c 2012 Society for Industrial and Applied Mathematics

SIAM J. NUMER. ANAL. Vol. 5, No., pp. 544 573 c Society for Industrial and Applied Mathematics ARBITRARILY HIGH-ORDER ACCURATE ENTROPY STABLE ESSENTIALLY NONOSCILLATORY SCHEMES FOR SYSTEMS OF CONSERVATION LAWS ULRIK S. FJORDHOLM, SIDDHARTHA MISHRA, AND EITAN TADMOR Abstract. We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative flues and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical eperiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes. Key words. entropy stability, ENO reconstruction, sign property, high-order accuracy AMS subject classifications. 65M6, 35L65 DOI..37/83696. Introduction. Systems of conservation laws are ubiquitous in science and engineering. They encompass applications in oceanography (shallow water equations), aerodynamics (Euler equations), plasma physics (MHD equations), and structural mechanics (nonlinear elasticity). In one space dimension, these PDEs are of the form (.) u t + f(u) = (, t) R R +, u(, ) = u () R. u : R R + R m is the vector of unknowns and f is the (nonlinear) flu vector. It is well known that solutions of (.) contain discontinuities in the form of shock waves, even for smooth initial data [6]. Hence, solutions of (.) are sought in a weak sense. A function u L (R R + )isaweak solution of (.) if (.) uϕ t + f(u)ϕ ddt + u(, )ϕ(, ) d = R R + R for all compactly supported smooth test functions ϕ Cc (R R + ). Weak solutions might not be unique and need to be supplemented with etra admissibility criteria, termed entropy conditions, in order to single out a physically relevant solution [6]. Assume that there eists a conve function E : R m R and a function Q : R m R such that u Q(u) =v u f(u), where v := u E(u). The functions E, Q, andv are termed the entropy function, entropy flu function, and entropy variables, respectively. Multiplying (.) by the entropy variables v shows that smooth solutions of (.) satisfy the entropy identity (.3) E(u) t + Q(u) =. Received by the editors June, ; accepted for publication (in revised form) December 8, ; published electronically March 7,. http://www.siam.org/journals/sinum/5-/83696.html Seminar for Applied Mathematics (SAM), ETH Zürich, Rämistrasse, Zürich-89, Switzerland (ulrikf@sam.math.ethz.ch, smishra@sam.math.ethz.ch). Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 74 (tadmor@cscamm.umd.edu). This author s research was supported by National Science Foundation grant DMS -8397 and Office of Naval Research grant ONR N438. 544

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 545 However, the solutions of (.) are not smooth in general and the entropy has to be dissipated at shocks. This translates into the entropy inequality (.4) E(u) t + Q(u) (in the sense of distributions). Formally, integrating (.4) in space and asserting a periodic or no-inflow boundary, we obtain the bound d (.5) E(u)d E (u(, T )) d E(u ())d dt R R for all T>. As E is conve, the above entropy bound can be converted into an a priori estimate on the solution of (.) in suitable L p spaces [6]... Numerical schemes. The design of efficient numerical schemes for the approimation of hyperbolic conservation laws has undergone etensive development. Finite volume (conservative finite difference) methods are among the most popular discretization frameworks. We consider a uniform Cartesian mesh { i } i Z in R with mesh size i+ i =Δ. The midpoint values are defined as i+ / := i+i+ and the domain is partitioned into intervals I i =[ i /, i+ /]. The conservative finite difference (finite volume) method updates point values (cell averages in I i )ofthe solution u and has the general form d (.6) dt u i(t) = ( Fi+ /(t) F i /(t) ), Δ where the numerical flu F i+ / = F(u i (t), u i+ (t)) is computed from an (approimate) solution of the Riemann problem at the interface i+ / []. Second-order spatial accuracy can be obtained with nonoscillatory TVD methods [8], and an even higher order of accuracy can be obtained with ENO [3] and an weighted ENO (WENO) [6] piecewise polynomial reconstructions. An alternative approach to a high order of spatial accuracy is the discontinuous Galerkin (DG) method of [3]. The DG method requires total variation bounded (TVB) limiters to suppress oscillations near discontinuities. Time integration for the semidiscrete scheme (.6) is performed with strong stability preserving Runge Kutta methods [] or with ADER schemes [34]... Accuracy and stability. For scalar conservation laws in one space dimension, monotone (first-order) schemes were shown to be TVD in [] and consistent with any entropy condition in [5]. Hence, these schemes converge to the entropy solution. E-schemes for scalar conservation laws that preserve a discrete version of the entropy inequality (.4) were designed by Tadmor [9] and Osher [4]. Convergence results for monotone schemes for multidimensional scalar conservation laws were obtained in []. Second-order accurate limiter-based schemes for scalar conservation laws were shown to be stable in the space of bounded variation (BV) functions in [8]. Secondorder entropy stable schemes for scalar conservation laws were presented in [5]. Stability results in BV for second-order and third-order accurate central schemes in the scalar case were shown in [3] and [], respectively. Very few stability results eist for schemes that approimate scalar conservation laws with even higher ( 3) order of accuracy. We mention [6] in which WENO schemes were shown to converge for smooth solutions of scalar conservation laws. This result is quite limited as solutions of the conservation law (.) have discontinuities. Convergence results for a streamline diffusion finite element method were shown in R

546 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR [7]. The arbitrary-order DG methods were shown in [4] to satisfy a global entropy estimate, i.e., a discrete version of (.5) for scalar conservation laws. Note that these methods might not satisfy a local version of the discrete entropy inequality (.4). DG methods must be limited by a TVD or TVB limiter in order to obtain BV bounds. Entropy stable limited DG methods are not currently available. Convergence results for numerical schemes (even first-order schemes) approimating nonlinear systems of conservation laws are difficult to obtain, as a global well-posedness theory for such equations is not currently available. It is reasonable to require that numerical schemes are entropy stable, i.e., satisfy a discrete version of the entropy inequality (.4). In particular, such a scheme satisfies a discrete form of the entropy bound (.5) and will be stable in a suitable L p space. No entropy stability results for high-order numerical schemes for approimating systems of conservation laws, based on the TVD, ENO, WENO, and DG procedures, are available. Entropy stable streamline diffusion finite element methods were proposed in [4]..3. Scope and outline of the paper. In view of the above discussion, it is fair to claim that none of the currently available high- and very-high-order schemes for systems of conservation laws have been rigorously shown to be stable. Given this background, we present a class of schemes in this paper that are (i) (formally) arbitrarily order accurate; (ii) entropy stable for any system of conservation laws; (iii) essentially nonoscillatory around discontinuities; (iv) convergent for linear symmetrizable systems; (v) computationally efficient. We recall that entropy stability automatically provides an a priori estimate on the scheme in L p (R). Our schemes do not contain any tuning parameters. Our schemes are based on the following two ingredients:. Entropy conservative flues. The first step in the construction of entropy stable schemes is to use entropy conservative flues, introduced by Tadmor in [3, 3]. More recent developments on entropy conservative flues are described in [7, 9, 5, 3, 33]. These papers construct second-order accurate entropy conservative schemes. Even higher-order accurate entropy conservative flues were proposed in [9, 3]. We utilize the procedure of [9] along with eplicit formulas obtained in [7, 5] to construct computationally efficient, arbitrarily high-order accurate entropy conservative flues.. Numerical diffusion operators. Following [7, 3], we add numerical diffusion in terms of entropy variables to an entropy conservative scheme to obtain an entropy stable scheme. Arbitrary order of accuracy is obtained by using piecewise polynomial reconstructions. We rely on a subtle nonoscillatory property, the so-called sign property of the ENO reconstruction procedure, to prove entropy stability. The sign property of the ENO reconstruction procedure was shown in a recent paper []. We call this combination of the entropy conservative and ENO reconstruction procedures TeCNO schemes and show that they are entropy stable while having a (formally) arbitrarily high order of accuracy. The TeCNO schemes are easily etended to several space dimensions. The rest of the paper is organized as follows. In section, we describe the procedure of [9] and the two-point entropy conservative flues of [7, 5] and construct high-order accurate entropy conservative schemes. The entropy stable numerical diffusion operators of an arbitrarily high order of accuracy are proposed in section 3. The TeCNO schemes are presented in section 4. Numerical eperiments are presented in section 5, and the etension to several space dimensions is provided in section 6. These results were announced earlier in [8].

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 547. Entropy conservative flues. In this section we review theory on entropy conservative schemes. These are schemes whose computed solutions satisfy a discrete entropy equality d (.) dt E(u i(t)) = ( Qi+ / Δ Q i /) for some numerical entropy flu Q i+ / consistent with Q. We introduce the following notation: [a] i+ / = a i+ a i, a i+ / = (a i + a i+ ). We will also use entropy potential ψ(u) :=v(u) f(u) Q(u). Theorem. (Tadmor [3]). Assume that a consistent numerical flu F i+ / satisfies (.) [v] i+ F / i+ / =[ψ ]. i+ / Then the scheme with numerical flu F i+ / is second-order accurate and entropy conservative solutions computed by the scheme satisfy the discrete entropy equality (.) with numerical entropy flu (.3) Qi+ / = v i+ F / i+ / ψ i+ /. We note that the condition (.) provides a single algebraic equation for m unknowns. In general, it is not clear whether a solution of (.) eists. Furthermore, the solutions of (.) will not be unique ecept in the case of scalar equations (m =). In [3], Tadmor showed the eistence of a solution for (.) for a general system of conservation laws by the following procedure: for ξ [ /, /], define the following straight line in phase space: (.4) v i+ /(ξ) = (v i + v i+ )+ξ(v i+ v i ). The numerical flu is then defined as the path integral (.5) Fi+ / = / / f(v i+ /(ξ))dξ. However,itmaybeveryhardtoevaluatethepath integral (.5) ecept in very special cases [7]. An eplicit solution of (.) was devised in [3]. Take any orthogonal eigensystem r k,l k for k =,,...,m. At an interface i+ /, we have the two adjacent entropy variable vectors v i and v i+. Define v = v i, v k = v k + v m = v i+. ( ) [v] i+ l / k r k (k =,...,m ), We are replacing the straight line joining the two adjacent states in the flu (.5) by a piecewise linear path along basis vectors. The resulting entropy conservative flu is given by n ψ(v k ) ψ(v k ) (.6) Fi+ / = [v] i+ l l k. / k k=

548 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR This construction is very general and works for any system of conservation laws. However, the computation of (.6) may be both epensive and numerically unstable [7]. Therefore, we follow a different approach and find eplicit algebraic solutions of (.) for specific systems... Eamples. We consider specific hyperbolic conservation laws and describe eplicit and computationally inepensive entropy conservative flues satisfying (.).... Scalar conservation laws. Consider the scalar version of (.) and denote u = u, f = f. Any conve function E can serve as an entropy function. Let v and ψ be the corresponding entropy variable and potential, respectively. It is straightforward to compute the unique entropy conservative flu F in this case as ψ i+ ψ i if u i u i+, (.7) F (ui,u i+ )= v i+ v i f(u i ) otherwise.... Linear symmetrizable systems. Let f(u) =Au with A being a (constant) m m matri. Assume that there eists a symmetric positive definite matri S such that SA is symmetric. Then (.8) E(u) = u Su, Q(u) = u SAu constitute an entropy-entropy flu pair for the linear system. The entropy variables and potential are given by v = Su, ψ(u) = u SAu. Inserting into (.5), one easily finds the entropy conservative flu (.9) Fi+ / = (Au i + Au i+ )...3. Shallow water equations. The shallow water equations model a body of water under the influence of gravity and have conservative variables and flu [ ] [ ] h hu (.) u =, f(u) = hu hu +. gh Here, h and u are the depth and velocity of the water, respectively. The (constant) acceleration due to gravity is denoted by g. The entropy in this case is the total energy: (.) E(u) = hu + gh, Q(u) = hu3 + guh. The corresponding entropy variables and potential are given by (.) v = gh u, ψ(u) = u guh.

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 549 An eplicit solution of (.) for the shallow water equations was proposed in the recent paper [7]: h i+ /u i+ / (.3) Fi+ / = h i+ /(u i+ /) + g. h i+ / The above flu is clearly consistent, very simple to implement, and computationally inepensive...4. Euler equations. Let (.4) u = ρ ρu ρu, f(u) = ρu + p. E (E + p)u Here, ρ, u, andp are the density, velocity, and pressure of the gas. The total energy E is related to other variables by the equation of state, (.5) E = p γ + ρu, where γ is the gas constant. Let s = log(p) γ log(ρ) be the thermodynamic entropy. An entropy-entropy flu pair for the Euler equations is (.6) E = ρs γ, Q = ρus γ. The corresponding entropy variables and potential are γ s (.7) v = γ ρu p ρu/p, ψ(u) =ρu. ρ/p In a recent paper [5], Ismail and Roe have constructed an eplicit solution of (.) for the Euler equations. Defining the parameter vectors z as (.8) z = z = z 3 z ρ u p, p the entropy conservative flu of [5] is F i+ / = F i+ / = z i+ / ( z 3 ) ln i+ /, [ F i+ / F i+ / F 3 i+ /] with (.9) F i+ = z3 i+ / + z i+ / F / z i+ / z i+, / i+ / ( ) F 3 i+ = z i+ / γ + z 3 ln i+ / / z i+ γ / (z ) ln i+ / + F i+. /

55 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR Here, a ln is the logarithmic mean, defined as a ln i+ = [a] i+ /. / [log(a)] i+ / The above eamples show that we can obtain eplicit and computationally inepensive epressions of entropy conservative flues for a large class of systems. In case such eplicit formulas are not available, we can use (.6) to compute the two-point entropy conservative flu... High-order entropy conservative flues. The entropy conservative flues defined above are only second-order accurate. However, following the procedure of LeFloch, Mercier, and Rohde [9], we can use these flues as building blocks to obtain pth-order accurate entropy conservative flues for any p N. These consist of linear combinations of second-order accurate entropy conservative flues F and have the form p (.) Fp i+ = r α p / r F(u i s, u i s+r ). r= s= Theorem. (see [9, Theorem 4.4]). For p N, assume that α p,...,αp p solve the p linear equations p rα p r =, r= p i s α p r = i= (s =,...,p), and define F p by (.). Then the finite difference scheme with flu F p is (i) pth-order accurate, in the sense that for sufficiently smooth solutions u we have ( Fp (u i p+,...,u i+p ) Δ F p (u i p,...,u i+p )) = f(u i )+O ( Δ p) ; (ii) entropy conservative it satisfies the discrete entropy identity d dt E(u i(t)) + ( ) Qp p Δ i+ Q / i =, / where p (.) Qp i+ = / r α p r r= s= Q(u i s, u i s+r ). As an eample, the fourth-order (p = ) version of the entropy conservative flu (.) is (.) F4 = 4 i+ / 3 F(u i, u i+ ) ) ( F(ui, u i+ )+ F(u i, u i+ ) 6 and the sith-order (p =3)versionis F 6 i+ = 3 / F(u i, u i+ ) 3 ) ( F(ui, u i+ )+ F(u i, u i+ ) (.3) + ( F(ui, u i+ )+ F(u i, u i+ )+ F(u i, u i+3 )). 3

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 55 Remark.3. Since the high-order entropy conservative flues (.) are based on linear combinations of two-point second-order flues F, they are computationally tractable only if computationally inepensive two-point flues like those described in the previous section are available. 3. Numerical diffusion operators. The entropy of solutions of hyperbolic conservation laws is conserved only if the solution is smooth. However, the solutions develop discontinuities where entropy is dissipated, which is reflected in the entropy inequality (.4). The entropy conservative schemes described in the previous section will produce high-frequency oscillations near shocks. (See [7] for numerical eamples.) Consequently, we need to add some dissipative mechanism to ensure that entropy is dissipated. This is achieved by designing entropy stable schemes schemes whose computed solutions satisfy a discrete entropy inequality (3.) d dt E(u i)+ ( Qi+ / Δ Q i /) for some numerical entropy flu function Q i+ / consistent with Q. 3.. First-order numerical diffusion operator. We begin with the secondorder entropy conservative flu F (.) and add a numerical diffusion term to define (3.) F i+ / = F i+ / D i+ / [v] i+ /. Here, D is any symmetric positive definite matri. Lemma 3. (Tadmor [3]). The scheme with flu (3.) is entropy stable its solutions satisfy (3.3) where d dt E(u i)+ ( Qi+ / Δ Q i /) = 4Δ ( ) [v] i+ D / i+ / [v] +[v] i+ / i D / i / [v] i, / Q i+ / = Q i+ / + v i+ / D i+ / [v] i+ / and Q i+ / is the numerical entropy flu function of the flu F i+ /. As a corollary, we can sum (3.3) over all i to obtain the entropy dissipation estimate d dt i E(u i )= [v] i+ Δ D / i+ / [v]. i+ / i Although the above lemma holds for any symmetric positive definite D i+ /, we will use diffusion matrices of the form (3.4) D i+ / = RΛR. Here, R is the matri of eigenvectors of the flu Jacobian u f and Λ is a positive diagonal matri that depends on the eigenvalues of the flu Jacobian. Two eamples of the matri Λ are the following:

55 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR Roe-type diffusion operator: (3.5) Λ = diag ( λ,..., λ m ), where λ,...,λ m are the eigenvalues of u f(u i+ /); Rusanov-type diffusion operator: (3.6) Λ = ma ( λ,..., λ m ) I, where I is the identity matri in R m m. As the term [v] i+ is of the order of Δ, the scheme with flu (3.) is in general / only first-order accurate. This remains true even if we replace the entropy conservative flu F in (3.) with the very-high-order entropy conservative flu (.). 3.. High-order diffusion operators. In order to obtain a higher-order accurate scheme, we need to perform a suitable reconstruction of the entropy variables v. A kth-order (k N) reconstruction produces a piecewise (k )th-degree polynomial function v i (). Denoting (3.7) v i = v i ( i /), v + i = v i ( i+ /), v i+ / = v i+ v+ i, we define our higher-order (depending on the order of the reconstruction) numerical flu as (3.8) F k i+ / = F p i+ / D i+ / v i+ / (compare to (3.)). The number p N is chosen as p = k/ if k is even and p = (k +)/ if k is odd. The flu F p is the high-order entropy conservative flu given by (.). The scheme with numerical flu (3.8) is kth-order accurate its truncation error is O(Δ k ) for smooth solutions. However, the scheme with numerical flu (3.8) might not be entropy stable. We need to modify the reconstruction procedure to ensure entropy stability. Lemma 3.. For each i Z, letr i+ / R m m be nonsingular, let Λ i+ / be any nonnegative diagonal matri, and define the numerical diffusion matri (3.9) D i+ / = R i+ /Λ i+ /R i+ /. Let v i () be a polynomial reconstruction of the entropy variables in the cell I i such that for each i, there eists a diagonal matri B i+ / such that (3.) v i+ / = ( R i+ /) Bi+ /R i+ / [v] i+ /. Then the scheme with numerical flu (3.8) is entropy stable its computed solutions satisfy the entropy dissipation estimate d (3.) dt E(u i)+ ( Qk Δ Q k i+ / i /), where the numerical entropy flu function Q k is defined as Q k i+ / and Q p is defined in (.). = Q p i+ / v i+ / D i+ / v i+ /

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 553 Proof. Multiplying the finite difference scheme (.6) by v i and imitating the proof of Theorem. (see [3]), we obtain d dt E(u i)= ( Qp Δ i+ / = Δ 4Δ ) p Q i / ( Qk Q k i+ / i /) + ( v Δ i D i+ / v i+ / vi D i / v i /) ( ) [v] i+ D / i+ / v i+ / +[v] i D / i / v i /. Suppressing vector and matri indices i + / for the moment, we have by (3.9) and (3.) [v] D v =[v] RΛR RBR [v] = [v] RΛBR [v] = ( R [v] ) ΛB ( R [v] ) (since B i+ / ), and so d dt E(u i) ( Qk Δ Q k i+ / i /). Remark 3.3. If the reconstructed variables satisfy (3.), then the numerical flu (3.8) admits the equivalent representation F k i+ / = F p i+ / R i+ /Λ i+ /B i+ /R i+ / [v] i+ /. This reveals the role of B i+ / as limiting the amount of numerical diffusion: in smooth parts of the flow, we have B i+ /, and we are left with the entropy conservative flu. Although any R, Λ in (3.8) gives an entropy stable scheme, we choose Λ to be either the Rusanov (3.6) or Roe (3.5) matrices. Similarly, R is chosen as the matri of eigenvectors of the flu Jacobian. The rationale for doing so is as follows. The Roe diffusion operator has the form R Λ R [u], where R and Λ are evaluated at the Roe average. In many cases there is some (possible different) intermediate state u i+ / such that [u] = v i+ / u [v],wherev i+ / u = u v(u i+ /). Moreover, by a theorem due to Merriam [, section 7.3] (see also [, Theorem 4]), there eists a scaling of the column vectors of R = R(u i+ /) such that v u = RR.Then R Λ R [u] R Λ R v u [v] = R Λ R [v]. This is precisely the form of our diffusion operator. 3.3. Reconstruction procedure. Lemma 3. provides sufficient conditions on the reconstruction for the scheme to be entropy stable. In this section, we will describe reconstruction procedures that satisfy the crucial condition (3.). Assume for the moment that v i, v i+, v + i, v i+ are given. Define the scaled entropy variables w ± i = R i± / v i, w ± i = R i± / v± i. Given entropy variables {v i } i, we will reconstruct in scaled entropy variables {w ± i }, obtaining reconstructed values { w ± i }; then, the values of v± i := (R i± / ) w ± i are

554 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR inserted into the numerical flu (3.8). Note that by the definition of the numerical diffusion term in (3.8), we have D i+ / v i+ / = R i+ /Λ i+ / w i+ /, sowedonot have to actually compute the inverse (R i± / ). The condition (3.) now reads w i+ / = B i+ / w i+ /. This is a componentwise condition; denoting the lth component of w i and w i by wi l and w i l, respectively, the above condition is equivalent to if w l i+ / >, then w l i+ /, (3.) if w l i+ / <, then w l i+ /, if w l i+ / =, then w l i+ / =. We abbreviate this by writing (3.3) sign w l i+ / = sign w l i+ /. We call this highly nonlinear structural property of the reconstruction the sign property. Reconstruction procedures that satisfy the sign property are presented in the following sections. 3.4. Second-order TVD reconstruction. We begin with the second-order case, which involves reconstruction with piecewise linear functions. For a fied l, we denote the lth component of the scaled entropy variable as w and define the undivided differences (3.4) δ i+ / = w i+ /. Let φ be some slope limiter with the symmetry property φ(θ )=φ(θ)θ (see []). Define the quotients θ i = δ i+ / δ i / We denote the slope in grid cell I i by and θ + i = δ i /. δ i+ / σ i = Δ φ(θ i )δ i / = Δ φ(θ+ i )δ i+ /. (The second equality follows from the symmetry of φ.) Hence, the reconstructedvalues at the left and right cell interfaces of grid cell I i are given by w i = w i φ(θ i )δ i / and, respectively, w + i = w + i + φ(θ+ i )δ i+ /. Weobtain w i+ / = w i+ w+ i ( φ(θ + i )+φ(θ i+ )) δ i+ / ( = ( φ(θ + i )+φ(θi+ ))) δ i+ /. Recalling the definition of δ (3.4), we find that the sign property (3.) is satisfied if and only if φ(θ) for all θ R. It is easily seen that the minmod limiter, given by if θ<, (3.5) φ mm (θ) = θ if θ, otherwise

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 555 satisfies φ(θ). In fact, the minmod limiter is the only symmetric TVD limiter that satisfies the sign property. However, non-tvd limiters might satisfy this condition. One eample is the second-order version of the ENO reconstruction procedure [3], which can be epressed in terms of the flu limiter { θ if θ, (3.6) φ(θ) = else. This limiter is both symmetric and satisfies φ(θ), thus ensuring the sign property. Indeed, the ENO limiter may be viewed as a symmetric etension of the minmod limiter (3.5) into the negative θ-ais. 3.5. ENO reconstruction procedure. The above discussion reveals that the second-order version of the ENO reconstruction procedure satisfies the sign property, encouraging us to investigate whether the sign property holds for higher-order versions of the ENO procedure. As described in [3], the ENO procedure for kth-order accurate reconstructions of point values w i amounts to selecting a stencil of k points { i ri,..., i ri+k}. The integer r i {,...,k } is the left shift inde of the stencil. We may determine the left-displacement inde r i for the grid cell I i by using values of the undivided differences {δ j+/ } i+k j=i k+ of w i. The question of whether the ENO procedure satisfies the sign property was answered in the recent paper []. Theorem 3.4 (Fjordholm, Mishra, and Tadmor []). Let k N and let ω i +,ω i+ betheleftandrightvaluesatthecellinterface i+ /, obtained through a kth-order ENO reconstruction of the point values ω i of a function ω. Then the reconstruction satisfies the sign property: (3.7) sign ( ω i+ ω+ i ) = sign ( ωi+ ω i ). Furthermore, we have (3.8) ω i+ ω+ i ω i+ ω i C k for a constant C k that only depends on k. As the values we are reconstructing, the scaled entropy variables, are centered at cell interfaces, we must modify the reconstruction method somewhat. Given the interface values of each component w of the scaled entropy variables w for a fied grid cell I i, define the point value μ i i = w i and inductively μ i j+ = μ i j + δ j+/ (j = i, i +,...), μ i j = μi j δ j / (j = i, i,...). Similarly, we define ν i i = w+ i and νj+ i = νi j + δ j+/ (j = i, i +,...), νj i = νi j δ j / (j = i, i,...). Then μ and ν retain the cell interface jumps of w, (3.9) [μ i ] j+/ =[ν i ] j+/ = δ j+/ = w j+/ j.

556 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR As ν and μ have the same jump at a cell interface, we have (3.) μ i+ j = νj i j. Since the divided differences of μ and ν coincide with those obtained with δ j+/ as described above, the ENO stencil selection procedure will yield eactly the same stencil (in other words, the same left-displacement inde r i )whetherμor ν is provided as input data for the procedure. Let p i () andq i (), respectively, be the unique (k )th-order polynomial interpolations for the values μ i and ν i on the above stencil. Since μ i j = ν i j +(μ i i ν i i) j, we have p i () =q i ()+(μ i i νi i ). Hence, the interpolation polynomial need only be computed once for both the left and the right interfaces. Finally, we obtain left and right reconstructed values: w i = p i ( i /) and w + i = q i ( i+ /). This process is repeated in each grid cell I i and for each component of w i ±. Corollary 3.5. The reconstructed values w ± i satisfy the sign property (3.). Proof. Fii Z and denote the (standard) ENO reconstructed polynomial of point values {μ i+ j } j Z in grid cell j by h j (). Because of (3.), the polynomial q i is precisely equal to h j. Obviously, p i+ = h i+. Hence, w i+ / = p i+ ( i+ /) q i ( i+ /) =h i+ ( i+ /) h i ( i+ /), which by Theorem 3.4 has the same sign as μ i+ w i+ /. i+ μi+ i, which by definition equals 4. Arbitrarily high-order accurate entropy stable schemes. We combine the high-order accurate entropy conservative flues (.) with a numerical diffusion operator based on ENO reconstruction of the scaled entropy variables. This defines an arbitrarily high-order accurate entropy stable scheme. Theorem 4.. For any k, letp = k (if k is even) or p = k + (if k is odd). Define the entropy conservative flu F p by (.). Let v in (3.8) be defined by the kth-order accurate ENO reconstruction procedure (as outlined in section 3.5). Then the finite difference scheme with numerical flu (3.8) is (i) (k )th-order accurate for smooth solutions, (ii) entropy stable computed solutions satisfy the discrete entropy inequality (3.). Proof. For (i) it suffices to show that w i+ / =(h i+ h i )( i+ /) =O(Δ k ). The functions h i and h i+ are kth-order approimations to some underlying function μ() (c.f. the notation in Corollary 3.5), and so h i+ ( i+ /) h i ( i+ /) =μ( i+ /) μ( i+ /)+O(Δ k )=O(Δ k ). (ii) is a direct consequence of Lemma 3.; condition (3.) of that lemma follows from Corollary 3.5. Remark 4.. Note that we are only able to prove that the scheme is (k )thorder accurate there is a nonzero term of order Δ k in the diffusion operator that does not vanish. However, in practice we see behavior of order Δ k and therefore chose not to alter our scheme.

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 557 Our scheme combines entropy conservative flu (.) with ENO-based numerical diffusion operator in (3.8); hence, we term them as TeCNO schemes. We have the following convergence result for TeCNO schemes approimating linear symmetrizable systems. Theorem 4.3. Consider a linear system, i.e., f(u) =Au for a constant m m matri A, and assume that there eists a symmetric positive definite matri S such that SA is symmetric. Let u i (t) be the solution computed with the scheme with flu (3.8), based on the two-point entropy conservative flu (.9), and define u Δ (, t) = u i (t) for I i.thenu Δ u (uptoasubsequence)inl ([,T],L (R)) as Δ, where u istheuniqueweaksolutionofthelinearsystem. Proof. Assume for simplicity that A is symmetric. An entropy/entropy flu pair for (4.3) is then E(u) = u u, Q(u) = u Au, with corresponding entropy variables v(u) =E (u) =u and entropy potential ψ(u) = u Au. The simplest entropy conservative flu for this entropy is the second-order accurate central scheme (.9). To this flu we add a diffusion operator to obtain entropy stability. A simple choice would be a La Friedrichs type operator of the form D i+ / = ai, wherei is the identity matri and a is any number Δ Δt a Δ Δt. The resulting flu is then F i+ / = A(u i + u i+ ) a [u] i+ / (recall that v i = u i ). Higher-order reconstruction of v = u would give the flu (4.) F i+ / = F k i+ / a u i+ /, where k is chosen so that k p. We remark that since the diffusion matri is a constant, diagonal matri, the ENO-type reconstruction procedure described in section 3.5 is reduced to a standard componentwise ENO reconstruction of u. In particular, each component of the reconstructed values satisfies the sign property (3.7) and the upper jump bound (3.8). To proceed, we prove that the computed solution, u Δ (, t) := i u i(t) Ii (), satisfies the following entropy and entropy production bounds: T (4.) u Δ (T ) L (R) + u Δ i+ u / L (R), Indeed, from the proof of Lemma 3. one obtains the eplicit entropy decay rate d dt ( (u Δ i ) ) + ( ) Qk k Δ i+ Q / i / = a 4Δ i ( [u Δ ] i+ / uδ i+ / +[u Δ ] i / uδ i / and after integrating over i Z, t [,T], u Δ (T ) L + a T [u Δ ] i+ uδ / i+ / = u Δ () L. i By the sign property (3.7), the integrands summed up in the second term on the lefthand side are nonnegative, and by (3.8) they are bounded from below by u Δ i+ /, and (4.) follows. ),

558 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR By the L bound (4.), the sequence {u Δ } is uniformly bounded in L ( [,T],L (R) ) by T u L. Hence, after etracting a subsequence if necessary, it converges weakly to some u L ( [,T],L (R) ). We claim that the limit u is a weak solution of (4.3). Indeed, letting φ C (R (,T)), multiplying the finite difference scheme by φ i (t) =φ( i,t), and integrating by parts, we get T = Δ d ( Fk ) φ i dt uδ k i + φ i Δ i+ F / i / i a ( φ i u Δ i+ / u Δ ) i / dt Δ T = Δ u Δ k φ i+ φ i i t φ i + F i+ a u Δ φ i+ φ i / i+ / dt. Δ Δ i By a standard La Wendroff type argument the first two terms converge to T R uφ t+ Auφ, while the third term vanishes: T Δ a u Δ φ i+ φ T i i+ / dt Δ a φ L Δ u i+ /dt i Z i S ( )/ T a φ L T S Δ u i+ dt / C Δ (where S = {i Z : i supp(φ)}) by Cauchy Schwarz and (4.). Theorem 4.3 tells us that the TeCNO method converges weakly when applied to the constant-coefficient symmetrizable system (4.3) u t + Au =. Note that this convergence result holds even when the solution of the linear system is discontinuous. It is straightforward to generalize this result to linear symmetrizable systems with variable (but smooth) coefficients. We epect similar convergence results to hold for scalar nonlinear conservation laws, but we are unable to obtain such convergence result for high-order (>) TeCNO schemes. However, using a specific choice of an entropy function, we obtain the following L bound. Lemma 4.4. Let u = u denote the solution of the scalar conservation law (.) subject to initial data a<u () <bfor all R. Let{u i } be the solution computed with the TeCNO flu (3.8), associated with the conve entropy (4.4) E(u) := log(b u) log(u a). Then, the following L estimate holds: (4.5) a<u i (T ) <b i Z, T. Proof. Integrating the entropy inequality (3.) over i Z and t [,T]gives Δ i E(u i(t )) Δ i E(u ( i )). In particular, since E(u) log ( ) b a for all u (a, b), we have E(u i (T )) C for all i Z. Since E(u) as u a, b, there must necessarily be some C > such that u i (T ) C for all i Z. i S

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 559.4. u t + u = with u () = sin(π ) at t= ENO4 ENO5 TeCNO4 TeCNO5 Eact u..4 Fig.. Solution at t =computed with fourth- and fifth-order accurate ENO and TeCNO schemes for the linear advection equation. 5. Numerical eperiments. We test the schemes ENOk kth-order accurate standard ENO scheme in the MUSCL formulation [3] and TeCNOk kth-order accurate entropy stable scheme with numerical flu (3.8) for k =, 3, 4, and 5 on a suite of numerical eperiments involving scalar equations as well as systems. The ENO-MUSCL and TeCNO schemes are semidiscrete and are integrated in time with a second-, third-, or fourth-order eplicit Runge Kutta method. In all eperiments we use a CFL number of.45. 5.. Linear advection equation. We consider the linear advection equation (5.) u t + au = with wavespeed a = in the domain [, ] with periodic boundary conditions. The initial data is u () =sin(π). The entropy function in this case is the total energy E(u) =u / and the entropy variable is v = u. The entropy conservative flu F is the average flu (.9). We use the advection velocity a as the coefficient of diffusion by setting D a in (3.8). The fourth- and fifth-order ENO and TeCNO schemes are compared in Figure. To illustrate the differences between the schemes, the solutions are computed on a very coarse mesh of points and the simulation is performed for a large time T =. The results show that the fifth-order schemes are more accurate than the fourth-order schemes. Furthermore, the TeCNO scheme is clearly more accurate than the corresponding standard ENO scheme of the same order. 5.. Burgers equation. Net we consider Burgers equation ( ) u (5.) u t + =. The computational domain is [, ] with periodic boundary conditions, and we use the initial data u () =+ sin(π). We choose to use the logarithmic entropy function E(U) = log(b u) log(u a) with constants a =andb = in order to bound the initial data. The entropy conservative flu is given in terms of the entropy variable, v i = u, i(u i ) (5.3) Fi+ / = ψ i+ ψ i v i+ v i, ψ i := u i u i +u i + log( u i ).

u u u 56 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR Burgers equation with u () = +sin(π ) at t=. Burgers equation with u () = +sin(π ) at t=. Burgers equation with u () = +sin(π ) at t=..5 ENO3 TeCNO3 Eact.5 ENO4 TeCNO4 Eact.5 ENO5 TeCNO5 Eact Fig.. Approimate solutions computed with third-, fourth-, and fifth-order accurate ENO and TeCNO schemes for Burgers equation at time t =. on a mesh of points. Errors for wave equation with u ()=sin(4π ). Errors at t=. 3 4 L error in h 5 6 7 8 ENO3 TeCNO3 ENO4 TeCNO4 ENO5 TeCNO5 3 Number of grid points Fig. 3. L errors in h for the wave equation with third-, fourth-, and fifth-order ENO and TeCNO schemes for the wave equation with sine initial data. Numerical results are shown in Figure. The initial sine wave breaks down into a shock and a rarefaction wave. In this eample, the ENO and TeCNO schemes show comparable resolution at the discontinuities. There is no visible gain in using a higher-order scheme at this mesh size. 5.3. The wave equation. We consider the one-dimensional wave equation (5.4) h t + cm =, m t + ch = and let c = (. The wave equation is a linear symmetric system and has the energy E(u) = h + m ) as entropy function with entropy variables v = u. The resulting entropy conservative flu is the average flu (.9). We use the diffusion matri [ ] c D = c in the numerical diffusion operator in (3.8). The ENO-MUSCL scheme uses reconstruction along characteristics. 5.3.. Smooth waves. Consider the wave equation (5.4) with initial data h(, ) =sin(4π) andm(, ) in the domain [, ] with periodic boundary conditions. We compute L errors for all the schemes (computed with respect to the eact solution) at time t = and show the convergence plot in Figure 3. The figures show that both ENO and TeCNO schemes converge at the claimed orders of accuracy. The TeCNO schemes have consistently lower error amplitudes than the ENO schemes at the same order.

h h h u u u ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 56 ENO3 TeCNO3 Eact ENO4 TeCNO4 Eact ENO5 TeCNO5 Eact Fig. 4. The height for the wave equation with discontinuous initial data, computed with the third-, fourth-, and fifth-order ENO and TeCNO schemes at time t =.5 onameshof points..5. ENO3 TeCNO3 Eact.5. ENO4 TeCNO4 Eact.5. ENO5 TeCNO5 Eact.5.5.5....5.5.5 Fig. 5. Computed heights for the shallow water dam break problem with third-, fourth-, and fifth-order ENO and TeCNO schemes at t =.4 onameshof points. 5.3.. Contact discontinuities. We consider the wave equation (5.4) with initial data { if <, h(, ) = m(, ) =, if >, on the domain [, ] with periodic boundary conditions. The solution features an initial jump discontinuity at = which breaks into two linear (contact) discontinuities. Computed solutions at time t =.5 for each scheme, on a mesh of points, are displayed in Figure 4. The two methods resolve the flow with a comparable level of accuracy. 5.4. The shallow water equations. We consider the shallow water equations (.) with entropy function, entropy flu, and entropy variables given in (.), (.). For the TeCNO schemes we use the two-point entropy conservative flu (.3). The numerical diffusion operator in (3.8) is of the Roe type (3.5) with eigenvalues and eigenvectors of the Jacobian evaluated at the arithmetic average of the left and right states. The ENO schemes use a MUSCL approach with the Rusanov numerical flu. The gravitational constant is set to g =. 5.4.. A dam break problem. We consider a dam break problem for the shallow water equations with initial data {.5 if <., h(, ) = hu(, ) =, if >., for [, ] with periodic boundary conditions. The eact solution consists of two shocks separated by two rarefactions. We display computed heights in Figure 5. The figure reveals that the TeCNO schemes are comparable to the ENO schemes of corresponding order: the TeCNO schemes approimate the shocks more sharply than the ENO schemes, whereas the ENO schemes resolve the rarefactions more accurately, albeit with small oscillations. The TeCNO schemes resolve the rarefactions without any noticeable oscillations.

56 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR rho.9.7.4.3.. 5 5 (a) ENO3 rho.9.7.4.3.. 5 5 (b) ENO4.9.7.9.7 rho.4.3.. 5 5 rho.4.3.. 5 5 (c) TeCNO3 (d) TeCNO4 Fig. 6. Comparing ENO (blue circles) and TeCNO (red circles) with the reference solution (black line) for the Sod shock tube. Density at t =.3 onameshof points is plotted. 5.5. Euler equations. We consider the Euler equations, as described in section..4. We define the TeCNO scheme with entropy conservative flu given by (.9) and diffusion matri of the Roe type (3.5). The eigenvalues and eigenvectors of the Jacobian are computed at the arithmetic average of the left and right states. The ENO-MUSCL schemes use the standard Roe numerical flu. 5.5.. Sod shock tube. The Sod shock tube eperiment is the Riemann problem { u L if <, (5.5) u(, ) = otherwise u R with ρ L u L =, p L ρ R u R =.5. p R in the computational domain [ 5, 5]. The initial discontinuity breaks into a left-going rarefaction wave, a right-going shock wave, and a right-going contact discontinuity. The computed density with the ENO3, ENO4, TeCNO3, and TeCNO4 schemes at time t =.3 on a mesh of points is shown in Figure 6. The results show that the ENO and TeCNO schemes are quite good at resolving the waves. The ENO4 scheme is slightly oscillatory behind the contact, whereas the TeCNO3 and TeCNO4 schemes resolve all the waves without any noticeable oscillations.

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 563 rho rho.6.4..4. 5 5.6.4..4 (a) ENO3. 5 5 (c) TeCNO3 rho rho.6.4..4. 5 5.6.4..4 (b) ENO4. 5 5 (d) TeCNO4 Fig. 7. Comparing ENO (blue circles) and TeCNO (red circles) with eact solution (black line) for the La shock tube. The density at time t =.3 onameshof points is plotted. 5.5.. La shock tube. We consider the Euler equations in the computational domain [ 5, 5] with Riemann initial data (5.5) given by ρ L u L =.445 98, 3.58 p L ρ R u R =. 7 The computed density at t =.3 on a mesh of points is shown in Figure 7. The resultsforenoandtecnoschemes are very similar in this eperiment. There are slight oscillations behind the shock for the TeCNO schemes. 5.5.3. Shock-entropy wave interaction. This numerical eample was proposed by Shu and Osher in [6] and is a good test of a scheme s ability to resolve a comple solution with both strong and weak shocks and highly oscillatory but smooth waves. The computational domain is [ 5, 5] and we use with initial data u(, ) = { u L u R p R if < 4, otherwise with ρ L u L = 3.85743.69369, p L.33333 ρ R +εsin(5) u R =. p R

564 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR rho 5 4.5 4 3.5 3.5.5 5 5 (a) ENO3 rho 5 4.5 4 3.5 3.5.5 5 5 (b) ENO4 rho 5 4.5 4 3.5 3.5.5 5 5 (c) ENO5 rho 5 4.5 4 3.5 3.5.5 5 5 (d) TeCNO3 rho 5 4.5 4 3.5 3.5.5 5 5 (e) TeCNO4 rho 5 4.5 4 3.5 3.5.5 5 5 (f) TeCNO5 Fig. 8. Comparing ENO (blue circles) and TeCNO (red circles) with a reference solution (black line) on the Shu Osher shock-entropy wave interaction problem. The plotted quantity is the density at time t =.8 onameshof points. As a reference solution, we compute with the ENO3 scheme on a mesh of 6 grid points. The approimate solutions are computed on a mesh of grid points, corresponding to about 7 grid points for each period of the entropy waves. The solutions computed by the ENO and TeCNO schemes are displayed in Figure 8. There are very minor differences between the ENO and TeCNO schemes of the same order. The test also illustrates that the higher-order schemes perform better than the low-order schemes. In conclusion, the above one-dimensional numerical eperiments show that the TeCNO schemes achieve the claimed orders of accuracy for smooth solutions and resolve shocks and other waves robustly. They are comparable to the standard ENO schemes of the same order. 6. Multidimensional problems. The arbitrary-order entropy stable TeCNO schemes can easily be etended to rectangular meshes in several space dimensions. We present a brief description of such schemes and omit details as they are very similar to the one-dimensional case. 6.. Continuous setting. For simplicity, we concentrate on systems of conservation laws in two space dimensions: (6.) u t + f(u) + g(u) y = (, y, t) R R R +, u(, y, ) = u (, y) (, y) R R. Here, u : R R R + R m is the vector of unknowns and f, g are flu vectors in the - andy-directions, respectively. We assume that there eists a conve function E : R m R and functions Q,Q y : R m R such that (6.) u Q (u) =v u f(u), u Q y (u) =v u g(u).

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 565 Again, the entropy variables are defined as v = u E(u). Entropy solutions of (6.) satisfy the entropy inequality (6.3) E(u) t + Q (u) + Q y (u) y (in the sense of distributions). We will design arbitrary-order accurate finite difference schemes that satisfy a discrete version of (6.3). 6.. Entropy stable finite difference schemes. We consider a (uniform) Cartesian mesh in R consisting of mesh points ( i,y j )=(iδ, jδy) fori, j Z and Δ, Δy >. Denoting the midpoints as i+ /,j = i + i+, y i,j+/ = y j + y j+, a semidiscrete conservative finite difference scheme for (6.) solves for point values u i,j u( i,y j ) and can be written as (6.4) d dt u i,j(t)+ ( Fi+ /,j(t) F Δ i /,j(t) ) + ( Gi,j+ /(t) G Δy i,j /(t) ) =. Here, F, G are numerical flu functions that are consistent with f and g, respectively. We suppress the t dependence of all quantities below for notational convenience. We will use the notation [a] i+ /,j = a i+,j a i,j, [a] i,j+ / = a i,j+ a i,j, as a i+ /,j = a i,j + a i+,j, a i,j+ / = a i,j + a i,j+. For any integer k, the kth-order accurate TeCNO numerical flues are defined (6.5) F i+ /,j = G i,j+ / = F p i+ /,j D i+ /,j v i+ /,j, G p i,j+ / Dy i,j+ / v i,j+ /. The flues F p, G p, matrices D, D y, and vectors v i+ /,j, v i,j+ / are described below. 6... High-order entropy conservative flues. The setting of entropy conservative schemes is completely analogous to the one-dimensional case [3]. Two-point entropy conservative flues F, G are chosen so that they satisfy (6.6) [v] i+ /,j F i+ /,j =[ψ ] i+ /,j, [v] i,j+ / G i,j+ / =[ψ y ] i,j+ / for all i, j, where the entropy potentials are defined as (6.7) ψ = v f Q, ψ y = v g Q y. Analogously to the one-dimensional case, solutions computed with the entropy conservative flues (6.6) satisfy the entropy equality d dt E(u i,j)+ ( Q Δ i+ /,j Q i /,j) + ( Qy Δy i,j+ Q ) y / i /,j =,

566 U. S. FJORDHOLM, S. MISHRA, AND E. TADMOR where Q i+ /,j = Q (u i,j, u i+,j )= (u i,j + u i+,j ) F(ui,j, u i+,j ) (ψ i,j + ψi+,j), Q y i,j+ / = Q y (u i,j, u i,j+ )= (u i,j + u i,j+ ) G(ui,j, u i,j+ ) (ψy i,j + ψy i,j+ ). The relations (6.6) are identical to the relation (.) in one space dimension. Hence, two-point entropy conservative flues like (.5) and (.6) can be easily adapted to this setting. We can obtain eplicit and algebraically simple solutions of (6.6) in a manner similar to section. Given an integer k, let p = k if k is even and p = k +ifk is odd. The high-order entropy conservative flues F p, G p are (6.8) F p i+ /,j = p G p i,j+ / = r α p r r= s= p r α p r r= s= F(u i s,j, u i s+r,j ), G(u i,j s, u i,j s+r ), where the constants α p r are the same as in (.). Solutions computed with these flues satisfy an entropy equality with numerical entropy flues p Q,p i+ /,j = r α p r Q (u i s,j, u i s+r,j ), Q y,p i+ /,j = p r= s= r α p r r= s= Q y (u i,j s, u i,j s+r ). 6... ENO-based numerical diffusion operators. The multidimensional reconstruction procedure is performed precisely as in the one-dimensional case, dimension by dimension. For each pair (i, j), let Ri+ /,j, Ry i,j+ be the eigenvector / matrices of u f(u i+ /,j), u g(u i,j+ /), where u i+ /,j and u i,j+ / are any intermediate states. For each fied j, we reconstruct entropy variables {v i,j } i Z along Ri+ /,j as in section 3.5, obtaining jumps in reconstructed values v i+ /,j. Net, i is kept fied, and {v i,j } j Z is reconstructed along R y i,j+ to obtain jumps v / i,j+ /. This completes the description of the TeCNO numerical flues (6.5). Theorem 6.. The TeCNO scheme (6.4), (6.5) is (i) kth-order accurate for smooth solutions, (ii) entropy stable computed solutions satisfy d (6.9) dt E(u i,j)+ ( ) Q,p,p Δ i+ /,j Q i /,j + ( ) Qy,p y,p Δy i,j+ Q / i /,j with Q,p,p i+ /,j = Q i+ /,j v D i+ /,j i+ /,j v i+ /,j, Q y,p y,p i,j+ = Q / i,j+ / v i,j+ D y / i,j+ v / i,j+ /. The proof follows analogously to that of Theorem 4.. We call the scheme with flues (6.8) the two-dimensional kth-order TeCNO scheme. It is straightforward to etend the TeCNO schemes to three dimensions on Cartesian meshes.

ESSENTIALLY NONOSCILLATORY ENTROPY STABLE SCHEMES 567 6.3. Numerical eperiments for two-dimensional Euler equations. We test the TeCNO schemes for the two-dimensional Euler equations (6.) ρ t +(ρu) +(ρv) y =, (ρu) t +(ρu + p) +(ρuv) y =, (ρv) t +(ρuv) +(ρv + p) y =, E t +((E + p)u) +((E + p)v) y =. Here, the density ρ, velocity field (u, v), pressure p, and total energy E are related by the equation of state E = p γ + ρ(u + v ). The entropy function, flues, variables, and potentials are given by E(u) = ρs γ, Q (u) = ρus γ, Qy (u) = ρvs γ, γ s γ ρ(u + v ) p v = ρu/p ρv/p, ψ (u) =ρu, ψ y (u) =ρv, ρ/p with s being the thermodynamic entropy. Defining the parameter vectors ρ (6.) z = p u, ρ p v ρp ρ p entropy conservative flues for the Euler equations are given by F i+ / =[ F i+ / F 3 i+ F 4 / i+ ] and G / i+ / = G i+ G / i+ G 3 / i+ G 4 / i+ ] with / F i+ / F i+ /,j = (z ) i+ /,j (z 4) ln i+ /,j, F i+ /,j = (z 4) i+ /,j + (z ) i+ /,j F i+ (z ) i+ /,j (z ) /,j, i+ /,j F 3 i+ /,j = (z 3) i+ /,j F i+ (z ) /,j, i+ /,j ( F 4 i+ /,j = γ + (z ) i+ γ /,j F i+ /,j (z ) ln i+ /,j ) + (z ) F i+ /,j i+ /,j + (z 3) F i+ /,j 3 i+ /,j,