Novel entropy stable schemes for 1D and 2D fluid equations

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1 Novel entropy stable schemes for D an D flui equations Eitan Tamor an Weigang Zhong Department of Mathematics, Center for Scientific Computation an Mathematical Moeling (CSCAMM, Institute of Physical Science an Technology, University of Marylan, College Park, MD 0 tamor@cscamm.um.eu Department of Mathematics, Center for Scientific Computation an Mathematical Moeling (CSCAMM, University of Marylan, College Park, MD 0 wzhong@math.um.eu Summary. We present a systematic stuy of the novel entropy stable approimations of a variety of nonlinear conservation laws, from the scalar Burgers equation to D Navier-Stoke an D shallow water equations. This new family of secon-orer ifference schemes avoi using artificial numerical viscosity in the sense that their entropy issipation is ictate solely by physical issipation terms. The numerical results of D compressible Navier-Stokes equations equations provie us a remarkable evience for ifferent roles of viscosity an heat conuction in forming sharp monotone profiles in the immeiate neighborhoos of shocks an contacts. Further implementation in D shallow water equations is realize imension by imension. The Invisci Burgers Equation. Entropy conservative schemes We begin with the invisci Burgers equation, u t + f(u =0, f(u = u. ( It is equippe with a family of entropy functions, U p (u =u p, p =,,, such that solutions of ( satisfy, at the formal level, t U p(u+ F p(u =0. ( These are aitional conservation laws balance by the corresponing entropy flu functions F p (u =pu p+ /(p +. Spatial integration then yiels (ignoring bounary contributions

2 Eitan Tamor an Weigang Zhong u p (, t = u p (, 0. (3 We turn to the iscrete framework. Discretization in space yiels the semiiscrete scheme, t u ν(t+ ( f ν+ f ν =0. ( Here, u ν (t enotes the iscrete solution along the griline ( ν,t with ν := ν, being the uniform meshsize, an f ν+ := f(u ν r+,,u ν+r isa consistent numerical flu base on a stencil of r + neighboring gri values, that makes ( conservative in the sense that ν u ν(t = ν u ν(0. Fi p. We seek a semi-iscrete scheme that conserves the entropy U p (u =u p in the sense of satisfying the iscrete analogue of ( p, t U p(u ν (t + (F ν+ F ν =0. Here, F ν+ is a consistent numerical entropy flu. Accoring to [Ta, Theorem.], such 3-point scalar entropy conservative schemes are uniquely etermine by the entropy conservative numerical flu f ν+ = f given by, ν+ f ν+ = f ν+ = p (p + u ν (u ν+/u ν p+ (u ν+ /u ν p. ( The resulting scheme (, ( is entropy conservative in the sense that the iscrete analogue of total entropy conservation (3 is satisfie, ν u p ν (t = ν u p ν (0 Of course, all the above manipulations are at the formal level. To recover the physical relevant entropy inequality, t U p (u+ F p (u 0, one can a numerical issipation, t u ν(t+ ( f ν+ f ν = ɛ ( (u ( ν+ (u ν +(u ν, ɛ > 0. This serves as an approimation to the vanishing viscosity regularization u t + f(u = ɛ(u, (u > 0. Sum this scheme against v ν := U p(u ν =pu p : the resulting entropy balance that follows reas, t ν U p (u ν (t = ɛ ν+ ( v v ν+ 0, ( ν ν+ since ν+ := (uν+ (uν v ν+ v ν+ v ν > 0 for (u > 0. Observe that the amount of entropy issipation on the right is completely etermine by the issipation

3 Novel entropy stable schemes for D an D flui equations 3 term ɛ(u. No artificial viscosity is introuce by the convective term. If we eclue any issipative mechanism (ɛ = 0, the entropy conservative solutions amit ispersive oscillations interesting for their own sake, consult [La, LL]. Scalar Burgers equation,entropy consertive scheme w/ U(u=u, 0, =0.00. Scalar Burgers equation,entropy consertive scheme w/ U(u=u, 0, =0.00. t= t= t= t= u( 0 u( (a p= (b p=. Scalar Burgers equation,entropy consertive scheme w/ U(u=u 3, 0, =0.00. Scalar Burgers equation,entropy consertive scheme w/ U(u=u, 0, =0.00 t= t= t= t= u( 0 u( (c p= ( p=3 Fig.. Scalar Burger s equation, sine initial conition, entropy-conservative schemes, 00 spatial gris, U(u =u p, t= 0 3, = 0 3. Numerical eperiments The semi-iscrete entropy conservative scheme (,( is integrate with the following thir-orer Runge-Kutta (RK3, consult [GST00].

4 Eitan Tamor an Weigang Zhong u ( = u n + tl(u n, u ( = 3 un + u( + tl(u(, ( u n+ = 3 un + 3 u( + 3 tl(u(, where [L(u] ν = (f f. We note that this eplicit RK3 time iscretization prouces a negligible amount of entropy issipation. For a general ν+ ν framework of entropy conservative fully iscrete schemes, consult [LMR00]. We solve the invisci Burgers equation with the sine initial conition, u(0, = sin(π an perioic bounary. In Fig., we isplay the numerical solutions for ( with the numerical flu ( for ifferent choices of p. Observe that the amplitue of the spurious ispersive oscillations ecreases as p increases. Inee, as we increase p, the control of a constant U p entropy, [ ν up ν (t ] p approaches the control of L -norm. The D Navier-Stokes Equations. Entropy issipation We consier the Navier-Stokes equations (NSE governing the ensity ρ = ρ(, t, momentum m = m(, t, an energy E = E(, t, t u + f(u=ɛ (u, u =[ρ, m, E]. ( They are riven by the convective flu f(u = [ m, qm + p, q(e + p ], together with the issipative flu ɛ(u =(λ +µ [ 0,q,q / ] [ ] + κ 0, 0,θ which stans for the combine viscous an heat flues. Here, ɛ represents the amplitues of the viscosity an conuctivity. These flues involve the velocity q := m/ρ, the pressure p = p(, t =(γ e where e := E m ρ, an the absolute temperature, θ = θ(, t > 0, such that C v ρθ = e. Here γ>,λ,µ are fie an κ κ/c v with C v =. The viscous an heat flues are issipative in the sense that they are responsible for the issipation of total entropy, U(u = ρs with S =ln(pρ γ being the specific entropy, t ( ρs + ( ( ms + κ(ln θ = (λ +µ q θ κ θ 0. ( θ Spatial integration of ( then yiels the secon law of thermoynamics, q ( ρs = (λ +µ t θ κ ( θ 0. (0 θ

5 Novel entropy stable schemes for D an D flui equations In fact, (0 specifies the precise entropy ecay rate. In the case of the Euler equations, λ = µ = κ = 0, total entropy is precisely conserve, ρs(, t = ρs(, 0. This correspons to the scalar entropy conservation (3.. Entropy stable schemes for the Navier-Stokes equations We now turn our attention to the construction of entropy stable schemes for NSE (. We use the conservative ifferences for convective flu an stanar centere ifferences for the issipative terms on the RHS. t u ν(t+ ( f ν+ f ν = ɛ ( ( ν+ ν + ν, ( Here, ν := (u ν. As in the scalar case, we seek entropy conservative flu f, so that the entropy ecay will be ictate solely by the issipation terms ν+ on the right of (. The construction of the entropy conservative flu f ν+ follows [Ta00] an [TZ00], an is summarize in the following Algorithm If u ν = u ν+ then f = f(v ν+ ν ; else Set u ν+ := u ν+ u ν. Starting with u := u ν+ ν, compute recussively the intermeiate states, u j+ ν+ = u j ν+ + lj ν+, u ν+ r j, j =,, 3,. ( ν+ Here, { l j ν+ } an { rj } are the left an right eigensystems of the Roe ν+ matri A(u ν, u ν+ (see [Roe]. Set r j := v(u j+ v(u j an let {l j } 3 ν+ ν+ ν+ j= be the corresponing orthogonal system. Compute the entropy-conservative numerical flu, f ν+ =(γ 3 j= m j+ m j ν+ ν+ l j, l j ν+, v ν+ ν+ l j, v j+ v j = δ ν+ ν+ ν+ jk (3 Here, v(u :=U u (u =[ E/e S + γ +,q/θ, /θ] are the entropy variables, an {m j } are intermeiate momentum values along the path. We now arrive at our main result of NSE corresponing to the Burgers statement (. Theorem ([TZ00], Theorem 3.. The semi-iscrete scheme ( with the entropy conservative numerical flu f in (-(3 an (u ν+ ν being the issipative NS flu, is entropy stable in the sense that 3 3 We let ẑ ν+ = ( z ν + z ν+ / an zν+ = z νz ν+.

6 Eitan Tamor an Weigang Zhong t [ ρ ν (ts ν (t] = ν ν ɛ = (λ +µ ν v ν+, ν+ ( qν+ v ν+ v ν+ ( /θ ν+ ( κ ν ( θν+ ( /θ 0. ν+ This statement is a iscrete analog of the entropy balance statement (0. Here is our main point: we introuce no ecessive entropy issipation ue to spurious, artificial numerical viscosity. Accoring to (, the semi-iscrete scheme contains the precise amount of numerical viscosity to enforce the correct entropy issipation ictate by NSE. More can be foun in [Ta00, TZ00]..3 Numerical eperiments We consier ieal polytropic gas equations as an approimation of air with γ =., C v =, κ =0.03, λ+µ =. 0. We simulate the So s shocktube problem, where the Euler an NSE are solve over the interval [0, ] subject to Riemann initial conitions { (.0, 0.0,. 0 < (ρ, m, E t=0 = (, 0.0, <<. In Fig., we isplay the numerical solutions for the fully iscrete scheme ( with RK3 metho ( an the numerical flu (3. The ensity fiels of four ifferent cases are recore. Density fiel of the Euler equations (a emonstrates the purely ispersive character of the entropy conservative schemes. Dispersive oscillations on the mesh scale are observe in shocks an contact regions ue to the absence of any issipative mechanism. These oscillations approach a moulate wave envelop, consult [La, LL] for more iscussions on ispersive oscillations. For the results of NSE in (b-(, the presence of heat flu causes the oscillations to be ramatically reuce aroun the contact iscontinuity an the shock in (b. The viscous flu in NSE, on the other han, is oing a better job than heat flu in amping oscillations aroun the shock in (c, but we still can observe an oscillatory behavior aroun the contact iscontinuity. In (, not only the oscillations aroun the shock are ampe out by viscosity, but the oscillations aroun the contact iscontinuity are significantly reuce ue to the heat flu.

7 ρ ρ ρ ρ Novel entropy stable schemes for D an D flui equations D Euler,ensity,γ=.,C v =,κ=0,entropy= ρ ln(pρ γ, t=.e 00, =0.000,T ma =0. D Euler,ensity,γ=.,C v =,κ=0.03,entropy= ρ ln(pρ γ, t=.e 00, =0.000,T ma =0. 0. t=0. 0. t= (a Euler (b Navier-Stokes with heat only 0. D N S,ensity,λ+µ=.e 00,γ=.,C v =,κ=0,entropy= ρ ln(pρ γ, t=.e 00, =0.000 t=0. 0. D N S,ensity,λ+µ=.e 00,γ=.,C v =,κ=0.03,entropy= ρ ln(pρ γ, t=.e 00, =0.000 t= (c Navier-Stokes with viscosity only ( Navier-Stokes with viscosity an heat Fig.. Density fiel So problem with 000 spatial gripoints, U(u = ρ ln ( pρ γ, t=. 0, =. 0 3 The D Shallow Water Equations We turn to D shallow water equations, t u + f(u+ g(u =η y ( h (u + η y (h y (u, ( with u =[h, uh, vh] being the vector of conserve variables balance by the flu vectors f =[uh, u h + gh /, uvh], g =[vh, uvh, v h + gh /],

8 Eitan Tamor an Weigang Zhong an the viscous flu vector = [0,u,v]. Here, h = h(, t is the total water height, (u(, t, v(, t are the epth-average velocities along an y irection. Finally, g is the constant acceleration ue to gravity, an η > 0 is the constant ey viscosity which moels the turbulence stress in the flow. The total energy U(u =(gh +u h+v h/ serves as an entropy function, t U(u+ F (u+ y G(u = ηh(u + u y + v + v y, ( where F (u =guh + u3 h+uv h huu an G(u =gvh + u vh+v 3 h hvv y are the entropy flues. Spatial integration of ( yiels U(u y = η h(u + u y + v + v t y y. ( y y For the invisci case (η = 0, the global entropy conservation is satisfie, U(u(, t y = U(u(, 0 y y Arguing along the same line as the above NSE imension by imension, we obtain the entropy-stable semi-iscrete schemes (recall ẑ ν+ := (z ν+ +z ν / t u ν, µ(t+ (f ν+,µ f ν,µ+ y (g ν, µ+ g ν, µ = η ( h ν+,µ ν, µ ν+,µ ν, µ ν,µ h ν,µ + η y ( h ν, µ+ ν, µ ν, µ+ ν, µ ν, µ h ν, µ, (a with the entropy-conservative flues f ν+,µ an g outline in Algorithm ν, µ+ along an y irection, respectively, f ν+,µ = g g ν, µ+ = g 3 j= 3 j= (h j+ u j+ ν+,µ (h j+ ν, µ+ y ν+,µ (hj u j ν+,µ ν+,µ l j l j ν+,µ, v ν+,µ u j+ (h j u j ν, µ+ ν, µ+ ν, µ+ l yj, v ν, µ+ ν, µ+ ν+ l yj ν, µ+,µ, (b, (c Here, u ν, µ (t enotes the iscrete solution at the gri point ( ν,y ν,t, ν, µ := (u ν, µ, an v := U u =[gh (u + v,u,v] is the entropy variable. Numerical flu f an g are constructe separately along two ifferent phase pathes ictate by two sets of vectors {l j } an {l yj }. {h j } an {u j } are intermeiate values of height an velocity along the path. The above ifference scheme (a-(c is an entropy stable scheme with no artificial viscosity in the sense that the following iscrete entropy balance is satisfie,

9 t Novel entropy stable schemes for D an D flui equations { [ ( uν+ ( ] vν+ h ν+,µ + ν, µ U(u ν, µ (t y = η ν, µ + h ν, µ+ [ ( uν, µ+ y,µ,µ ( ]} vν, µ+ + y. ( y ( is a iscrete analogue of the entropy balance statement (. Equippe by RK3, we test the entropy stable scheme (a-(c by the D partial Dam-Break problem with free-slip bounary escribe in [FC0]. Both the invisci an viscous case are teste. The water surface profiles at time t =.s are recore in Fig.. Height at t= Height at t= (a η =0 (b η =0m s Fig. 3. Shallow water equations, Dam-Break, m basin, free-slip bounary, = y =m, t = 0 3 s Comparing 3(b to 3(a, we observe the improvements in smoothness of the numerical solutions. There is no analytical reference solution for this test case, but other numerical results are available in [FC0]. References [FC0] Fennema, R.J., Chauhry, M.H.: Eplicit methos for D transient freesurface flows. J. Hyraul. Eng. ASCE,, (0 [GST00] Gottlieb, S., Shu, C.-W., an Tamor, E.: Strong stability-preserving high-orer time iscretization methos. SIAM Review, 3,, (00 [La] La, P.D.: On ispersive ifference schemes. Phys. D,, 0 ( [LMR00] LeFloch, P.G., Mercier, J.M., an Rohe, C.: Full iscrete, entropy conservative schemes of arbitrary orer. SIAM J. Numer. Anal., 0,, (00

10 0 Eitan Tamor an Weigang Zhong [LL] Levermore, C.D., Liu, J.-G.: Oscillations arising in numerical eperiments. Phys. D,, ( [Roe] Roe, P.L.: Approimate Riemann solvers, parameter vectors an ifference schemes. J. Comp. Phys., 3, 3 3 ( [Ta] Tamor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp.,, 03 ( [Ta00] Tamor, E.: Entropy stability theory for ifference approimations of nonlinear conservation laws an relate time-epenent problems. Acta Numerica, (00 [TZ00] Tamor, E. an Zhong, W.: Entropy stable approimations of Navier- Stokes equations with no artificial numerical viscosity. J. of Hype. Diff. Equa., 3, 3, (00