u+ ( k T ) Dt t p= p(, T ), e=e (,T ) Alternate forms of energy equation
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationTOPICAL PROBLEMS OF FLUID MECHANICS 53 ON THE PROBLEM OF SINGULAR LIMITS IN A MODEL OF RADIATIVE FLOW
TOPICAL PROBLEMS OF FLUID MECHANICS 53 ON THE PROBLEM OF SINGULAR LIMITS IN A MODEL OF RADIATIVE FLOW Bernard Ducomet, Šárka Nečasová 2 CEA, DAM, DIF, F-9297 Arpajon, France 2 Institute of Mathematics
More informationCapSel Euler The Euler equations. conservation laws for 1D dynamics of compressible gas. = 0 m t + (m v + p) x
CapSel Euler - 01 The Euler equations keppens@rijnh.nl conservation laws for 1D dynamics of compressible gas ρ t + (ρ v) x = 0 m t + (m v + p) x = 0 e t + (e v + p v) x = 0 vector of conserved quantities
More informationTo study the motion of a perfect gas, the conservation equations of continuity
Chapter 1 Ideal Gas Flow The Navier-Stokes equations To study the motion of a perfect gas, the conservation equations of continuity ρ + (ρ v = 0, (1.1 momentum ρ D v Dt = p+ τ +ρ f m, (1.2 and energy ρ
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationLinear stability of MHD configurations
Linear stability of MHD configurations Rony Keppens Centre for mathematical Plasma Astrophysics KU Leuven Rony Keppens (KU Leuven) Linear MHD stability CHARM@ROB 2017 1 / 18 Ideal MHD configurations Interested
More informationHyperbolic 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 informationAME 513. " Lecture 8 Premixed flames I: Propagation rates
AME 53 Principles of Combustion " Lecture 8 Premixed flames I: Propagation rates Outline" Rankine-Hugoniot relations Hugoniot curves Rayleigh lines Families of solutions Detonations Chapman-Jouget Others
More informationProjection Dynamics in Godunov-Type Schemes
JOURNAL OF COMPUTATIONAL PHYSICS 142, 412 427 (1998) ARTICLE NO. CP985923 Projection Dynamics in Godunov-Type Schemes Kun Xu and Jishan Hu Department of Mathematics, Hong Kong University of Science and
More informationShock on the left: locus where cars break behind the light.
Review/recap of theory so far. Evolution of wave profile, as given by the characteristic solution. Graphical interpretation: Move each point on graph at velocity c(ρ). Evolution as sliding of horizontal
More informationRelativistic Hydrodynamics L3&4/SS14/ZAH
Conservation form: Remember: [ q] 0 conservative div Flux t f non-conservative 1. Euler equations: are the hydrodynamical equations describing the time-evolution of ideal fluids/plasmas, i.e., frictionless
More informationWaves in plasma. Denis Gialis
Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous.
More informationThe Euler Equation of Gas-Dynamics
The Euler Equation of Gas-Dynamics A. Mignone October 24, 217 In this lecture we study some properties of the Euler equations of gasdynamics, + (u) = ( ) u + u u + p = a p + u p + γp u = where, p and u
More informationNumerical Methods for Modern Traffic Flow Models. Alexander Kurganov
Numerical Methods for Modern Traffic Flow Models Alexander Kurganov Tulane University Mathematics Department www.math.tulane.edu/ kurganov joint work with Pierre Degond, Université Paul Sabatier, Toulouse
More informationThe Euler Equations! Advection! λ 1. λ 2. λ 3. ρ ρu. c 2 = γp/ρ. E = e + u 2 /2 H = h + u 2 /2; h = e + p/ρ. 0 u 1/ρ. u p. t + A f.
http://www.nd.edu/~gtryggva/cfd-course/! Advection! Grétar Tryggvason! Spring! The Euler equations for D flow:! where! Define! Ideal Gas:! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ E = e + u / H = h + u
More informationModelling and numerical simulation of bi-temperature Euler Equations in toroidal geometry
Modelling and numerical simulation of bi-temperature Euler Equations in toroidal geometry E. Estibals H. Guillard A. Sangam elise.estibals@inria.fr Inria Sophia Antipolis Méditerranée June 9, 2015 1 /
More informationDeforming Composite Grids for Fluid Structure Interactions
Deforming Composite Grids for Fluid Structure Interactions Jeff Banks 1, Bill Henshaw 1, Don Schwendeman 2 1 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore,
More informationThe RAMSES code and related techniques 4. Source terms
The RAMSES code and related techniques 4. Source terms Outline - Optically thin radiative hydrodynamics - Relaxation towards the diffusion limit - Hydrodynamics with gravity source term - Relaxation towards
More informationImplicit kinetic relaxation schemes. Application to the plasma physic
Implicit kinetic relaxation schemes. Application to the plasma physic D. Coulette 5, E. Franck 12, P. Helluy 12, C. Courtes 2, L. Navoret 2, L. Mendoza 2, F. Drui 2 ABPDE II, Lille, August 2018 1 Inria
More informationNumerical 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 informationSpace-time Discontinuous Galerkin Methods for Compressible Flows
Space-time Discontinuous Galerkin Methods for Compressible Flows Jaap van der Vegt Numerical Analysis and Computational Mechanics Group Department of Applied Mathematics University of Twente Joint Work
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 31 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Linearization and Characteristic Relations 1 / 31 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
More informationConservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion
http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation
More informationSimple waves and a characteristic decomposition of the two dimensional compressible Euler equations
Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations Jiequan Li 1 Department of Mathematics, Capital Normal University, Beijing, 100037 Tong Zhang Institute
More informationModeling using conservation laws. Let u(x, t) = density (heat, momentum, probability,...) so that. u dx = amount in region R Ω. R
Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...) so that u dx = amount in region R Ω. R Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...)
More informationA general well-balanced finite volume scheme for Euler equations with gravity
A general well-balanced finite volume scheme for Euler equations with gravity Jonas P. Berberich, Praveen Chandrashekar, Christian Klingenberg Abstract We present a second order well-balanced Godunov-type
More informationLarge time-step and asymptotic-preserving numerical schemes for gaz dynamics with stiff sources
Lagrange-Projection strategy for the gas-dynamics equations Relaxation for the Lagrangian step Source terms and notion of consistency in the integral sense Propertie Large time-step and asymptotic-preserving
More informationDifferentiability with respect to initial data for a scalar conservation law
Differentiability with respect to initial data for a scalar conservation law François BOUCHUT François JAMES Abstract We linearize a scalar conservation law around an entropy initial datum. The resulting
More informationHervé Guillard, INRIA Projet Smash, B.P. 93, Sophia-Antipolis Cedex, France,
TRAVELING WAVE ANALYSIS OF TWO-PHASE DISSIPATIVE MODELS Hervé Guillard, INRIA Projet Smash, B.P. 93, 06902 Sophia-Antipolis Cedex, France, Herve.Guillard@sophia.inria.fr Joint work with : Mathieu Labois,
More informationRadiation hydrodynamics of tin targets for laser-plasma EUV sources
Radiation hydrodynamics of tin targets for laser-plasma EUV sources M. M. Basko, V. G. Novikov, A. S. Grushin Keldysh Institute of Applied Mathematics, Moscow, Russia RnD-ISAN, Troitsk, Moscow, Russia
More informationMath 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 informationPROBLEM SET. Heliophysics Summer School. July, 2013
PROBLEM SET Heliophysics Summer School July, 2013 Problem Set for Shocks and Particle Acceleration There is probably only time to attempt one or two of these questions. In the tutorial session discussion
More informationQuasi-linear first order equations. Consider the nonlinear transport equation
Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x < Quasi-linear first order equations Consider the nonlinear transport equation u t +
More informationAdmissibility and asymptotic-preserving scheme
Admissibility and asymptotic-preserving scheme F. Blachère 1, R. Turpault 2 1 Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes, 2 Institut de Mathématiques de Bordeaux (IMB), Bordeaux-INP
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationNUMERICAL METHODS IN ASTROPHYSICS An Introduction
-1 Series in Astronomy and Astrophysics NUMERICAL METHODS IN ASTROPHYSICS An Introduction Peter Bodenheimer University of California Santa Cruz, USA Gregory P. Laughlin University of California Santa Cruz,
More informationFluid Equations for Rarefied Gases
1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 21 May 2001 with E. A. Spiegel
More informationOn 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 informationYAN GUO, JUHI JANG, AND NING JIANG
LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION YAN GUO, JUHI JANG, AND NING JIANG Abstract. We revisit the classical ork of Caflisch [C] for compressible Euler limit of the Boltzmann equation. By using
More informationThe semi-geostrophic equations - a model for large-scale atmospheric flows
The semi-geostrophic equations - a model for large-scale atmospheric flows Beatrice Pelloni, University of Reading with M. Cullen (Met Office), D. Gilbert, T. Kuna INI - MFE Dec 2013 Introduction - Motivation
More informationWaves in a Shock Tube
Waves in a Shock Tube Ivan Christov c February 5, 005 Abstract. This paper discusses linear-wave solutions and simple-wave solutions to the Navier Stokes equations for an inviscid and compressible fluid
More informationDiffusion limits in a model of radiative flow
Institute of Mathematics of the Academy of Sciences of the Czech Republic Workshop on PDE s and Biomedical Applications Basic principle of mathematical modeling Johann von Neumann [193-1957] In mathematics
More informationCapSel 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 informationCurved spacetime tells matter how to move
Curved spacetime tells matter how to move Continuous matter, stress energy tensor Perfect fluid: T 1st law of Thermodynamics Relativistic Euler equation Compare with Newton =( c 2 + + p)u u /c 2 + pg j
More informationLinear Hyperbolic Systems
Linear Hyperbolic Systems Professor Dr E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro October 8, 2014 1 / 56 We study some basic
More informationρ c (2.1) = 0 (2.3) B = 0. (2.4) E + B
Chapter 2 Basic Plasma Properties 2.1 First Principles 2.1.1 Maxwell s Equations In general magnetic and electric fields are determined by Maxwell s equations, corresponding boundary conditions and the
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationModeling uncertainties with kinetic equations
B. Després (LJLL- University Paris VI) thanks to G. Poette (CEA) and D. Lucor (Orsay) B. Perthame (LJLL) E. Trélat (LJLL) Modeling uncertainties with kinetic equations B. Després (LJLL-University Paris
More informationLecture 5: Kinetic theory of fluids
Lecture 5: Kinetic theory of fluids September 21, 2015 1 Goal 2 From atoms to probabilities Fluid dynamics descrines fluids as continnum media (fields); however under conditions of strong inhomogeneities
More informationÉquation 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 informationWave propagation methods for hyperbolic problems on mapped grids
Wave propagation methods for hyperbolic problems on mapped grids A France-Taiwan Orchid Project Progress Report 2008-2009 Keh-Ming Shyue Department of Mathematics National Taiwan University Taiwan ISCM
More informationNumerical Simulation of Rarefied Gases using Hyperbolic Moment Equations in Partially-Conservative Form
Numerical Simulation of Rarefied Gases using Hyperbolic Moment Equations in Partially-Conservative Form Julian Koellermeier, Manuel Torrilhon May 18th, 2017 FU Berlin J. Koellermeier 1 / 52 Partially-Conservative
More informationEULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS
EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS ML Combrinck, LN Dala Flamengro, a div of Armscor SOC Ltd & University of Pretoria, Council of Scientific and Industrial Research & University
More informationThe one-dimensional equations for the fluid dynamics of a gas can be written in conservation form as follows:
Topic 7 Fluid Dynamics Lecture The Riemann Problem and Shock Tube Problem A simple one dimensional model of a gas was introduced by G.A. Sod, J. Computational Physics 7, 1 (1978), to test various algorithms
More informationThe 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 informationRadiative & Magnetohydrodynamic Shocks
Chapter 4 Radiative & Magnetohydrodynamic Shocks I have been dealing, so far, with non-radiative shocks. Since, as we have seen, a shock raises the density and temperature of the gas, it is quite likely,
More information2. Basic assumptions for stellar atmospheres
. Basic assumptions for stellar atmospheres 1. geometry, stationarity. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1 1. Geometry Stars as gaseous spheres
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationContents of lecture 2b. Lectures 2a & 2b. Physical vs. computational coordinates [2] Physical vs. computational coordinates [1]
Contents of lecture b Lectures a & b P. A. Zegeling Mathematical Institute Utrecht University The Netherlands Parameter-free non-singular moving grids in D: Theory & properties Application to resistive
More informationMATANIA BEN-ARTZI, JIEQUAN LI AND GERALD WARNECKE
A DIRECT EULERIAN GRP SCHEME FOR COMPRESSIBLE FLUID FLOWS MATANIA BEN-ARTZI JIEQUAN LI AND GERALD WARNECKE Abstract. A direct Eulerian generalized Riemann problem (GRP) scheme is derived for compressible
More informationComputations of non-reacting and reacting two-fluid interfaces
Computations of non-reacting and reacting two-fluid interfaces Kunkun Tang Alberto Beccantini Christophe Corre Lab. of Thermal Hydraulics & Fluid Mechanics (LATF), CEA Saclay Lab. of Geophysical & Industrial
More informationPalindromic Discontinuous Galerkin Method
Palindromic Discontinuous Galerkin Method David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, Laurent Navoret To cite this version: David Coulette, Emmanuel Franck, Philippe Helluy,
More informationPhysical Diffusion Cures the Carbuncle Phenomenon
Physical Diffusion Cures the Carbuncle Phenomenon J. M. Powers 1, J. Bruns 1, A. Jemcov 1 1 Department of Aerospace and Mechanical Engineering University of Notre Dame, USA Fifty-Third AIAA Aerospace Sciences
More informationChapter 1 Direct Modeling for Computational Fluid Dynamics
Chapter 1 Direct Modeling for Computational Fluid Dynamics Computational fluid dynamics (CFD) is a scientific discipline, which aims to capture fluid motion in a discretized space. The description of the
More informationHilbert Sixth Problem
Academia Sinica, Taiwan Stanford University Newton Institute, September 28, 2010 : Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem:
More informationAdvanced Newtonian gravity
Foundations of Newtonian gravity Solutions Motion of extended bodies, University of Guelph h treatment of Newtonian gravity, the book develops approximation methods to obtain weak-field solutions es the
More informationNotes: Outline. Diffusive flux. Notes: Notes: Advection-diffusion
Outline This lecture Diffusion and advection-diffusion Riemann problem for advection Diagonalization of hyperbolic system, reduction to advection equations Characteristics and Riemann problem for acoustics
More informationRadiative Transfer Chapter 3, Hartmann
Radiative Transfer Chapter 3, Hartmann Shortwave Absorption: Clouds, H 2 0, O 3, some CO 2 Shortwave Reflection: Clouds, surface, atmosphere Longwave Absorption: Clouds, H 2 0, CO 2, CH 4, N 2 O Planck
More informationFundamentals of compressible and viscous flow analysis - Part II
Fundamentals of compressible and viscous flow analysis - Part II Lectures 3, 4, 5 Instantaneous and averaged temperature contours in a shock-boundary layer interaction. Taken from (Pasquariello et al.,
More informationInternational Engineering Research Journal
Special Edition PGCON-MECH-7 Development of high resolution methods for solving D Euler equation Ms.Dipti A. Bendale, Dr.Prof. Jayant H. Bhangale and Dr.Prof. Milind P. Ray ϯ Mechanical Department, SavitribaiPhule
More informationShock and Expansion Waves
Chapter For the solution of the Euler equations to represent adequately a given large-reynolds-number flow, we need to consider in general the existence of discontinuity surfaces, across which the fluid
More informationA Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations
A Space-Time Expansion Discontinuous Galerkin Scheme with Local Time-Stepping for the Ideal and Viscous MHD Equations Ch. Altmann, G. Gassner, F. Lörcher, C.-D. Munz Numerical Flow Models for Controlled
More informationMulti-D MHD and B = 0
CapSel DivB - 01 Multi-D MHD and B = 0 keppens@rijnh.nl multi-d MHD and MHD wave anisotropies dimensionality > 1 non-trivial B = 0 constraint even if satisfied exactly t = 0: can numerically generate B
More informationApproximate Harten-Lax-Van Leer (HLL) Riemann Solvers for Relativistic hydrodynamics and MHD
Approximate Harten-Lax-Van Leer (HLL) Riemann Solvers for Relativistic hydrodynamics and MHD Andrea Mignone Collaborators: G. Bodo, M. Ugliano Dipartimento di Fisica Generale, Universita di Torino (Italy)
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More information