Entropy and the numerical integration of conservation laws

Transcription

1 Pysics Procedia Pysics Procedia ) 1 28 Entropy and te numerical integration of conservation laws Gabriella Puppo Dipartimento di Matematica, Politecnico di Torino Italy) Matteo Semplice Dipartimento di Fisica e Matematica, Università dell Insubria Como, Italy) Abstract In tis paper, we review recent results on te role of entropy in te numerical integration of conservation laws. It is well known tat weak solutions of systems of conservation laws may not be unique. Pysically relevant weak solutions possess a viscous profile and satisfy entropy inequalities. In te discrete case entropy inequalities are used as a tool to prove convergence to entropy dissipating weak solutions. We start wit classical results, in wic entropy stability is used to prove te convergence of numerical solutions. We continue wit entropy stable scemes, wic are designed in order to produce entropy dissipation and can be proven to be convergent. Next we consider entropy as an error/regularity indicator permitting a local control on te beavior of te sceme, and tus driving grid and sceme adaptivity. Keywords: conservation laws, entropy, ig order, error indicator 1. Introduction We consider systems of m conservation laws in d dimensions: u t + d x f u) = 0, A u) = f ) u u). 1) =1 We suppose tat te system is yperbolic, i.e. eac Jacobian matrix A u) as m real eigenvalues and a complete set of eigenvectors u. Many matematical models describing pysical penomena caracterized by signals travelling wit finite speed of propagation can be written in tis form. Applications include gas dynamics, traffic flow, magnetoydrodynamics, astropysics, to name ust a few. It is well known tat solutions of 1) may exibit a very ric structure, wit discontinuities socks) arising in a finite time even from smoot initial data. Tis possibility obliges to enric te set of possible solutions of 1) Corresponding autor addresses: gabriella.puppo@polito.it Gabriella Puppo ), matteo.semplice@uninsubria.it Matteo Semplice)

2 G. Puppo, M. Semplice / Pysics Procedia ) introducing te class of weak solutions. Te set of weak solutions contains piecewise C 1 functions, wit step-like discontinuities wose speed is imposed by te weak form itself, troug te so-called Rankine-Hugoniot conditions. Te problem is tat enlarging te class of admissible solutions, uniqueness is lost, and retrieving uniqueness turns out to be an extremely ard task, wic up to now as not been settled in te general case, [1]. In many models of te form 1) coming from pysics, te yperbolic system is obtained disregarding iger order terms, multiplied by a small parameter. In many cases, te iger order term is parabolic, and te small parameter is a diffusive parameter, wic, in analogy wit fluid dynamics, is called viscosity. In tese cases, uniqueness can be retrieved if weak solutions of 1) are required to be limits of viscous solutions as te viscosity tends to zero. On te oter and, uniqueness in pysics is often ensured imposing irreversibility on admissible socks. Tis results in entropy conditions tat a weak solution sould satisfy. Te link between tese two criteria is complex. Weak solutions obtained wit te vanising viscosity metod satisfy entropy conditions [1, 2]. Te reverse is not true in general, and is linked to te ricness of te family of entropies tat a system possesses. Te best understood case, is te multidimensional scalar case, were entropy conditions are enoug to ensure uniqueness, see [3], were Kruzkov s uniqueness result is clearly described. Te numerical integration of 1) clearly must deal wit te same difficulties encountered in te analysis of systems of conservation laws. It is easy to design scemes tat compute weak solutions propagating wit te speed prescribed by te Rankine Hugoniot conditions. Tese are te so called conservative scemes, and teir introduction as permitted to construct scemes wic preserve exactly te correct speeds, wit a very easy to implement condition on te numerical fluxes. Te issue of ensuring convergence to te correct unique solution as proven to be muc arder. Usually, one relies on discrete entropy conditions, wic owever is satisfactory only in te scalar case. However, te numerical solution of yperbolic systems, beside te difficulties arising from te analysis of tese equations, introduces new ones, wic are linked to te need of dealing wit discontinuous solutions. First order scemes ave te most favourable teoretical results. Te class of monotone scemes produces solutions wic are perturbed by a viscous-like error term, and tus tey converge to solutions possessing a viscous profile, [3, 4]. Moreover, tey satisfy discrete entropy inequalities. However, tey converge slowly, tus requiring very fine grids, and tey smear discontinuous profiles. Increasing te order is conceptually quite easy, but numerical difficulties increase tremendously. Hig order scemes tend to produce spurious oscillations on socks wic may interact non linearly wit te solution, generating unreliable results. To prevent te onset of spurious oscillations, a uge amount of ideas as been spent, and te literature is very extensive, see [3, 4] for second order scemes and [5] for iger order, and teir references. Higer order scemes do not produce numerical solutions wit an error term caracterized by a neat parabolic structure, as is te case for first order scemes. Tus it is ard to prove tat teir numerical solutions converge to vanising viscosity solutions. Initially, te construction of ig order scemes as concentrated on te need to avoid spurious oscillations, introducing te concept of TVD Total Variation Diminuising) scemes, wic are prevented from oscillating by brute force. But tis concept cannot be applied in 2D, and it cannot be applied even in 1D for te solution of systems of equations in conservative variables. For tese reasons, exploring te consequences of imposing entropy conditions at te discrete level as been a fruitful tool. Tis program of work as started at te beginning of te 80s, wit a series of works [6], [7], [8], continued wit a study of te link between entropy and numerical viscosity in [9], and more recently summarized and furter expanded in [10]. In te present work, we describe tecniques for te construction of numerical scemes wic are linked to te concept of entropy for systems of conservation laws. Te paper starts wit a brief summary of te basic analytical results involving te entropy, and of te main difficulties of numerical scemes for conservation laws in 2. Ten, 3 contains te construction of entropy stable scemes from [10], exploring te link between numerical diffusion and entropy stability. Next te generalization to iger order scemes designed in [11, 12] is described, and we sow numerical results illustrating te entropy dissipation of tese scemes, especially in te fully discrete case, were te entropic beavior of very common time discretizations is not well known yet. Next in 4 our work [13, 14] is described on te use of entropy residuals to obtain a local estimate of te truncation error of te sceme, based on te computed numerical solution. Tis tool permits to construct a reliable a posteriori error indicator to drive te construction of a locally adaptive grid, following te unsteady structure of te numerical solution. It is also possible to use te error indicator to modify te metod locally sceme adaptivity), applying ig order scemes only on smoot portions of te solution, and/or using te costly non linear devices tat prevent te onset of spurious oscillations only were tey are actually needed.

3 G. Puppo, M. Semplice / Pysics Procedia ) Several important results ave not found space in tis review. A posteriori error control based on entropy residuals can also be found in [15], [16] and recently summarized in [17]. Te error indicator proposed and analyzed in tese works relies on te family of Kruzkov s entropies for multidimensional scalar conservation laws. Tese results are rigorous, but tey lack te simplicity and te localization of te indicator proposed in [13, 14] and described below. Moreover, tese results depend eavily on te family of entropies for scalar conservation laws, and cannot be generalized easily to yperbolic systems of equations. An original approac to te role of entropy in te analysis of systems of conservation laws can be found in [18] and its copious references. Here entropy inequalities are derived minimizing te entropy of kinetic BGK models converging to given systems of conservation laws, as relaxation parameters go to zero. Te entropy pairs of systems of conservation laws are naturally obtained studying te convergence towards equilibrium of te BGK system. Tis approac follows extensive work on te connection between systems of conservation laws and kinetic models, wic provides a promising alternative to te idea of caracterizing weak solutions troug viscous profiles. Note also tat troug kinetic approximations many numerical scemes for systems of conservation laws ave been derived, see [19, 20]. Finally, in [21], a measure of te entropy residual is used to modify te sceme locally. Te sceme proposed is a ig order linear sceme, witout te usual non linear devices preventing te onset of spurious oscillations. Here, te non oscillatory beavior of te solution is preserved adding a degenerate parabolic term wic is turned on wen te entropy residual surpasses a given tresold. 2. Conservative scemes and entropy dissipation In tis section, we recall te main properties of te numerical solution of conservation laws. Classical references are [2] and [22] and, more recently, [1] for te analysis of solutions of systems of conservation laws. Several textbooks contain descriptions of te treatment of numerical integration of conservation laws, in particular see [4] and [3]. We start considering te single scalar conservation law wit initial condition: u t + f u) = 0, ux, t = 0) = u 0 x). 2) x As long as te solution is smoot, tere is a unique classical solution, wic can be constructed wit te metod of caracteristics. However, it is easy to see tat discontinuities may form in finite time, even from smoot initial data. Te metod of caracteristics in fact breaks down if two or more) caracteristics intersect at te same point and a sock develops. In tis case, te solutions of te conservation law must be understood in te weak sense. Multiply te conservation law by a smoot test function φ wit compact support, and integrate in space and time for t [0, ) = R + and x R. Integrating by parts, te boundary terms pick up only te initial condition, and te weak form of te conservation law is found: + 0 [ φt u + φ x f u) ] dx dt = + φx, 0)ux, 0) dx 3) φ C0 1R R+ ). Te weak solutions obtained from 3) admit discontinuities satisfying te Rankine-Hugoniot ump conditions wic dictate te speed at wic suc solutions can travel. To see on an easy example tat te Rankine Hugoniot condition derives from te weak form of te conservation law, we consider te following initial data wit a ump: u L x < 0 ux, t = 0) = u 0 x) = 4) u R x 0. Te situation is depicted in Fig. 1. Clearly by similarity, tis step will travel wit constant speed s, so tat te solution will be: u L x < st ux, t) = x st. u R

4 G. Puppo, M. Semplice / Pysics Procedia ) u L u L u R s t u R Figure 1: Advection of a step function Substituting tis information in te weak form, one finds: st 0 φ t u L dx dt + wic, after some algebra, reduces to: 0 st φ t u R dx dt + [ f u L ) f u R ) ] φst, t) dt = 0 + [ sul u R ) f u L ) f u R )) ] φst, t) dt = 0, φ C0 1 R R+ ). Since tis equation olds for all test functions, te speed of te ump must satisfy te equation In te scalar case te above relation reduces to 0 φx, 0)ux, 0) dx, su L u R ) = f u L ) f u R ). 5) s = f u L) f u R ) u L u R, 6) wic dictates te speed at wic a discontinuity will move. Te Rankine Hugoniot condition is linked to te conservation of te amount mass) of u present in te flow. Let Mt) = b ux, t) dx be te amount of u present in an interval a a, b) containing te origin, and consider again te step-like initial condition given above. From Fig. 1, one can see tat te mass at time t + t is given by Mt + t) = Mt) + s tu L u R ). On te oter and, te variation of mass is given by te net flux leaving te interval a, b) in te time t, so Mt + t) = Mt) + t f u L ) f u R )), and equating tese two expressions te Rankine Hugoniot condition results. Te weak form 3) extends te class of possible solutions of 2), but uniqueness is lost. To recover uniqueness, some furter constraints must be imposed. Typically, many systems of conservation laws derive from models of matematical pysics, and are approximations of parabolic systems of equations, wit small viscous coefficients. Tus, one is interested in a weak solution possessing a viscous profile, tat is a solution tat can be obtained as te vanising viscosity limit of a solution of te parabolic system. It is possible to prove [2, Capter 20] tat vanising viscosity solutions satisfy entropy inequalities wic can be constructed in te following fasion. Suppose te conservation law is endowed wit an entropy pair ηu), ψu) suc tat η is convex and ψ satisfies η u) f u) = ψ u). Note tat at least in te scalar case infinite entropies can be found. Multiplying te conservation law by η one finds tat for smoot flows: η t + ψu) = 0. x If socks develop, te vanising viscosity weak solution satisfies te inequality: η t + ψu) 0 x

5 wic again sould be understood in te weak sense: + 0 G. Puppo, M. Semplice / Pysics Procedia ) [ φt ηu) + φ x ψu) ] dx dt + φx, 0)ηux, 0)) dx 7) φ C 1 0 R R+ ), φ 0. Note tat tis time te sign of te test function must be prescribed, so tat te sign of te expression being integrated does not depend on te sign of φ. Considering again te step like initial condition 4), te weak form of te entropy inequality reduces to: ηu R ) ηu L ))s ψu R ) ψu L )) 0, 8) were again s is te speed of te discontinuity. Te inequality above selects wic initial steps result in entropic socks, according to te entropy η. See also [23]. Te vanising viscosity solutions satisfy entropy inequalities 7) for all admissible entropies, tat is for all entropy pairs satisfying te compatibility condition η f = ψ, tus te entropy inequality is a necessary condition for uniqueness. In te scalar case, all convex functions are admissible entropies, and Kruzkov in 1970 proved tat a weak solution of te conservation law satisfying te entropy inequality for all admissible entropies is unique, [3, capt 2]. Tis important result olds also in te multidimensional case. For systems of conservation laws, te existence of an entropy pair is not guaranteed, even toug systems deriving from pysics usually admit entropy inequalities, as in gas dynamics. However, even for systems wic ave entropies, te entropy condition is not sufficient to establis uniqueness in te general case, see [1]. Naturally, in te numerical integration of conservation laws, it is desirable to preserve te properties outlined above. Te main tools are te Lax-Wendroff teorem, to prove convergence to weak solutions, and te discrete version of te entropy inequality, to prove tat a certain sceme produces numerical solutions satisfying te entropy condition. Te computational domain is discretized wit a grid in space and time wic, in tis section, we suppose to be uniform. Let x, t n ) denote te grid points, and suppose tat te mes is caracterized by a widt in space and a time step k. Introduce te control volumes V n = x /2, x + /2) [t n, t n + k), and te cell averages u n at time t n defined as: u n = 1 x +/2 x /2 ux, t n ) dx. Integrating te conservation law in space and time on te control volume V n, te finite volume formulation for te cell averages u is obtained: u n+1 = u n 1 [ k f 0 ux + ) k 2, tn + t) dt f ux ) ] 0 2, tn + t) dt, 9) Note tat tis relation is exact, and must old for all control volumes. On te cosen grid, te conservation law reduces to a system of finite volume equations for te cell averages. Note owever tat tis system of equations is not closed, because te equations provide te evolution of te cell averages in terms of fluxes evaluated at te cell interfaces. To obtain te evolution of te cell averages from te finite volume equations, one sould extract pointwise information from te cell averages, to compute te integral of te fluxes at te cell interfaces. Numerical scemes in conservation form mimic tis structure. Let U denote te numerical solution. Ten a numerical sceme in conservation form can be written as: U n+1 = U n k F n +1/2 ) Fn 1/2 10) were F n +1/2 is te numerical flux function. For te time being, we consider very simple scemes, for wic te solution U n+1 at time t n+1 depends only on te numerical solution at te previous time step t n at tree grid points. In tis case, te numerical flux function will depend on two grid points, namely F n +1/2 = FUn, U n +1), and te function F must satisfy te following properties: FU, U) = f U), consistency)

6 FU, U +1 ) is at least Lipscitz continuous. G. Puppo, M. Semplice / Pysics Procedia ) Numerical solutions obtained wit conservative scemes are good solutions, in te sense tat, if tey converge, tey converge to weak solutions. Tis is te content of te Lax Wendroff teorem: Teorem 2.1. Lax Wendroff teorem. Let U x, t) be a numerical solution obtained on a grid of widt. Suppose tat: U is bounded in L uniformly in ; U U as goes to zero in L 1 on every bounded rectangle in R R +, wit k/ fixed; U is obtained wit a conservative sceme. Ten te limit solution is a weak solution of te conservation law. Te teorem assumes te existence of a sequence of numerical solutions, wic are indexed by te mes widt. Te functions in te sequence must enoy some sort of stability, and te sequence must be convergent. If moreover all functions in te sequence are obtained wit a conservative sceme, ten te limit solution is guaranteed to satisfy te weak form of te conservation law 3), and tus to contain socks satisfying te Rankine Hugoniot ump conditions. Te proof is interesting, see [3], and is based on te following ingredients: multiply te conservative form of te sceme by a test function φ, summing over all grid points in space and time; sum by parts, discarging te differences from U and F +1/2 on φ; pass to te limit for 0, using te boundedness of U, te convergence of U, and te consistency of te numerical flux F, and get te weak form of te conservation law. Note tat wen summing by parts all interior terms can be grouped togeter in pairs, because, tanks to te conservative form, te fluxes are evaluated at te interfaces, and terefore tey always appear twice: once as flux exiting from te cell V n, and te second time as flux entering te cell Vn +1. Tis point is important, and we illustrate it using again te initial condition 4). Suppose te computational domain is te interval [a, b] wic is large enoug to contain te initial step up to a final time T. Let x a = a + 2 and x b = b 2 be te grid points at te end points of te interval. Ten te total mass contained in te interval [a, b] at time t n < T is given by: Mt n ) = b = a U n Applying te conservative sceme 10), te total mass at time t n+1 < T is given by: Mt n + k) = = b U n+1 = a b = a [ U n k F n +1/2 1/2) ] Fn = Mt n ) k F n b +1/2 Fn a 1/2), were all interior terms cancel out because te conservative structure of te sceme ensures tat watever flows out a cell, flows into te following cell. At te boundary te consistency of te sceme can be applied. Here by ypotesis te solution is constant, so: F n b +1/2 = FUn b, U n b +1) = Fu R, u R ) = f u R ) F n a 1/2 = FUn a 1, U n a ) = Fu L, u L ) = f u L )

7 G. Puppo, M. Semplice / Pysics Procedia ) Substituting tis information above, it can be seen tat te sceme provides te exact evolution of te total mass of te system, and tus te speed of propagation of socks is also exact. In more general cases, te propagation speed of socks will not be exact, because te end points of te ump may depend on time and tey will not be evolved exactly, but we wis to stress te fact tat no error in te propagation speed will occur if te end points of te ump are evolved correctly, even toug te sock may be spread on several cells. Tis in turn is guaranteed by te fact tat evaluating te total mass, te numerical fluxes cancel out for interior cell boundaries. In oter words, to preserve global properties, suc as convergence, te numerical flux functions do not need to be sopisticated, as long as tey are consistent and tey are evaluated at cell boundaries, as in 10). Convergence means tat te numerical solution approximates te true solution as 0, but ow small sould be before te error between te numerical and te true solution lies below a given tresold remains unknown. Naturally, one is interested in te beavior of te solution for a finite mes size. But for a finite value of convergence teorems are not elpful. In tis case, error estimates are needed, as in [24] or [17], and a posteriori error estimators [17, 25, 14]. As a matter of fact, a convergent sceme may be quite lowsy locally, for a finite mes widt. To transfer te entropy condition to te discrete level, integrate te entropy inequality over te control volumes V n defined above as in 9), obtaining: 1 x +/2 x /2 [ ηux, t n+1 )) ηux, t n )) ] dx + 1 k [ ψux +1/2, t n + t)) ψux 1/2, t n + t)) ] dt 0. 11) 0 Tis inequality involves te unknown exact solution, and it sould be extended to te numerical solution. Te finite volume conservative sceme 10) constructs te solution as a sequence of point values wic coincide wit te cell averages of te numerical solution. Tese point values are used to define a piece-wise constant function Ux, t) as follows: Ux, t) = U n x, t) [x 2, x + 2 ) [tn, t n+1 ). 12) Note tat tis reconstruction is conservative, in te sense tat it preserves te amount of mass U: x +/2 x /2 Ux, t) dx = U n t [t n, t n+1 ). Wit tis assumption, one can evaluate te space integral in te entropy inequality. For te integral in time, wic involves te entropy fluxes, numerical entropy fluxes are constructed. For a 3-point sceme wit numerical flux F n +1/2 = FUn, U n +1), te numerical entropy flux will be a function Ψ n +1/2 = ΨUn, U n +1), satisfying te following properties: ΨU, U) = ψu), consistency) ΨU, U +1 ) is at least Lipscitz continuous. Substituting tis information in te finite volume formulation of te entropy inequality 11), it is found tat te numerical solution sould satisfy te discrete entropy inequality: η U n+1 ) η U n ) k [ + Ψ n +1/2 1/2] Ψn 0. 13) A sceme satisfying te discrete entropy condition and te ypoteses of te Lax-Wendroff teorem converges to an entropy satisfying weak solution. In te scalar te following teorem olds [3]: Teorem 2.2. Convergence to te entropic solution. widt. Suppose tat: Let U x, t) be a numerical solution obtained on a grid of U is bounded in L uniformly in ; U U as goes to zero in L 1 on every bounded rectangle in R R +, wit k/ fixed; U is obtained wit a conservative sceme.

8 G. Puppo, M. Semplice / Pysics Procedia ) U satisfies te cell entropy inequality 13) wit a consistent entropic flux Ψ constructed for all admissible entropy pairs η, ψ and for all. Ten te limit solution is te unique entropic weak solution of te conservation law. Te proof is very similar to te proof of te Lax Wendroff teorem, and sows tat te limit solution U satisfies te weak form of te entropy inequality 7) for all admissible entropies. Again, te proof relies on te fact tat te numerical entropy fluxes are evaluated at te cell interfaces, see [3]. 3. Numerical diffusion and entropy stable scemes In tis section we will start our review of numerical scemes obtained using te entropy as a tool to design numerical scemes. Tis section starts wit te notion of entropy conservative and entropy stable scemes introduced in [9] and expanded in [10]. Next, we sow a few typical results in te one-dimensional scalar case. Te section ends wit a description of te ig order entropy stable scemes of [12]. Te easiest conservative sceme one can tink of as te simple numerical flux F n +1/2 = 1 n 2 f U ) + f U n +1) ), leading to U n+1 = U n k n f U 2 +1) f U n 1) ). Tis sceme is owever disastrous, since it is unstable for all k. Using discrete Fourier analysis for te equation u t + au x = 0, it is easy to see tat tis sceme actually introduces a negative numerical diffusion, since its modified equation as te form: u t + au x = 1 2 ka2 u xx. Te viscosity in tis equation is ν = 1 2 ka2, and te numerical solution will blow up exponentially for every coice of te timetep k. Tus, te sceme must be modified adding a diffusive term. Te simplest coice is: U n+1 = U n k n f U 2 +1) f U n 1) ) + k 2 Q U n +1 2U n + U n ) 1 Let λ = k denote te mes ratio, wic, in tis section, will be constant. Te parameter Q must be determined in order to ensure tat te sceme preserves te monotonicity of te solution, in order to avoid spurious oscillations. Scemes satisfying tis request are called monotone, and tey can be caracterized by te constraints U n+1 U n +l 0 l = 1, 0, 1. It is easy to see tat te sceme above is monotone, provided tat Q max f u), wit λq 1, wic implies also te CFL condition. Tese scemes can be proven to satisfy all entropy inequalities, and tus tey converge to te unique entropy solution in te scalar case, [3, 26]. More generally, te following modified numerical flux is introduced: F +1/2 = 1 2 n f U ) + f U n +1) ) 1 2 Q n +1/2 U +1 U n ). 14) Tis sceme is monotone if Q +1/2 f U +1 ), Q 1/2 f U 1 ) and 1 2 λq +1/2 + Q 1/2 ) 1. Most first order numerical scemes can be written in tis form. Te Local Lax Friedrics sceme for instance as a coefficient of numerical viscosity given by Q +1/2 = max f U +1 ), f U ) ). Monotone scemes are nice, because tey converge to te exact solution, but tey are limited to first order accuracy [4]. Tis means tat to get an accurate solution very fine grids are needed, and umps in te numerical solution, especially for contact discontinuities, are smeared on several grid points. We illustrate tis beavior on te simple advection equation u t + u x = 0 wit periodic boundary conditions, considering te evolution of a square wave using

9 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 2: Linear advection of a square wave wit te Lax Friedrics sceme, N = 100, N = 200, N = 400, N = 800 cells. a) T = 8, b) T = 32. te Lax Friedrics sceme, wit Q +1/2 = 1 wic gives te minimum amount of numerical dissipation in tis case, and λ = 0.9, see Fig. 2. It is clear tat te profile is smeared as time increases, even for very fine grids. Clearly tis sceme is too diffusive. In tis case, te same results would be obtained wit te Godunov sceme wic, in te linear case, as a coefficient of numerical viscosity Q +1/2 = a, were a is te propagation speed. In order to reduce drastically te artificial diffusion, a possibility is to increase te order of te sceme. Fig. 3 sows te effect of increasing te order of accuracy, on a smoot solution for long integration times ere te initial ump completes its period 8 times). Wile te first order sceme is still far away from te exact solution wit N = 800 grid points, te second order solution is almost coinciding wit te exact solution wit only 200 grid points. However, using a standard second order sceme on a discontintinuity, te solution starts oscillating. Te Lax Wendroff sceme for te linear advection equation u t + au x = 0 can be written in viscosity form 14) wit a coefficient Q +1/2 = λa 2. Since for stability λ < 1/ a, te Lax Wendroff sceme indeed as a smaller viscosity tan te Godunov sceme, but te problem is tat it as too little viscosity. Te problem tan is: is it possible to say wat is te minimum amount of viscosity tat a numerical sceme sould ave? Entropy conservative scemes address exactly tis issue. Since it is known tat for convergence a numerical sceme sould dissipate entropy, at least on socks, a starting point is to construct scemes wic dissipate no entropy. Ten te numerical diffusion inerent in tese scemes can be used as a bencmark to set te minimum numerical diffusion tat a sceme sould ave. Tis idea as been developed by Tadmor in a series of papers [8], [9] and te review [10]. Here we illustrate te beavior of te scemes for scalar conservation laws, but te main setting will be described for systems of equations. In tis section, we consider one-dimensional systems of m conservation laws: u t + f x u) = 0, Au) = f u u). 15) Again, te system is yperbolic, i.e. te Jacobian matrix Au) as m real eigenvalues and a complete set of eigenvectors u. Recall tat an entropy pair η, ψ) for te system 15) is a couple of functions ηu) and ψu) suc tat η is convex i.e. te Hessian η uu is a positive definite symmetric matrix) and ψ satisfies te compatibility relation For example gas dynamic equations ave a family of entropy pairs of te form η u ) T Au) = ψ u ) T. 16) ηu) = ρs ), ψu) = ms ), u = [ρ, m, E] T, S = lnpρ γ ) 17)

10 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 3: Linear advection of a smoot function wit te Lax Friedrics sceme, and wit te Lax Wendroff sceme N = 100, N = 200, N = 400, N = 800 cells, up tp T = 32. a) Lax-Friedrics, b) Lax-Wendroff. were ρ is te density, m is momentum, E is te density of total energy, S is te specific entropy, p is te pressure and γ is te ratio of specific eats. Te scalar function must satisfy te properties γ > 0, > 0, [6]. In particular for S ) = S we recover te standard entropy used in gas dynamics. Introduce te entropy variables vu) = η u. Since te Jacobian of te transformation is η uu, wic is invertible, te map associating te conservative variables u to te entropy variables is invertible. Entropy variables are used to rewrite te sytem of conservation laws in symmetric form: In fact, carrying out differentiation, te system becomes t uv) + x gv) = 0, gv) = f uv)). 18) u v t v + f u u v x v = η uu ) 1 t v + Au)η uu ) 1 x v = 0, in wic te matrix Au)η uu ) 1 is symmetric, because of te compatibility condition 16). However, ere te interesting point is tat entropy variables permit to introduce te entropy potentials φ and θ: φv) = v uv) ηuv)) 19) θv) = v gv) ψuv)), 20) were u v denotes te inner product between te vectors u and v. Note tat φ v) = uv) wile θ v) = gv) Semidiscrete entropy stable scemes Entropy stable scemes are first defined in te semidiscrete case, in wic only te space discretization is considered, wile time is left as a continuous variable. Te original system 15) is re-written in terms of entropy variables, and on tis system a semidiscrete conservative sceme is applied: d dt uv) + 1 ) G +1/2 G 1/2 = 0 21) were G +1/2 denotes a numerical flux consistent wit te flux function guv)). Te aim is to compute a numerical flux G +1/2 suc tat te sceme above is entropy conservative, in te sense tat a local cell entropy equality is satisfied in eac cell: d dt ηu) + 1 ) Ψ +1/2 Ψ 1/2 = 0 22)

11 G. Puppo, M. Semplice / Pysics Procedia ) for some consistent numerical entropy flux Ψ. In particular, let [10]: Ψ +1/2 = Ψv, v +1 ) = 1 2 v + v +1 ) G +1/2 1 2 θv ) + θv +1 ) ). 23) Note tat, on grid values, u = uv ), and f u ) = gv ). It is easy to ceck tat te expression above defines a consistent entropy flux, provided G +1/2 is also consistent wit te flux gv). Now te entropy dissipation due to te sceme can be computed: d dt ηu) + 1 ) d Ψ +1/2 Ψ 1/2 = v dt u + 1 ) Ψ +1/2 Ψ 1/2. Substituting te numerical entropy flux 23) wit G +1/2, and te expression for d dt uv) using 21), te entropy dissipation becomes 1 v G +1/2 G 1/2) 1 + v + v +1 ) G +1/2 2 θv ) + θv +1 ) ) v + v 1 ) G 1/2 + θv ) + θv 1 ) )) = 0 Rearranging terms 1 [ v +1 v ) G +1/2 2 θv +1 ) θv ) ) + v v 1 ) G 1/2 θv ) θv 1 ) )] = 0 and clearly te equality olds, provided tat G +1/2 v +1 v ) = θv +1 ) θv ). 24) So, wit tis coice te entropy dissipation in 22) is zero. On te oter and, if te numerical flux G +1/2 satisfies G +1/2 v +1 v ) θv +1 ) θv ), 25) te left and side of 22) is negative and entropy dissipation occurs witin te cell. Definition 3.1. Entropy conservative and entropy stable scemes A numerical sceme is entropy conservative if its numerical flux G +1/2 satisfies 24). It is entropy stable if its numerical flux satisfies 25). Note tat a numerical flux G +1/2 in te entropy variables induces a corresponding numerical flux in te conservative variables: F +1/2 = FU, U +1 ) = Gv, v +1 ) = G +1/2. In te scalar case, te entropy conservative flux is unique: G +1/2 = θv +1) θv ) v +1 v. For systems of equations, several coices are possible, since 24) is a single scalar constraint. Example 3.2. Burgers equation For Burgers equation f u) = 1 2 u2. Coosing te entropy ηu) = 1 2 u2, te entropy flux is ψu) = 1 3 u3. Te entropy variable in tis case is simply vu) = u and te entropy potential is θu) = 1 6 u3. Tus te entropy conservative flux is F +1/2 = F U, U +1 ) = 1 6 ) U U +1 U + U 2. As te example sows, te entropy conservative flux is linked to te particular entropy cosen. Tis is not a problem in te scalar case for convex flux functions, since in tis case entropy stability for te quadratic entropy is enoug to select te unique entropic solution, see [10, 27]. Now, it is possible to rewrite te entropy conservative sceme in viscous form, to empasize te numerical viscosity of tese scemes. It is convenient to rewrite te viscous form of te sceme in terms of entropy variables: G +1/2 = 1 2 f U n ) + f U n +1) ) 1 2 Q +1/2 v n +1 v n ).

12 G. Puppo, M. Semplice / Pysics Procedia ) Substituting tis expression in 25), and using 24), one finds tat te viscosity of an entropy stable sceme must satisfy 1 2 Q +1/2 v +1 v Q +1/2 v +1 v 2, wic gives Q +1/2 Q +1/2, 26) tat is any entropy stable sceme must contain more numerical viscosity tan an entropy conservative one. Note tat in te scalar case Q +1/2 = Q +1/2 U +1 U v +1 v, and since vu) is a monotone increasing function of u, te fraction on te rigt and side is always non-negative. Tus te viscosity coefficients of entropy stable scemes must satisfy Q +1/2 Q +1/2. 27) Te entropy conservative coefficient of numerical viscosity Q +1/2 depends on te particular entropy cosen. It is interesting to see wat is te maximum of Q +1/2 over all possible entropies. To compute tis value, rewrite Q +1/2 as Q +1/2 = f U ) + f U +1 ) 2F +1/2 U +1 U and compute te supremum over all possible entropies. It as been proven [8, 7] tat te supremum is acieved by Godunov flux. Tus an entropy stable sceme, wic is uniformly stable for all entropies, as a coefficient of numerical viscosity Q +1/2 wic is larger tan te coefficient of viscosity of te Godunov sceme. Tese are te so called E-scemes [7], wic are again first order scemes. However, it is easy to see, at least in te scalar case, tat entropy conservative scemes are second order accurate. In fact, te entropy conservative flux G +1/2 can be written as G +1/2 = 1 v +1 θ v) dv = v +1 v v 1 v +1 v v +1 v gv) dv, were te definition of te entropy potential 19) as been applied. Introduce te cange of variables v +1/2 ξ) = 1 2 v + v +1 ) + ξv +1 v ). Te expression above becomes Integrating by parts, one obtains 1/2 G +1/2 = gv +1/2 ξ)) dξ. 1/2 G +1/2 = 1 2 gv +1) + gv )) 1/2 1/2 Substituting te viscous coefficient Q +1/2, tis expression gives ξg v +1/2 ξ))v +1 v ) dξ. 1/2 Q +1/2 = 2ξg v +1/2 ξ))v +1 v ) dξ. Integrating by parts again, te following expression for Q is finally found: 1/2 Q +1/2 = v +1 v ) 1/2 1/2 ) ) 1 4 ξ2 g v +1/2 ξ)) dξ,

13 G. Puppo, M. Semplice / Pysics Procedia ) wic sows tat Q +1/2 = Ov +1 v ) = O). Terefore te resulting sceme du dt = λ 2 f U +1 ) f U 1 ) ) + λ 2 Q +1/2 v +1 v ) Q 1/2 v v 1 ) ) is second order accurate in space and it conserves entropy. To obtain a second order entropy stable flux terefore one possibility is to increase te artificial diffusion of an entropy conservative sceme wit a term of order v +1 v. Tus te numerical flux [9] is modified as were G +1/2 = G +1/2 1 2 D +1/2, 28) D +1/2 = α +1/2 v +1 v ) 29) and α +1/2 is a positive coefficient. For instance one can take α +1/2 = max f U ), f U +1 ) ). Te entropy dissipation due to tis sceme can be computed defining te numerical entropy flux as in 23) and it is given by: d dt ηu) + 1 ) 1 [ Ψ +1/2 Ψ 1/2 = D +1/2 v +1 v ) + D 1/2 v v 1 ) ] 30) 4 = 1 [ α +1/2 v +1 v ) 2 + α 1/2 v v 1 ) 2]. 4 Note tat te entropy dissipation as te correct sign if α +1/2 0, tus yielding an entropy stable sceme. Tis construction can be easily extended to systems of equations. Now te space contribution to te cell entropy dissipation can be computed as S x) = 1 [ v +1 v ) T α +1/2 v +1 v ) + v v 1 ) T α 1/2 v v 1 ) ], 31) 4 were α +1/2 will be positive definite matrices. In te scalar case, te entropy stable sceme constructed above as an artificial diffusion coefficient Q +1/2 = Q +1/2 + D +1/2. v +1 v Terefore, if te numerical diffusion term is defined as in 29), Q +1/2 = O1) and tus te sceme is only first order accurate. To obtain iger order scemes te diffusion term defined in 29) must be modified, following [12]. See Te fully discrete problem Numerical solutions are obtained wit fully discrete scemes, in wic te time derivative of u is discretized usually wit Runge-Kutta scemes or multistep metods. Terefore, in order to ensure entropy stability of te final solution, te entropy production due to time discretizations must be computed. Tis problem is addressed in [10], were it is proven tat te Explicit Euler sceme creates spurious entropy, wile te second order implicit Crank Nicolson sceme is entropy conservative, and te Implicit Euler sceme dissipates entropy, tus improving te stability of entropy stable space discretizations. A more general approac can be found in [28, 11] were implicit ig order entropy conservative scemes are constructed. Most scemes for yperbolic systems of conservation laws are explicit. Te teoretical results imply tat in order to acieve entropy stability it is necessary to counterbalance te entropy production due to te explicit time integration wit te entropy dissipation of te space part. It is possible to follow te bounds on [10] for te Explicit Euler sceme to obtain entropy stable scemes for a first order in time explicit integration. However te entropy beavior of iger order explicit time discretizations is less known. Te analysis in [10] considers a few prototype cases. We start from te Backward Euler sceme. In tis case, te cell entropy dissipation of te fully discrete sceme can be defined as S = 1 k ηv) n+1 ηv) n ) 1 + Ψ n+1 +1/2 1/2) Ψn+1. 32)

14 G. Puppo, M. Semplice / Pysics Procedia ) Adding and subtracting te quantity 1 k vn+1 U n+1 U n ), S can be broken up in te two contributions S = S x) + S t), defined below: S x) S t) = 1 k vn+1 U n+1 U n ) + 1 Ψ n+1 +1/2 ) Ψn+1 1/2 = 1 vn+1 G n+1 +1/2 Gn+1 = 1 k Ψ n+1 +1/2 Ψn+1 1/2 ) 33) 1/2 ) + 1 ηv n+1 ) ηv n )) 1 k vn+1 U n+1 U n ). 34) S x) as already been computed in 31). Note tat te diffusion terms will also be evaluated implicitly. For S t), te first term, dropping te index, can be rewritten as: ηv n+1 ) ηv n ) = v n+1 v n η v v) dv = v n+1 v n η u u)u v dv. Let Hv) = η uu ) 1 = u v, wic is a symmetric positive definite matrix. Recall tat v = η u u), and cange variables as v n+1/2 ξ) = 1 2 vn+1 + v n ) + ξv n+1 v n ): ηv n+1 ) ηv n ) = 1/2 1/2 Te second term, dropping again te index, can be written as v n+1 U n+1 U n ) = v n+1 v n+1 1/2 v n+1/2 Hv n+1/2 )v n+1 v n ) dξ. 35) v n u v dv = v n+1 1/2 1/2 Hv n+1/2 )v n+1 v n ) dξ. Subtracting tis expression from te one above, te entropy dissipation due to te time discretization can be written as S t) = 1 1/2 ) 1 k 2 ξ v n+1 v n ) T Hv n+1/2 )v n+1 v n ) dξ 0. 36) Terefore, te Backward Euler differentiation introduces a negative term to te cell entropy dissipation, tus enancing entropy stability. For te Forward Euler sceme, te cell entropy dissipation can be defined as S = 1 k ηv) n+1 Tis time, add and subtract te quantity 1 k vn Un+1 two contributions S = S x) + S t) defined as S x) S t) ηv) n ) 1 + Ψ n +1/2 1/2) Ψn. 37) U n ). Te total entropy dissipation S will now be given by te = 1 vn Gn +1/2 Gn 1/2 ) + 1 Ψ n +1/2 ) Ψn 1/2 38) = 1 ) ηv) n+1 ηv) n 1 k k vn Un+1 U n ). 39) Again, S x) is already known from 31), except tat now te diffusion terms will be computed explicitly. For S t), te first term as already been computed in 35), and te following expression results: S t) = 1 k 1/2 1/2 ) ξ v n+1 v n ) T Hv n+1/2 )v n+1 v n ) dξ 0. 40) Tus, for te explicit Euler differentiation te entropy dissipation is positive, and terefore to acieve entropy stability, te time step k must be reduced, in order to ensure tat S t) < S x). Tis is found in [10] at te price of a suboptimal

15 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 4: Linear advection of a smoot function wit entropy conservative space differencing, N = 100, N = 200, N = 400, N = 800 cells, up to T = 32. a) Crank Nicolson, b) Backward Euler. CFL condition. Finally, consider te Crank Nicolson sceme, wic, unlike [10], we use in its standard form. Te cell entropy production will be given by: S = 1 k ηv n+1 ) ηv n )) + 1 Ψ n+1 +1/2 2 + Ψn +1/2 Ψn+1 1/2 ) Ψn 1/2 Using te same tecniques illustrated above, and 31), te space and time entropy dissipations will be given by: S x) S t) = 1 [ v n vn+1 ) T α +1/2 v n+1 +1 vn+1 ) + v n +1 vn )T α +1/2 v n +1 vn ) + v n+1 v n+1 1 )T α 1/2 v n+1 v n+1 1 ) + vn vn 1 )T α 1/2 v n vn 1 )], = 1 k 1/2 1/2 ξ v n+1 v n )T Hv n+1/2 )v n+1 v n ) dξ. For te scalar case, one can coose te quadratic entropy ηu) = 1 2 u2, wic gives Hv) 1, and S t) = 0. Tis sows, tat for te scalar case, te Crank Nicolson sceme is entropy conservative. Now, we want to sow a few numerical results illustrating te beavior of entropy conservative scemes in te scalar case. We consider te linear advection problem u t + au x = 0, wit te quadratic entropy ηu) = 1 2 u2. As already noted, in tis case vu) = u. Te entropy flux is ψu) = 1 2 au2 and te entropy potential is θv) = 1 2 av2. Consequently te entropy conservative numerical flux is G +1/2 = 1 2 v +1 v ), wic gives te centered sceme. Clearly, tis space discretization is unstable, wen coupled wit te Forward Euler sceme. In fact te entropy production due to te explicit time differencing is not counterbalanced by entropy dissipation from te space discretization. Wen te entropy conservative sceme is coupled wit Crank Nicolson time differencing, a fully discrete entropy conservative sceme is obtained. Te results obtained in tis case on a smoot problem are compared wit an entropy stable in time sceme, as te Backward Euler time differencing in Fig. 4. It is clear tat te fully discrete entropy conservative sceme is muc more accurate tan Backward Euler. Te error for te entropy conservative sceme is due mainly to a dispersion error caracteristic of second order scemes, wile te error of te entropy stable implicit sceme is due to a very ig numerical diffusion. In bot cases, te grid parameter λ = k = 1, altoug bot scemes are unconditionally stable. On te oter and, te fully discrete entropy conservative sceme introduces spurious oscillations on non smoot solutions, see Fig. 5. Note tat tese are contact discontinuities, so tat no entropy dissipation occurs in te exact solution. Still, entropy dissipation is clearly necessary to obtain a solution witout oscillations. However, te sceme does ave some weak form of stability, wic is illustrated by te fact tat te Total Variation of te numerical solution remains bounded in time. 41)

16 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 5: Linear advection of a non smoot function wit a fully discrete entropy conservative sceme, N = 100, N = 200, N = 400, N = 800 cells, up to T = 32. On te rigt, we sow te beavior of te total variation and te total entropy as functions of time bottom curve), wic remains perfectly constant for all grids. a) Solution up to T = 32, b) Total variation and entropy in time Hig order scemes Te construction of ig order entropy stable scemes follows [11] and [12]. First, ig order entropy conservative space discretizations are constructed, wic are later modified adding diffusion terms to preserve ig order and ensure entropy stability. Te construction starts from te basic entropy conservative flux computed in 24). Tis numerical flux is based on two points, and iger order entropy conservative scemes are obtained enlarging te stencil and considering linear combinations of te basic two point entropy conservative flux: G p, +1/2 v p+1,..., v +p ) = p 1 k,r=0 β r,k G v k, v +1+r ). 42) Note tat for p = 1 te basic entropy conservative flux is recovered, wic is second order accurate. In [12], it is proven tat, coosing appropriately te coefficients β r,k, it is possible to obtain numerical fluxes wic are 2p order accurate. For example: G 2, +1/2 = 4 3 G v, v +1 ) 1 6 G v 1, v +1 ) 1 6 G v, v +2 ), 43) G 3, +1/2 = 3 2 G v, v +1 ) 3 [ G v 1, v +1 ) + G v, v +2 ) ] 10 44) + 1 [ G v 2, v +1 ) + G v 1, v +2 ) + G v, v +3 ) ]. 30 Entropy stable numerical fluxes wic are 2p order accurate are obtained modifying 28) and designing igly accurate diffusion terms D +1/2. It as already been noted tat to obtain a q order sceme te diffusion terms must be of order q. Tus te diffusion terms written in te form 29) are modified applying diffusion to reconstructed data. For tis purpose, a piecewise polynomial reconstruction operator of degree q 1 is applied to te grid values of te entropic variables, obtaining te function Vx), and te extrapolated values at te cell interfaces are computed as v + +1/2 = lim Vx) and v x x + +1/2 = lim Vx). 2 x x +1 2 If te function v is smoot, ten v + +1/2 v +1/2 = Oq + 1), and tus te new diffusion terms will be given by D +1/2 = α +1/2 v + +1/2 v +1/2). 45)

17 G. Puppo, M. Semplice / Pysics Procedia ) It is straigtforward to prove tat te space entropy dissipation 31) for te semidiscrete sceme becomes S x) = 1 [ ) )] v +1 v ) T α +1/2 v /2 v +1/2 + v v 1 ) T α 1/2 v + 1/2 v 1/2. 46) Since te scemes must be entropy stable, te entropy space dissipation must be negative. Terefore, te ump in te reconstructed values must ave te same sign as te ump in grid values of v, i.e. v +1 v ) v + +1/2 v +1/2 ) 0. Tis requirement is called sign preserving property in [12], were it is proven tat te sign preserving property is enoyed by te ENO Essentially Non Oscillatory) reconstruction algoritm of [29], and te standard piecewise linear reconstruction wit te MinMod limiter. However, te sign preserving property does not old for oter standard reconstruction algoritms, suc as te piecewise linear reconstruction wit te Superbee limiter, or te WENO reconstruction [5]. Not muc is known on te beavior of te entropy dissipation due to ig order time integrators. Usually, ig order fully discrete scemes for systems of conservation laws rely on Runge-Kutta scemes for te time discretization. Runge Kutta metods are defined by teir Butcer tableax A, b), were b is a vector of ν coefficients, and A is a ν ν matrix of coefficients; ν is te number of stages of te Runge-Kutta sceme. For explicit Runge-Kutta metods ERK) te matrix A is strictly lower triangular. Introducing a Runge-Kutta discretization in time in te conservative semidiscrete sceme d dt u + 1 ) F +1/2 F 1/2 = 0, 47) te numerical solution can be written as U n+1 = U n λ ν were, for an explicit sceme, te numerical fluxes F i) wic are given by i 1 = U n λ U i) i=1 +1/2 k=1 b i F i) +1/2 Fi) 1/2), 48) are computed only from te already known stage values, a i,k F k) +1/2 Fk) 1/2). 49) Witin tis framework, it is quite natural to define te cell entropy dissipation due to te fully discrete sceme as: were S = 1 k Ψ n +1/2 = ηu) n+1 ν i=1 b i Ψ i) +1/2 ηu) n p 1 Ψ i) +1/2 = Ψvi) p+1,..., vi) +p ) = β r,k Ψ v i) k, +1+r) vi), ) 1 + Ψ n +1/2 1/2) Ψn, 50) k,r=0 and te two-point numerical entropy flux Ψa, b) was defined in 23), te notation for te ig order flux is establised in 42) and clearly v i) = vu i) ). Since te expression for Ψ is known, te quantity S given by 50) is computable, once te numerical solution as been updated. Due to te ig non-linearity of te Runge-Kutta sceme, it is quite ard to separate te space from te time contributions to S, as was done for te simpler scemes before. However, in analogy wit te case of te Backward and te Forward Euler sceme, and keeping into account te structure of te Runge Kutta sceme, te space contribution to te entropy dissipation can be defined as S x) = 1 4 ν i=1 [ b i v i) +1 vi) ) T α +1/2 v +,i) +1/2 ) v,i) +1/2 + v i) v i) 1 )T α 1/2 v +,i) 1/2 1/2)] v,i). 51)

18 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 6: Linear advection of a smoot function. Numerical solution dased line). Time entropy residual for scemes of order 2, 3, and 4 from top to bottom). a) Entropy conservative, b) Entropy stable. In tis case, te time contribution to te cell entropy dissipation can be estimated only a-posteriori as S t) = S S x). 52) Te definition above of te contribution to te entropy dissipation given by te time discretization is not exact, because space and time are non linearly coupled in a ig order sceme wit Runge-Kutta time advancement, and at present it is not known ow to separate te two contributions. However, we can ave an idea of ow reliable te construction proposed really is, comparing te entropy dissipation due to an entropy conservative in space) semidicrete sceme integrated in time wit a Runge-Kutta metod, and te time entropy dissipation S t) of an entropy stable sceme, wit S t) defined in 52). In te first case, we find te exact entropy beavior of te time integrator, because te space entropy dissipation is zero by construction. Tis procedure can be applied only for smoot flows, because te numerical solution would start oscillating in te presence of discontinuities and/or ig gradients. In te second case, we are only estimating te time contribution to te entropy production, but we are working on a sceme tat can be applied also to non linear problems developing socks. Te differences in te two approaces are sown in Fig. 6. Bot plots sow te time entropy production due to several Runge-Kutta scemes, for a fixed number of grid points in space. On te left, we sow te total entropy production due to a fully discrete sceme built wit entropy conservative fluxes; note tat, by construction, te entropy production of tis sceme is due only to te time discretization. On te rigt, we plot S t) for a fully discrete sceme using entropy stable fluxes. As te figure sows, te two results do not coincide, but tey ave te same beavior. In particular, in bot cases, te explicit Runge-Kutta scemes give a positive contribution to te entropy production, wic terefore must be counterbalanced by entropy dissipation from te space discretization. However, te spurious entropy production due to te time discretization decreases very fast increasing te order of te time integrator. Note tat te two numerical solutions dased line in te background) are quite different: tese results were obtained wit a small number of grid points N = 50), and te second order sceme. Te solution on te left does not suffer from numerical diffusion, and is very close to te exact solution, on te rigt te effect of te diffusive fluxes, needed to acieve entropy stability, smears te profile. In te following part of tis section, we want to sow a few results concerning te entropy beavior of ig order Runge Kutta scemes for te time integration of scalar conservation laws, coupled wit ig order entropy stable fluxes. In eac cell entropy dissipation is computed wit 50), te space contribution is given by 51), and te time contribution is found by subtraction as in 52). We underline tat te splitting we propose between te space and te time dissipation is only approximate, since bot terms are intertwined in te Runge-Kutta time advancement. Still, te data we obtain permit to compute te total entropy dissipation, and to give ideas of wat is te order of magnitude of te terms involved. Now, we consider a non linear test. Te equation is Burgers equation wit initial data u 0 x) = sinx), and periodic boundary conditions. In tis problem, a sock develops at T = 1/2π) starting from zero strengt, reacing a

19 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 7: Burgers solution before sock formation. Rate of convergence of te entropy residuals for fully discrete Runge-Kutta scemes wit entropy stable fluxes. a) max S t), b) max S x). maximum strengt wic later decreases wit time. Fig. 7 sows te rate of convergence of te entropy residuals for fully discrete entropy stable scemes, from order 1 to order 4, before sock formation. Bot residuals converge to zero wit te expected rate, but it is clear tat te entropy dissipation due to te space discretization is orders of magnitude larger tan te entropy production due to te time integrators. Next, Fig. 8 sows te beavior of te entropy wit time as te solution develops a sock, for several scemes. Te solution is obtained wit N = 50 grid points. On te left, we sow te difference ηt) between te total entropy present in te computational domain as a function of time and te initial entropy. Since te sceme dissipates entropy, te quantity ηt) is negative, and it furter decreases wit time. Here te absolute value of tis quantity is sown. Te vertical line indicates te time of sock formation. On te rigt, te absolute value of te total entropy dissipated in eac time step appears. Te different curves correspond to te different scemes. As long as te flow remains smoot, te entropy dissipation decreases in absolute value as te order is increased. After sock formation owever, all curves converge towards te exact entropy dissipation. In particular, te exact entropy dissipation per time step for a single sock is given by, see 8): S exa t) = 1 f η u + were u denotes te ump in te variable u across te sock, since only te socked cell gives a contribution. Tis is te value te curves in te plots converge to. Note tat te entropy on te sock per time step canges wit time. Initially, in tis test problem, S exa t) increases as te sock develops starting from zero strengt, and later it decreases as te sock strengt slowly decays. 4. Entropy residual as an a posteriori error indicator In tis section we sow ow to compute an error indicator using te entropy production due to a finite volume sceme. Te indicator can ten be used as a tool to drive grid adaptivity. Tis tecnique was presented in [13] for central scemes based on staggered grids, and later extended to standard finite volume scemes in [14]. A more rigorous approac wic owever applies only to scalar conservation laws can be found in [15]. Te idea is to use te entropy production in eac cell, induced by a standard finite volume sceme, to estimate te local residual. Tis information can be exploited to build an a posteriori error indicator to drive te construction of an adaptive mes. For a standard finite volume sceme, te numerical solution satisfies equation 10) in all cells. Using te information needed to compute te numerical flux F, a consistent entropy flux Ψ can be computed, and te entropy production in eac cell corresponding to te sceme above can be defined as ψ η ), S n = ηu)n+1 ηu) n k + 1 Ψ +1/2 Ψ 1/2 ). 53)

20 G. Puppo, M. Semplice / Pysics Procedia ) a) b) Figure 8: Burgers solution and sock formation. Total entropy as a function of time left) minus initial entropy, and total entropy dissipation per time step rigt) for te fully discrete Runge-Kutta scemes wit entropy stable fluxes, for order from 1 to 4. a) Total entropy, b) Entropy production per time step. However, even on smoot flows S n 0. It will be sown tat on smoot flows S n = O)p, were p is te accuracy of te underlying numerical sceme. On socks, S n 1. Tus S n as te same size of te local error on smoot flows, wile it is large on discontinuities. Tese ideas can be applied on a standard ig order finite volume metod, based on Runge-Kutta time adavancement and piecewise polynomial reconstructions. Suppose te RK metod as ν stage values, and it is based on te Runge-Kutta tableaux defined by te coefficient matrices A, b). Ten, equation 48) is applied to te cell averages of te solution, obtaining te updated cell averages as U n+1 = U n λ ν i=0 b i F i) +1/2 Fi) 1/2), 54) were te numerical fluxes are computed wit a standard first order numerical flux Fa, b), suc as Lax-Friedrics or Godunov, to te boundary extrapolated data of te solution at te same stage: F i) +1/2 = FUi), +1/2, Ui),+ +1/2 ). Te boundary extrapolated data are obtained applying a non-oscillatory reconstruction procedure Rx) to te cell averages, obtaining te values at te cell edges: Tus te boundary extrapolated data U i),± are given by: R : {U} Rx), U + +1/2 = lim Rx), U x x + +1/2 = lim Rx). 2 x x /2 U i) are computed from te cell averages of te stage values at level i), wic = U n i 1 λ k=1 a ik F k) +1/2 Fk) 1/2). Terefore a single Runge-Kutta time step requires ν evaluations of te numerical fluxes and of te reconstruction R at eac cell interface. For ig order scemes, it is known tat te numerical flux cosen influences te solution less

21 G. Puppo, M. Semplice / Pysics Procedia ) and less as te order of te sceme is increased, [5]. So one cooses simple and fast numerical fluxes, suc as te local Lax Friedrics flux. On te oter and te reconstruction becomes more complex as te order of accuracy is increased. As a consequence, most of te computational effort is concentrated on te reconstruction of te boundary data. Once te solution U as been updated, te entropy production can be computed. Coose a numerical entropy flux Ψ, consistent wit ψu). At eac Runge-Kutta stage, evaluate Ψ i) +1/2 = ΨUi), +1/2, Ui),+ +1/2 ). 55) Note tat ere te numerical entropy flux is applied to te boundary extrapolated values wic ave already been reconstructed for computing F i) +1/2. Ten te numerical entropy production in te t cell is defined as S n = 1 k η U n+1) ν η 1/2) Un ) + λ b i Ψ i) +1/2 Ψi), 56) were η U n ) = 1 x +/2 x /2 i=1 η Rx)) dx. Tis integral will be estimated wit quadrature, using te same reconstruction operator used to compute te boundary extrapolated data. For order up to 2, ηu n ) can be evaluated wit te midpoint quadrature rule: η U n ) = η U n ) + O) 2. For iger accuracy, an improved quadrature rule must be cosen to evaluate ηu n ). It is important to note tat te evaluation of te entropy indicator 56) costs less tan te actual update of te solution 54), because no extra reconstructions are needed. A few properties of te entropy indicator defined in 56) now follow. Teorem 4.1. Rate of convergence. Consider a conservative sceme of order p, of te form 54). Coose a numerical entropy flux Ψ consistent wit te entropy flux ψ. Evaluate te entropy production S n as in 56). Ten, if te solution is smoot in te t cell: S n = Op ) for 0, If te solution is not smoot, in te t cell, S n = O1/) Tus te teorem ensures tat te entropy production is of te same order of te local truncation error of te sceme, were te solution is smoot, and te grid is fine enoug to resolve te accuracy of te sceme. On te oter and, S n C on socks. Te proof is based on te fact tat S n is te residual of a finite volume sceme for t ηu) + x ψu) = 0. We give te proof using te midpoint rule as a quadrature rule for te evaluation of ηu n ). Te general case is similar and it can be found in [14]. Proof. Using te midpoint rule for evaluating ηu), te time part of te entropy residual can be written as ηu n ) = ηu n ) η Ux ) U x x + O 3 ). Let ut, x) be te exact solution and take U n = u n, so tat te local error is U n+1 = u n+1 + O p+1 ) for a sceme of order p. Ten η U n+1) η Un ) = ηu n ) + 24 η U x ηu n ) 24 η U x + O 3 ) = ηu n+1 ) + O p+1 ) ηu n ) + O 3 ) = ηu) n+1 = 1 I ηu) n + O min3,p+1) ) = 1 t n+1 t n ηut, x)) dx dt + O min3,p+1) ). t I [ ηut n+1, x)) ηut n, x)) ] dx + O min3,p+1) )

22 G. Puppo, M. Semplice / Pysics Procedia ) Now te space part of S n is estimated. Since a Runge-Kutta sceme can be written as a time quadrature rule, and te numerical flux Ψ +1/2 is consistent wit te exact entropy flux ψ, λ ν i=1 b i Ψ i) +1/2 ) 1 t n+1 Ψi) 1/2 = ΨU + +1/2 t), U +1/2 t)) ΨU+ 1/2 t), U 1/2 t))) dt + Ok p+1 ) t n = 1 = 1 = 1 t n+1 t n [ Ψux +1/2, t), ux +1/2, t)) Ψux 1/2, t), ux 1/2, t)) ] dt + Ok p+1 ) + O p+1 ) t n+1 [ ψux +1/2, t)) ψux 1/2, t)) ] + Ok p+1 ) + O p+1 ) t n ψu) x dxdt + O p+1, k p+1), t n+1 t n I were k = λ was used togeter wit te assumption tat te sceme is p t order accurate on smoot solutions. Now, putting togeter te two contributions S n = 1 1 t n+1 ηv) k t n I + ψv) ) dxdt + O min3,p+1), k p+1) t x = Omin 2,p ) 57) Te proof can be extended to iger order scemes, improving te quadrature for η) as in [14], tus removing te O 2 ) term. If te cell contains a discontinuity, eac term contributing to te definition of S n is bounded. Tus S n C on socks. Te entropy indicator preserves te correct sign of te entropy dissipation, provided te numerical entropy flux is not only consistent, but accurately tailored to te particular numerical flux used to update te solution. Teorem 4.2. First order sceme wit Lax-Friedrics flux. Suppose a scalar conservation law is integrated wit te Forward Euler sceme and te Lax Friedrics flux, wit α = max f u). Coose Ten F +1/2 = 1 [ f U +1 ) + f U ) αu +1 U ) ], 2 Ψ +1/2 = 1 [ ψu +1 ) + ψu ) αηu +1 ) ηu )) ]. 2 S n 0 away from local extrema, wile small positive oversoots may occour in tose cells containing a local extremum, wit S n = O4 ) Te proof is quite tecnical and appears in [14]. However, te line of te proof can be appreciated from te proof of te following result. Teorem 4.3. First order sceme wit Upwind flux. Suppose a scalar conservation law wit f u) > 0 is integrated wit te Forward Euler sceme and te Lax Friedrics flux. Coose Ten S n 0. F +1/2 = f U ), Ψ +1/2 = ψu ).

23 G. Puppo, M. Semplice / Pysics Procedia ) Proof. Tanks to te particular coice of te numerical fluxes, te space part of te entropy production is wile, for te time part U n Ψ +1/2 Ψ 1/2 = ψu n ) ψun 1 ) = ψ u)du = U n 1 η ) ) U n+1 U n+1 η U n = η v)dv. U n U n U n 1 η u) f u)du. Introducing te cange of variables vu) = U n λ[ f Un ) f u)], te time part becomes η ) ) U n U n+1 1 η U n = η vu))λ f u)du. U n Adding te two terms, dropping te index n, and applying twice te mean value teorem one gets: ks n = U U 1 η ξ) [ 1 λ f η) ] λ f u)u u) du, were ξ and η are contained in te interval formed by U 1 and U. Finally, recalling tat η 0, tat te CFL condition ensures tat te first parentesis is positive, and tat f λ 1 by ypotesis, te sign of S n coincides wit te sign of U U 1 U u) du wic is negative, as required. Figure 9: Linear rotation of an initial rectangular patc. Solution wit limiters left), and witout limiters rigt) We illustrate tis result sowing te entropy production obtained integrating on a fixed grid, wit a second order sceme based on a piecewise linear reconstruction, te linear equation t u + x y 1 2 ) π 2 u) + y x 1 2 ) π 2 u) = 0, wit initial data consisting of a rectangular patc. Fig 9 sows te solution wit and witout limiters. As expected, te vertical sides of te patc are smooted by numerical diffusion, and te solution witout limiters presents spurious oscillations. Fig. 10 sows te entropy production corresponding to te numerical solutions sown in Fig. 9. Te indicator signals clearly te presence of te discontinuities, but it also detects te presence of spurious oscillations, wic correspond to te positive oversoots in te entropy production. Te non-oscillatory solution results in entropy dissipation wit te correct sign. On te oter and, Fig. 11 sows te error indicator by Karni-Kurganov-Petrova KKP) presented in [25], again for te numerical solutions sown in Fig. 9. In tis case te indicator detects te presence of singularities, but it does not give indications to distinguis te entropic solution from te solution wit spurious oscillations.

24 G. Puppo, M. Semplice / Pysics Procedia ) Figure 10: Linear rotation of an initial rectangular patc. Entropy production for te solution wit limiters left), and witout limiters rigt) Figure 11: Linear rotation of an initial rectangular patc. Karni-Kurganov-Petrova error indicator for te solution wit limiters left), and witout limiters rigt) Te rate of convergence of te entropy indicator is sown in Fig. 12, for a second and a for fourt order sceme, applied to a scalar conservation law. Te figure exibits te rate of convergence on a sock, on a contact discontinuity and on a smoot transition, togeter wit te reference slopes. It is clear tat S O 1 ) on a sock, S O 0 ) on a contact discontinuity, wile S O p ) on smoot solutions. Moreover, te figure illustrates clearly tat S varies by several orders of magnitude on te different solutions. Tis caracteristic is important for adaptive strategies, because it means tat an adaptive grid driven by te entropy indicator will not be too sensitive to te tresold cosen Adaptive algoritm In tis section, we sow results obtained applying te entropy indicator as an a posteriori error indicator to select an adaptive grid tat canges in time according to te beavior of te solution. Te grid used ere is based on mes widts tat cange as powers of 2 from a maximum lengt 0, so tat a cell at level l will be caracterized by a mes widt l = 0 2 l. Te non uniform grid on wic te numerical sceme is applied is stored in a tree structure, in order to ave easy access to eac given cell and its neigbors. Te memory structure wit wic te grid is stored is sown in Fig. 13. Cartesian grids are used in tese examples, as in te Conservation Laws Pack CLAWPACK) [30], but, unlike CLAWPACK, in tis case only te data for active cells is kept in memory, i.e. only one single grid wit cells of different sizes is stored, as opposed to patces of overlayed uniform grids, as in [30]. Te error indicator determines te local mes size. Te time step can be cosen following two strategies. dt-mode: use te same time step everywere. Wit tis coice, te CFL condition imposes tat te time step