Dedicated with appreciation to Tai-Ping Liu on his 60 th birthday

Transcription

1 ENTROPY STABLE APPROXIMATIONS OF NAVIER-STOKES EQUATIONS WITH NO ARTIFICIAL NUMERICAL VISCOSITY EITAN TADMOR AND WEIGANG ZHONG Abstract. We construct a new family of entropy stable difference schemes which retain the precise entropy decay of the Navier-Stokes equations, d/dt ρsd = λ +µq /θ + κθ /θ d. To this end we employ the entropy conservative differences of [Tadmor4] to discretize Euler convective flues, and centered differences to discretize the dissipative flues of viscosity and heat conduction. The resulting difference schemes contain no artificial numerical viscosity in the sense that their entropy dissipation is dictated solely by viscous and heat flues. Numerical eperiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts. Dedicated with appreciation to Tai-Ping Liu on his 6 th birthday Contents. Introduction. Entropy dissipation 4.. The entropy variables 4.. Eamples of entropy pairs for Navier-Stokes equations 6 3. Entropy stable schemes for Navier-Stokes equations Entropy conservative schemes Entropy stable semi-discrete schemes for Navier-Stokes equations 3.3. Time discretization 3 4. Numerical eperiments 4 References 5. Introduction We consider the full Navier-Stokes NS equations for compressible viscous flows in one-space dimension, ρ m + m qm + p =λ +µ t E qe + p q + κ q /.. θ Here, ρ = ρ, t is the density of the flow, m = m, t is the momentum and E = E, t stands for the total energy per unit volume. The three equations epress, respectively, conservation laws of mass, momentum and total energy for the flow, driven by convective flues on the left together with viscous Date: December, Mathematics Subject Classification. 76N,65M. Key words and phrases. Navier-Stokes equations, viscosity, heat conduction, entropy decay, entropy variables, difference schemes, entropy conservative schemes, entropy stability.

2 Eitan Tadmor and Weigang Zhong and heat flues on the right. These flues involve the velocity q := m/ρ, the pressure p = p, t which is determined by an ideal polytropic equation of state, p =γ e, e := E m ρ,. and the absolute temperature, θ = θ, t >, such that C v ρθ = e. The constant γ > is the specific heat ratio and e = e, t is the internal energy. On the RHS of. we have the viscous and heat flues, depending on the constant Lamé coefficients of the viscosity λ, µ > and the constant conductivity κ>. Finally, C v > is the specific heat at constant volume; for simplicity, we set C v = while rescaling κ κ/c v. If the heat flu is ecluded from the full NS equations, i.e. κ =, we obtain the viscous NS equations ρ m + m qm + p =λ +µ t E qe + p q..3 q / On the other hand, if we turn off the viscosity, i.e. λ = µ =, then the NS equations amount to t ρ m E + m qm + p qe + p = κ θ..4 If the heat flu and viscosity are both taken away, the system. is reduced to the compressible Euler equations, ρ m + m qm + p =..5 t E qe + p The additional viscous and heat flu terms on the RHS of the various NS equations.,.3 or.4, are dissipative terms in the sense that they are responsible for the dissipation of the total entropy. To this end, we now discuss the entropy balance associated with the above equations. We begin with the specific entropy S := ln pρ γ. A straightforward manipulation on.,. yields the transport equation, S t + qs = κ ρ ln θ +λ +µ q e + κ ρ θ..6 θ Multiplied by ρ,.6 becomes ρs t + ms = κln θ +λ +µ q θ + κ θ..7 θ On the other hand, pre-multiplying the continuity equation, ρ t + m =,bys and adding it to.7, ρs + ms + κln θ = λ +µ q t θ κ θ..8 θ Spatial integration of.8 then yields d ρs d = λ +µ dt q θ d κ θ d..9 θ Since the epression on the right is negative, we conclude that the total entropy, ρs d, is decreasing in time, thus recovering the second law of thermodynamics, e.g., [GrootMazur984]. In fact, equation.9 specifies the precise entropy decay rate, which is dictated by the viscous and heat flues through their dependence on the nonnegative κ, λ, and µ.

3 Entropy stable approimations for the Navier-Stokes equations 3 The purpose of this paper is to present a systematic study of difference schemes which respect the above entropy dissipation statements. The typical approach by practitioners in the field of Computational Fluid Dynamics is to address the general issue of entropy stability by adding enough artificial numerical viscosity often an ecessive amount of it, in order to mask various discretizations errors and enforce the decay of the total entropy ρs d. Our aim here is to construct more faithful approimations of the NS equations, with a discrete analogue for the precise entropy decay statement in.9. Our prototype result is the following. Theorem.. Consider the NS equations., t u + fu =ɛ du. Here, u = [ ρ, m, E ] is the vector of conservative variables, fu is the corresponding 3-vector of flues fu = [ m, qm + p, qe + p ], and ɛdu stands for the combined viscous and heat flues, ɛdu =λ +µ [,q,q / ] + κ [,,θ ] where ɛ signals the vanishing amplitudes of viscosity and heat conductivity. We approimate these NS equations by a semi-discrete scheme of the form d dt u νt+ f f = ɛ du ν+ du ν du ν du ν..a ν+ ν ν ν ν+ ν Let f = f u ν+ ν, u ν+ is the numerical flu given the by the eplicit formula, f ν+ =γ 3 j= m j+ m j [ l j, v ν+ v ν lj, v = vu := E e S + γ +, q ]. θ,.b θ Here, {l j = l j } 3 ν+ j= are three linearly independent directions in v-space at our disposal consult eamples 3., 3. and 3.3 below; {r j = r j } 3 ν+ j= is the corresponding orthogonal system and {mj = m j } 4 ν+ j= are the intermediate values of the momentum specified along the corresponding path, vj+ = v j + l j, v ν+ v ν r j, starting with v = v ν and ending with v 4 = v ν+. Then, the resulting scheme.a,.b is entropy stable and the following discrete entropy balance holds d dt ρ ν S ν ν = ν λ +µ qν+ /θ ν ν+ ν+ ν+ κ θν+ /θ ν ν+ ν+ ν+.. We briefly describe the content of the paper, leading to the above result. The entropy balance. is a precise discrete analogue of.9. The scheme.a,.b contains no artificial numerical viscosity in the sense that entropy dissipation is driven solely by the viscous and heat flues. In the particular case that viscosity and the heat conduction are absent, κ = λ = µ =, then the entropy balance.8 is reduced to the formal entropy equality of the Euler equations.5, ρs + t ms =,. We let ẑ ν+ and z ν+ denote the arithmetic and geometric means, ẑ ν+ = z ν + z ν+ / and zν+ = z νz ν+.

4 4 Eitan Tadmor and Weigang Zhong which in turn, implies the entropy conservation ρs,t d = ρs, d. Similarly, setting ɛd = we omit the dissipative terms in NS equations, and the difference scheme.a becomes entropy conservative, ρν S ν t ν = ρ ν S ν ν. Entropy conservative schemes are studied in section 3, following [Tadmor4]. The key ingredient here is the construction of their entropy conservative flues, such as f in.b. These flues employ ν+ the so called entropy variables, v = vu, which are discussed in section. The main results are then summarized in theorems 3. and 3.. Finally, in section 4 we present a series of numerical simulations with the new schemes. The entropy conservative approimations of Euler equations are purely dispersive and as such, their solutions eperience dispersive oscillations, interesting for their own sake, consult [La986, HFMM986, Tadmor986, LLV993, LevermoreLiu996, LeFlochRhode] and the references therein. Turning to the NS equations, our simulations provide a remarkable evidence for the different roles that viscosity and heat conduction have in removing the dispersive oscillations, to yield sharp monotone profiles of well-resolved shock and contact layers. No limiters were added, but instead, the viscous and heat conduction terms in NS equations are found to serve as accurate edge detectors Remark.. The viscous NS equations dissipate a general family of entropies, ρhs, where hs is an arbitrary increasing function. Indeed, arguing along the above lines we multiply the continuity equation by hs and adding it to.7 h S to find ρhs + t mhs+κln θ h S = θ..3 θ = κh θ SS θ h S λ +µ q θ + κ In the case that the heat conduction is absent, the first term on the RHS of.3 vanishes, and we are left with d q ρhs d = λ +µ dt θ h Sd..4 Thus, the viscous NS equations imply the dissipation of a family of entropies, ρhs d for all h S > ; consult [Harten983]. Each one of these entropies carries its own entropy conservative flu f. The eplicit construction of such flues is outlined in theorem 3. below. Combining ν+ these entropy conservative flues together with centered differencing of the additional viscous terms, λ +µ[,q,q /], yield a generalization of theorem. which recovers the precise entropy balance.4. We note in passing that when heat conduction is present, however, the negativity of the first term on the right of.3 requires h S =, so that we are left with one canonical entropy, hs S discussed in theorem.; consult [HFMM986]. Remark.. It is straightforward to generalize the recipe for faithful entropy stable approimations of multidimensional NS equations. The etension is carried out dimension by dimension and as indicated in the one-dimensional setup of theorem., one has the freedom of choosing different paths in phase space.. Entropy dissipation.. The entropy variables. We turn our attention to general systems of hyperbolic conservation laws t u + fu =,, t R [,,.

5 Entropy stable approimations for the Navier-Stokes equations 5 governing the N-vector of conserved variables, u =[u,...,u N ]. The entropy equality t Uu+ F u =.. is an additional conservation law for an entropy function, Uu, which is balanced by an entropy flu F u. The Euler equations.5 are viewed as a prototype eample for such systems, with the three conservative variables u = [ ρ, m, E ] [ ] balanced by the flu f = m, qm + p, qe + p and endowed with entropy pairs U, F = ρhs, mhs. We now briefly recall the circle of ideas linking the dissipation of the total entropy, Uu,td, and the realization of u as a vanishing viscosity limit, in analogy to the vanishing NS limits and their relation to entropic solutions of the Euler equations. We refer to e.g., [Liu99] and [Dafermos], for a more comprehensive discussion. Let Uu,Fu be a given entropy pair associated with.. Note that Uu satisfies the entropy equality. if and only if it is linked to an entropy flu function F u through the compatibility relation U u f u = F u..3 Indeed, multiplying. by Uu on the left, one recovers the equivalence between. and.3 for all u s solving.. Of course, these formal manipulations are valid only under the smooth regime. To justify these steps in the presence of shock discontinuities, the conservation law. is realized by appropriate vanishing viscosity limits. To this end, we define the entropy variables vu :=U u u. We make the additional assumption that the entropy Uu isconve, so that the mapping u v is a one-to-one. Following, [Godunov96, Mock98], we claim that the change of variables, u = uv, puts the system. into the equivalent symmetric form, t uv+ fuv =..4 The system.4 is symmetric in the sense that the Jacobian matrices of flues are, u v v = u v v and fv v = f v v..5 Indeed, a straightforward computation utilizing the compatibility relation.3, shows that uv and fv are, respectively, the gradients of the corresponding potential functions, φ and ψ, uv =φ v v, φv := v, uv Uuv,.6 fv =ψ v v, ψv := v, fv F uv..7 Hence the Jacobian matrices Hv :=u v v and Bv :=f v v in.5 are symmetric, being Hessians of the potentials φv and ψv. Moreover, the conveity of U implies that H is positive definite, H =U uu >. Physically relevant solutions of. are postulated as limits of the vanishing viscosity solutions, t uɛ + fuɛ =ɛ duɛ,.8 where du is any admissible dissipative flu, and ɛ stands for vanishing amplitudes such as the viscosity coefficients λ, µ, the heat conductivity κ, etc. Here, the admissibility of the dissipative flu requires the Jacobian d u to be H-symmetric positive-definite, that is, d H =d H, d H := d u H..9 For brevity of notations we often write fv for fuv whenever the different dependence of fu and fv is made clear by the distinction between the conservative variables u and entropy variables v.

6 6 Eitan Tadmor and Weigang Zhong If we epress the dissipation flu in terms of the entropy variables, dv = duv, then admissibility requires that the v-jacobian of this flu will be positive symmetric, d H = d v v =d v v. Thus, in this case.8 reads t uvɛ + fvɛ =ɛ dvɛ, and all v-dependent flues the temporal, spatial and the dissipation flu have symmetric Jacobains. We now integrate.8 against v = Uu, employ the compatibility relation U u f = F and use differentiation by parts on the admissible dissipation on the right to find t Uuɛ + F ɛ u ɛ ɛ v ɛ, du ɛ = ɛ v ɛ, d v v ɛ.. Letting ɛ, we obtain the entropy inequality, [La973] t Uu+ F u,. which is the generalization of the entropy decay statements for NS equations.8 and.3... Eamples of entropy pairs for Navier-Stokes equations. The NS equations admit the family of conve entropy pairs Uu = ρhs, Fu = mhs, h.. Here S =lnpρ γ is the specific entropy and the conveity of the corresponding Uu s as functions of u =ρ, m, E holds iff h S γh S >, [Harten983]. We consider two prototype eamples. Eample.. The simplest choice of hs is the specific entropy S itself, hs =S =ln pρ γ..3 Straightforward computation gives us the following entropy pair, entropy variables, and potentials. Entropy pair Uu = ρs and F u = ms; Entropy variable consult [Harten983] with the inverse mapping uv = p v 3 v = w γ v v 3 vu = v 3 v v v 3 E/e S + γ + q/θ /θ, where w =.4 γ γ S e v 3 γ γ, S = γ v + v ; v 3 Potential pair φ =γ ρ and ψ =γ m. In this case, the general statement of entropy balance in. with the entropy pair U, F = ρs, ms amounts to the one we have in.9, d dt ρs d = ɛ v ɛ, d vv ɛ d = λ +µ q θ d κ θ d..5 θ Eample.. A particularly convenient form of entropy variables is associated the entropy function consult [Harten983, Tadmor4], Uu = ρhs with hs = γ + γ e S γ+,.6 where we have the following.

7 Entropy stable approimations for the Navier-Stokes equations 7 Entropy pair Uu = +γ γ ρp +γ and F u = +γ γ qρp Entropy variable vu = u Uu = ρp γ +γ with the inverse mapping uv = ρp γ γ+ v 3 v v = E m ρ +γ ; [ ] γ γ v v 3 v γ v 3 v v Potential pair φ, ψ = ρp γ+,mpρ γ γ+ In case that the heat conduction is absent κ =, we apply the general statement of entropy balance. with the entropy pair, U, F = +γ γ ρp +γ,qρp +γ, obtaining ρp +γ d = ɛ v, ɛ d v v ɛ d d dt λ +µ γ h Sq d = λ +µ + γθ +γ ; e S/+γ q d..7 θ Remark.. As noted in [Harten983], the flu fv is a homogeneous function of degree η =: + γ/ γ, fαv =α η fv, α R. Homogeneity implies that f v vv = ηfv which in turn, enables us to rewrite the spatial flu in. in a skew-adjoint form, fu = f v v +f v v /η + ; consult [Tadmor984a]. 3. Entropy stable schemes for Navier-Stokes equations 3.. Entropy conservative schemes. We turn our attention to consistent approimations of., based on semi-discrete conservative schemes of the form d dt u νt = f ν+ f ν ν. 3. Here, u ν t denotes the discrete solution along the grid line ν,t, ν := ν+ ν is the possibly variable mesh spacing and f ν+ is the Lipschitz-continuous numerical flu which occupies a stencil of p + -gridvalues, = fu ν p+,, u ν+p. f ν+ The scheme is said to be consistent with the system. if f satisfies fu, u,, u =fu, u R N. By making the changes of variables u ν = uv ν, we obtain the equivalent form of 3. d dt uv νt = f ν+ f ν ν. 3. The essential difference lies with the numerical flu, f ν+, which is now epressed in terms of the entropy variables, f ν+ = f v ν p+,, v ν+p :=f u v ν p+,, u v ν+p, consistent with the differential flu, fv, v,, v =fv fuv. 3.3 The semi-discrete schemes 3. and 3. are completely identical. It proved useful, however, to work with the entropy variables rather than the usual conservative ones, since system. is symmetrized with respect to these entropy variables. The entropy variables-based formula 3. has the advantage that it provides a natural ordering of symmetric matrices, which in turn enables us to compare the

8 8 Eitan Tadmor and Weigang Zhong numerical viscosities of different schemes, consult [Tadmor984b, Tadmor987] for details. In particular, we will be able to utilize the entropy conservative discretization of [Tadmor4] for the Euler convective part of the equations, and thus recover the precise entropy balance dictated by viscous and heat flues in NS equations. Let U, F be a given entropy pair. We proceed with the construction of an entropy conservative scheme, in the sense of satisfying a discrete entropy equality analogous to., d dt U u νt + ν F ν+ F ν =. 3.4 Here, F ν+ = F u ν p+,, u ν+p is a consistent numerical entropy flu, such that F u, u,, u = F u, u R N. The numerical flu of such entropy conservative schemes will play an essential role in the construction of entropy stable schemes, by adding a judicious amount of physical viscosity. The key step in the construction of entropy conservative schemes is the choice of an arbitrary piecewise-constant path in phase space, connecting two neighboring gridvalues v ν and v ν+ through the intermediate states {v j } N ν+ j= at the spatial cell [ ν, ν+ ]. Let {r j } N ν+ j= be an arbitrary set of N linearly independent N-vectors, and let {l j } N ν+ j= be the corresponding orthogonal set. We introduce the intermediate gridvalues {v j } N ν+ j=, which define a piecewise constant path in phase space v ν+ v j+ ν+ v N+ ν+ = v ν = v j ν+ + = v ν+ l j, v ν+ ν+ r j, j =,,,N, ν+ v ν+ := v ν+ v ν. 3.5 Theorem 3. Tadmor4, Theorem 6.. Consider the system of conservation laws.. Given the entropy pair U, F, then the conservative scheme d dt u νt = f f ν+ ν ν 3.6 with a numerical flu f ν+ N ψ v j+ ψ v j f ν+ ν+ = ν+ j= l j, v ν+ ν+ l j ν+ 3.7 is an entropy-conservative approimation, consistent with.,.. Here, v are the entropy variables, v = U u u and ψv is the entropy potential.7 ψv = v, fuv F uv. The proof is based on the requirement of entropy conservation in [Tadmor987, Theorem 5.], v ν+, f = ψ ν+ ν+, ψ ν+ := ψv ν+ ψv ν. 3.8 The numerical flu 3.7 satisfies this entropy conservation requirement, for N ψ v j+ ψ v j v ν+, f ν+ ν+ = l j, v ν+ j= l j ν+ ν+, v ν+ ν+ = N j= ψ v j+ ψ v j ν+ ν+ = ψ v N+ ψ v ν+ ν+ = ψ ν+.

9 Entropy stable approimations for the Navier-Stokes equations 9 In addition, f ν+ path, v j+ ξ :=v j + v j+ /+ξ l j, v ν+ ν+ ν+ ν+ ν+ r j ν+ we have is a consistent flu satisfying 3.3. Indeed, if we let v j+ ξ denote intermediate ν+ ψ v j+ ψ v j = ν+ ν+ ξ= d dξ ψ v j+ ξ ν+ connecting v j ν+ dξ and v j+, then by.7, ν+ = f v j+ ξ dξ, r j l j, v ξ= ν+ ν+ ν+ ν+. Inserted into 3.7, we can rewrite the entropy-conservative flu 3.7 in the equivalent form N f = f v j+ ν+ ξ dξ, r j l j, 3.9 ν+ ν+ ν+ j= ξ= and the consistency relation 3.3 now follows, f v, v = N fv, r j l j = fv. ν+ ν+ j= We emphasize that the recipe for construction entropy-conservative flues in 3.7 allows an arbitrary choice of a path in phase space. We demonstrate this recipe with three eamples. Eample 3.. Set {r j } along the standard Cartesian coordinates, r j = e ν+ j, j =,,...,N. In this case we have [ v j vν+ = ν+,..., v ν+ j, v ν j,...,, v ν N] j =, 3,...,N, and the entropy conservative flu 3.7 is given by the particularly simple eplicit formula = ψ v ν+ ψ vν ψ v j+ vν+ ν+ ψ v j ψ v,..., ν+ ν+ ψ v N v ν vν+ j ν+,..., v ν vν+ N. 3. v ν f ν+ We carried out numerical eperiments with these flues for the approimate solution of Euler and NS equations. The formulation is particularly simple though the computed intermediate values might lie outside the physical space ρ, p >. Eample 3.. A more physically relevant choice than the Cartesian path is offered by a Riemann path which consists of {u j } N ν+ j=, stationed along an approimate set of right eigenvectors, { rj }, ν+ of the Jacobian f u u ν+. Set v j = vu j, j=,,...,n, and let l j s be the orthogonal system ν+ ν+ to {v j+ v j } N j=. This will be our choice of a path for computing entropy stable approimations of NS equations in section 3. below. The resulting flu, miing conservative and entropy variables, admits the somewhat simpler form N ψ u j+ ψ u j f ν+ ν+ = l j where ψu = U ν+ j= l j ν+ u u fu F u. 3., v ν+ ν+ Eample 3.3. If all r j s are chosen to approach the same direction of v ν+, then by 3.9 the flu 3.9 collapses to the entropy-conservative flu f ν+ = ξ= f v ν+ ξ dξ, v ν+ j ξ := v ν + v ν+ +ξ v ν+. 3. N

10 Eitan Tadmor and Weigang Zhong The resulting flu 3. was introduced in [Tadmor986] and was the forerunner for the family of entropy conservative flues outlined in theorem 3.. It has the drawback, however, that its evaluation requires a nonlinear integration in phase space. Thus, with the loss of linear independence, we lose here the eplicit evaluation of the entropy conservative flu offered in 3.7 and demonstrated in the previous two eamples. 3.. Entropy stable semi-discrete schemes for Navier-Stokes equations The compressible Euler equations. Let U, F be an admissible entropy pair associated with the Euler equations.5, let v = vu denote the corresponding entropy variables outlined in eamples. and. above. To conserve the total entropy Uu,td, we appeal to the semi-discrete scheme 3.6 with the entropy-conservative numerical flu 3.7, f ν+ = 3 j= ψ v j+ ν+ ψ l j, v ν+ ν+ v j ν+ l j. ν+ To compute f, we distinguish between two cases. If v ν+ ν = v ν+, we employ the equivalent form of the numerical flu in 3.9, f ν+ = 3 j= ξ= f v j+ ξ dξ, r j l j, ν+ ν+ ν+ which implies that all the intermediate gridvalues coincide, v ν = v = v = v 3 = v 4 = v ν+ ν+ ν+ ν+ ν+ and the entropy-conservative flu amounts to f = f ν+ ν = f ν+. Otherwise, if v ν v ν+, we choose to work along the path which is dictated by an approimate Riemann solver. Specifically, we use the eigensystem of the Roe matri, [A] ν+ = Au ν, u ν+, [Roe98], γ 3 [A] ν+ := q 3 γ q ν+ ν+ γ. 3.3a γ q3 ν+ q ν+ H ν+ H ν+ γ q ν+ γ q ν+ Here q and H are the average values of the velocity q and total enthalpy H =E +p/ρ at Roe-average state, q ν+ = q ν ρν + q ν+ ρν+ ρν +, Hν+ = H ν ρν + H ν+ ρν+ ρ ν+ ρν b ρ ν+ The r j s are the right eigenvectors { r j r j } 3 ν+ j= of the Roe matri 3.3a given by omitting the subscript ν+ of all averaged variables r = q c, r = q, r 3 = q + c, 3.3c H q c q / H + q c with the corresponding left eigenvector set { l j l j ν+ } 3 j= given by + δ/4γ l = + δ/ c, l = δ /γ δ/4γ δ/ c, l 3 = δ/ c γ / c γ / c γ / c. 3.3d

11 Entropy stable approimations for the Navier-Stokes equations Here δ := γ q/ c, and c is the average sound speed given by c =γ H q. We are now able to form the intermediate path in u space as in 3.5 u j+ = u j + lj, u r j, j =,, Since the mapping between u and v is one-to-one, then these intermediate gridvalues in u space, {u j } 4 j=, correspond to intermediate gridvalues {vj } 4 j= in v space. We let {rj } 3 j= be the right vectors connecting these v-values, r j := v j+ v j, and let {l j } 3 j= be the corresponding left orthogonal set. We summarize the algorithm of computing the entropy-conservative flu f in the ν+ following. Algorithm 3.. If u ν = u ν+ then f ν+ = fv ν ; else Set u ν+ := u ν and compute recursively the intermediate states, u j+ ν+ = u j + lj ν+ ν+, u ν+ r j, j =,, ν+ Here, { l j ν+ } and { rj } are the left and right eigensystems of the Roe matri in 3.3c, 3.3d. ν+ Set r j = vu j+ vu j and compute {l j } 3 ν+ ν+ ν+ j= as the corresponding orthogonal system, l j, r k = δ ν+ ν+ jk, r j := v j+ v j 3.6 ν+ ν+ ν+ 3 ψ v j+ ψ v j Compute the entropy-conservative numerical flu, f ν+ ν+ = l j ν+ j= l j ν+, v ν+ ν+ Remark 3.. Observe that if {u j+ u j } 3 j= are linearly independent then, since u v is symmetric positive definite, the corresponding set of directions in v-phase space, {v j+ v j } 3 j=, are also linearly independent, at least when u ν+ is in a small neighborhood of u ν. It guarantees the eistence of the orthogonal set {l j } 3 j=. But what happens when ˆl j, u = for certain j s? for eample, if u ν is connected to u ν+ through a k-shock, then the Roe matri [A] ν+ retains the perfect resolution of such a shock by enforcing ˆl j, u =, j k and we can omit the contribution of these sub-paths to the conservative flu f. The general approach is to construct a precise mirror image of the Roe-path in v-phase space in terms of the right and left orthogonal systems, r j := [H] ˆr j, l j := [H] ν+ ν+ ˆlj, j =,,..., 3.7 where [H] ν+ denotes an averaged symmetrizer such that u ν+ =[H] ν+ v ν+ and there are many such averages. Then, {r j } 3 j= forms the path in v-phase space, vj+ = v j + l j, vr j, which retains the desired Roe property of perfect resolution of shocks. Indeed, if u is a k-shock with speed s then it satisfies omitting subscripts f = s u = s[h] v. But the Roe matri in 3.3a is constructed so that [Roe98], f = [A] u = [A][H] v, and we conclude v = r k. Thus, l j, v =, j k. The corresponding entropy conservative numerical flu reads f = ψ v j + ξ j r j ψ v j ν+ ν+ ν+ l j, ξ j = ν+ ξ ν+ j {j ξj } l j, v ν+ ν+

12 Eitan Tadmor and Weigang Zhong 3... The Navier-Stokes equations. We turn to the construction of entropy-stable schemes for the full NS equations.. To this end, we rewrite the equation as a system of conservation laws t u + fu =ɛ du, u = ρ m m, fu = qm + p, 3.8a E qe + p with additional diffusive terms ɛdu :=λ +µ q q / + κ θ. 3.8b For the convection part on the LHS, we use the same entropy-conservative differencing used for the Euler equations. For the dissipative terms on the RHS, we employ standard centered differences. We arrive at our main result. Theorem 3.. Let U, F be a given entropy pair of the NS equations 3.8a,3.8b, which respect the entropy inequality.. Consider the semi-discrete approimation d dt u νt+ f f = ɛ d ν+ d ν d ν d ν. 3.9a ν+ ν ν ν ν+ ν Here f is an entropy conservative numerical flu 3.7, ν+ 3 ψ v j+ ψ v j f ν+ ν+ = l j, 3.9b ν+ j= l j ν+, v ν+ ν+ which is outlined in algorithm 3. above. {i} The resulting scheme 3.9a,3.9b is entropy-dissipative in the sense that d U ν t ν = ɛ v dt ν+, d ν+ v ν ν ν+ v ν+. 3. ν+ This entropy balance is a discrete analogue of the entropy balance statements.5 and.7. {ii} In the specific case of the canonical entropy pair U, F = ρs, ms, the entropy decay 3. amounts to. d U ν t ν = dt ν qν+ /θ ν+ ν+ ν+ κ ν θν+ /θ ν+ ν+ ν+ λ +µ. 3. ν Proof. We multiply 3.9a by [U u ] ν = v ν, then sum up all spatial cells to get the balance of the total entropy, d U ν t ν + v ν, f f = ɛ v dt ν+ ν ν, d ν+ d ν d ν d ν. 3. ν ν ν ν+ ν Since we chose f as the entropy conservative flu, a straightforward manipulation on the entropy ν+ conservation requirement 3.8 yields the conservative difference, v ν, f f = F ν+ ν ν+ F ν, 3.3

13 Entropy stable approimations for the Navier-Stokes equations 3 where F ν+ = v ν + v ν+, f ν+ ψv ν +ψv ν+. On the other hand, summation by parts on the RHS of 3. yields ɛ ν v ν, d ν+ d ν ν+ d ν d ν ν nuhf = ν = ν = ν ɛ ν+ ɛ ν+ ɛ ν+ By 3.3 and 3.4b, the semi-discrete entropy balance amounts to Here d dt U ν t ν = ν ν ɛ ν+ v ν+ v ν+ v ν, d ν+ d ν v ν+ v ν+, d ν+, d ν+ v v ν+ ν+, d ν+ v v ν+ ν+ 3.4a. 3.4b. 3.5 d ν+ v ν+ = ξ= d v v ν+ ξ dξ, where v ν+ ξ is given by 3.. By the admissibility of the dissipative NS flues d v and the RHS of 3.5 is indeed non-positive. Thus, the semi-discrete scheme 3.9a guarantees the total entropy dissipation. In the specific case of the entropy pair U, F = ρs, ms, the entropy variable are found in.4, and we eplicitly compute the inner products in 3.4a as omitting all subscripts, ν ɛ v, d = ν = λ +µ ν = λ +µ ν { q λ +µ q λ +µ θ q + θ ν+ θ ν q + θ ν+ θ ν q κ θ κ ν κ ν θ } θ θ θ θ. θ ν+ θ ν The discrete entropy balance. now follows. We emphasize the main point made here, namely, we introduce no ecessive entropy dissipation due to spurious, artificial numerical viscosity: by 3., the semi-discrete scheme contains the precise amount of numerical viscosity to enforce the correct entropy dissipation dictated by the NS equations Time discretization. To complete the computation of a semi-discrete scheme, it needs to be augmented with a proper time discretization. To enable a large time-stability region and maintain simplicity, the three-stage third-order Runge-Kutta RK3 method will be used, consult [GST], u ν t n + t =u ν t n + t 9 K ν, +3K ν, +4K ν,3, 3.6

14 4 Eitan Tadmor and Weigang Zhong where K ν, = {f ν+ u n ν, u n ν+ f ν u n ν, u n } ν ν K ν, = { f u n ν+ ν ν + t K ν,, u n ν+ + t K ν+, f u n ν ν + t K ν,, u n ν + t } K ν, K ν,3 = { f u n ν+ ν + 3 t ν 4 K ν,, u n ν+ + 3 t 4 K ν+, f u n ν ν + 3 t 4 K ν,, u n ν + 3 t } 4 K ν, The resulting fully-discrete schemes has a spatial stencil involving seven-point gridvalues, with two ghost boundary values on the left boundary and two ghost boundary on the right required to close the system. For simplicity, these ghost values are etrapolated from the given Dirichlet boundary values. We note in passing that though the fully eplicit RK3 time discretization need not conserve the entropy, it introduces a negligible amount of entropy dissipation; for a general framework of entropy-conservative fully discrete schemes consult [LMR]. 4. Numerical eperiments We consider ideal polytropic gas equations as an approimation of air with γ =.4, C v = 76, κ =.3, λ+µ =.8 5 We simulate the Sod s shocktube problem, [Sod978], where the Euler and NS equations are solved over the interval [, ] subject to Riemann initial conditions {.,.,.5 <.5 ρ, m, E t= =.5,.,.5.5 <<. In the following figures, we display the numerical solutions for the fully discrete scheme 3.6 with the numerical flu 3.7, or in its equivalent yet simpler form 3.. Uniform space and time grid sizes, and t, are used. Both viscous and inviscid cases are eplored. We use different spatial resolutions for the same problem, and adjust time step according to the CFL condition. Different choices of entropy function are also tested in the numerical eperiments. We group our results into four sets.. Euler equations. The first four sets of figures are devoted to the Euler equations with zero viscous and heat flues.5. With the choice of the entropy pair +γ Uu, Fu = γ ρp +γ, +γ γ qρp +γ, 4. Figure 4. depicts the density, velocity, and pressure fields at t =.5 and t =.; here we use =. and t =.5. Comparing these to the corresponding results of the canonical entropy pair Uu, Fu = ρ lnpρ γ, m lnpρ γ, 4. in figure 4., we see that the different choices of entropies do not affect the behavior of the numerical solutions. Figures 4.d and 4.d demonstrate the conservation of the total entropies: the negligible amount of entropy decay 4 is introduced by the RK3 time discretization. t Net, we make the same comparison for the refined the spatial mesh, taking =.5, =.. Figure 4.3 presents the computed solutions of density, velocity, and pressure fields at t =.5 and.

15 Entropy stable approimations for the Navier-Stokes equations 5 t =. with the entropy pair 4. while figure 4.4 depicts the solutions with the canonical entropy pair 4.. The total entropy is shown in figures 4.3d and 4.4d. The above results demonstrate the purely dispersive character of the entropy conservative schemes. Dispersive oscillations on the mesh scale are observed in shocks and contact regions, due to the absence of any dissipation mechanism, consult [La986, LevermoreLiu996]. The numerical solutions do not blow up. Actually, as we refine the mesh, these dispersive oscillations approach a modulated wave envelope. The study of these modulated waves in the conservative Euler equations would be an etremely challenging task. A similar entropy conservative Lagrangian formulation of Euler equations of [TrulioTrigger96] motivated the discussion in [La986]. Navier-Stokes equations with heat flu. We solve the Navier-Stokes equations.4. The results are summarized in the net three sets of figures We follow the same pattern of plotting density, velocity, pressure and total entropy. As before, the choice of entropy pairs 4. in figures 4.5 and 4.6 are very similar. The presence of heat flu causes the oscillations to be dramatically reduced around the contact discontinuity. Furthermore, oscillations are significantly damped around the shock; when the mesh is well-refined, figure 4.7 shows that heat conduction causes these oscillations to be well localized in the immediate neighborhood of the shocks. If the mesh is underresolved, a small portion of dispersive oscillations persist in the neighborhood of shocks. 3. Navier-stokes equations with viscosity and no heat flu. We solve the viscous NS equations.3. The results are summarized in figures Since the results are essentially independent of the choice of entropy, we chose to quote here only the results for the canonical pair 4.. The viscosity in NS equations is doing a better job than heat flu in damping oscillations around the shock discontinuity. The plots of total entropy, reveal a greater entropy decay than the NS equations with heat conduction. On the other hand, we still observe an oscillatory behavior around the contact discontinuity, even with the refined mesh in figure Full Navier-stokes equations with viscous and heat flues. In figures we record the results for the full NS equations.. As before, the difference due to different entropy functions is undetectable and we chose to record here only the canonical entropy. As epected, these numerical solutions are the smoothest ones found in our numerical eperiments. especially in very fine meshes, depicted in figure 4.. Small oscillations remain with underresolved meshes. Not only the oscillations around the shocks are damped out by viscosity, but the oscillations around the contact discontinuity are significantly reduced due to the heat flu. Compared with the results of NS equations with heat conduction.4 in figures , oscillations in the neighborhood of the shock are better damped here thanks to the viscosity terms. The remaining sharp spike at the tip of shock discontinuity is due to the relatively small viscosity coefficient of air. Acknowledgment. Weigang Zhong wants to thank Ning Jiang, Dongming Wei, and Bin Cheng for their very intriguing discussions. Research was supported in part by NSF DMS #DMS4-774 and ONR #N4-9-J-76. References [Dafermos] C. Dafermos, Hyperbolic Conservation Laws in Continuum Mechanics, Springer,. [Godunov96] S. K. Godunov, An interesting class of quasilinear systems, Dokl. Acad. Nauk. SSSR, 39, 96, [GST] S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43,,, 89. [GrootMazur984] S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Dover, 984. [Harten983] A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys , 5-64.

16 6 Eitan Tadmor and Weigang Zhong [HFMM986] T. J. R. Hughes, with L. P. Franca and M. Mallet, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-stokes equations and the second law of thermodynamics, Computer Methods in Applied Mechanics and Engineering, , 3 34; with M Mallet and A Mizukami, A new finite element formulation for computational fluid dynamics: II. Beyond SUPG Computer Methods in Applied Mechanics and Engineering, , ; with M. Mallet, A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems, Computer Methods in Applied Mechanics and Engineering, , 35 38; with M. Mallet, A new finite element formulation for computational fluid dynamics: IV. A discontinuity- capturing operator for multidimensional advective-diffusive systems, Computer Methods in Applied Mechanics and Engineering, , [La973] P. D. La, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS- NSF, SIAM, 973. [La986] P. D. La, On dispersive difference schemes, Physica D, 8, 986, [LLV993] P. D. La, C. D. Levermore and S. Venakides, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, in Important Developments in Soliton Theory 98-99, T. Fokas and V.E. Zakharov eds, Springer-Verlag, 993. [LeFlochRhode] P. G. LeFloch and C. Rohde, High-order schemes, entropy inequalities, and nonclassical shocks, SIAM J. Numer. Anal., 37, no. 6, 3 6. [LMR] P. G. LeFloch, J. M. Mercier, and C. Rohde, Full discrete, entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal., 4, no. 5, [LevermoreLiu996] C. D. Levermore and J. -G. Liu Oscillations arising in numerical eperiments, Physica D, 99, 996, 9. [Liu99] T. -P. Liu, Viscosity criterion for hyperbolic conservation laws, in Viscous Profiles and Numerical Methods for Shock Waves, SIAM. M. Shearer ed. 99, pp [Mock98] M. S. Mock, Systems of conservation of mied type, J. Diff. Eqns, 37, 98, [Roe98] P. L. Roe, Approimate Riemann solvers, parameter vectors and difference schemes, J. Comp. Phys., 43 98, [Sod978] G. Sod, A survey of several difference methods for hyperbolic systems of nonlinear conservation laws, JCP, 7 978, 3. [Tadmor984a] E. Tadmor, Skew-selfadjoint form for systems of conservation laws, J. Math. Anal. Appl., 3, 984, [Tadmor984b] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp., 43, 984, [Tadmor986] E. Tadmor, Entropy conservative finite element schemes in Numerical Methods for Compressible Flows - Finite Difference Element and Volume Techniques, Proc. winter annual meeting of the ASME AMD-Vol. 78 T. E. Tezduyar and T.J.R. Hughes, eds. 986, [Tadmor987] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp., 49, 987, 9 3. [Tadmor4] E. Tadmor, Entropy stability theory for difference approimations of nonlinear conservation laws and related time-dependent problems, Acta Numerica, 4, [TrulioTrigger96] J. G. Trulio and K. R. Trigger, Numerical solution of one dimensional hydrodynamical shock problem, UCRL Report 65, 96.

17 Entropy stable approimations for the Navier-Stokes equations 7.9 D Euler,density,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5 t=..6.4 D Euler,velocity,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5 t=...7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5 t= D Euler,entropy conservative scheme w/ 3rd R K mtd,entrop y vs time,entropy = +γ/ γ*ρ p /+γ p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.. Euler equations with spatial gridpoints, Uu = +γ γ pρ +γ, t =.5 5, = 3

18 8 Eitan Tadmor and Weigang Zhong.9 D Euler,density,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t=..6.4 D Euler,velocity,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t=...7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t= D Euler, entropy conservative scheme w/ 3rd R K mtd, entr opy vs time,entropy = ρ lnpρ γ p.5.3. U total entropy t c Pressure d Total entropy v.s. time Figure 4.. Euler equations with spatial gridpoints, Uu = ρ ln pρ γ and same t and as figure 4.

19 Entropy stable approimations for the Navier-Stokes equations 9.9 D Euler,density,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.5,T ma =. 5 t=..6.4 D Euler,velocity,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.5,T ma =. 5 t=...7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.5,T ma =. 5 t= D Euler,entropy conservative scheme w/ 3rd R K mtd,entrop y vs time,entropy = +γ/ γ*ρ p /+γ p U total entropy t c Pressure d Total entropy v.s. time Figure 4.3. Euler equations with 4 spatial gridpoints, Uu = +γ γ pρ +γ, t =.5 5, =.5 4

20 Eitan Tadmor and Weigang Zhong.9 D Euler,density,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t=..6.4 D Euler,velocity,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t=...7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t= D Euler, entropy conservative scheme w/ 3rd R K mtd, entr opy vs time,entropy = ρ lnpρ γ.38.7 p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.4. Euler equations with 4 spatial gridpoints, Uu = ρ ln pρ γ and same t and as Figure 4.3

21 Entropy stable approimations for the Navier-Stokes equations D Euler,density,γ=.4,C v =76,κ=.3,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5.9 t=..4. D Euler,velocity,γ=.4,C v =76,κ=.3,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5 t=..7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=.3,entropy=+γ/ γ*ρ p /+γ, t=.5e 5, =.,T ma =. 5 t= D Euler,entropy conservative scheme w/ 3rd R K mtd,entrop y vs time,entropy = +γ/ γ*ρ p /+γ p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.5. Navier-Stokes equations with heat conduction and no viscous term. spatial gridpoints, Uu = +γ γ pρ +γ, t =.5 5, = 3

22 Eitan Tadmor and Weigang Zhong.9 D Euler,density,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t=..4. D Euler,velocity,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t=..7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.,T ma =. 5 t= D Euler, entropy conservative scheme w/ 3rd R K mtd, entr opy vs time,entropy = ρ lnpρ γ p.5.3 U total entropy t c Pressure d Total entropy v.s. time Figure 4.6. Navier-Stokes equations with heat conduction and no viscous terms. spatial gridpoints, Uu = ρ ln pρ γ and same t and as Figure 4.5

23 Entropy stable approimations for the Navier-Stokes equations 3.9 D Euler,density,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t=..4. D Euler,velocity,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t=..7 ρ u a Density b Velocity.9 D Euler,pressure,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5,T ma =. 5 t= D Euler, entropy conservative scheme w/ 3rd R K mtd, entr opy vs time,entropy = ρ lnpρ γ p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.7. Navier-Stokes equations with heat conduction and no viscous terms. 4 spatial grids, Uu = ρ ln pρ γ, t =.5 5, =.5 4

24 4 Eitan Tadmor and Weigang Zhong.9 D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t=..4. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t=..7 ρ u a Density b Velocity.9 D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t= D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ lnpρ γ p.5.3 U total entropy t c Pressure d Total entropy v.s. time Figure 4.8. Navier-Stokes equations with viscous terms and no heat conduction. spatial gridpoints, Uu = ρ ln pρ γ, t =.5 5, = 3

25 Entropy stable approimations for the Navier-Stokes equations 5.9 D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t=.. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t=..7 ρ u a Density b Velocity.9 D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t= D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ lnpρ γ p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.9. Navier-Stokes equations with viscous terms and no heat conduction. 4 spatial grids, Uu = ρ ln pρ γ, t =.5 5, =.5 4

26 6 Eitan Tadmor and Weigang Zhong.9 D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t=..4. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t=..7 ρ u a Density b Velocity.9 D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =. 5 t= D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ lnpρ γ p.5 U total entropy t c Pressure d Total entropy v.s. time Figure 4.. Navier-Stokes equations with viscosity and heat conduction. spatial gridpoints, Uu = ρ ln pρ γ, t =.5 5, = 3

27 Entropy stable approimations for the Navier-Stokes equations 7.9 D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t=.. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t=..7 ρ u a Density b Velocity.9 D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ lnpρ γ, t=.5e 5, =.5 5 t= D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ lnpρ γ p U total entropy t c Pressure d Total entropy v.s. time Figure 4.. Navier-Stokes equations with viscosity and heat conduction. 4 spatial gridpoints, Uu = ρ ln pρ γ, t =.5 5, =.5 4

28 8 Eitan Tadmor and Weigang Zhong Department of Mathematics, Center for Scientific Computation and Mathematical Modeling CSCAMM and Institute for Physical Science and Technology IPST, University of Maryland, MD 74. address: Department of Mathematics, Center for Scientific Computation and Mathematical Modelling CSCAMM, University of Maryland, College Park, MD 74 address: