APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS FOR NONLINEAR SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

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1 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS FOR NONLINEAR SYSTEMS OF HYPERBOLIC CONSERVATION LAWS CLAUS R. GOETZ AND ARMIN ISKE Abstract. We study analytical properties of the Toro-Titarev solver for generalized Riemann problems GRPs, which is the heart of the flux computation in ADER generalized Godunov schemes. In particular, we compare the Toro-Titarev solver with a local asymptotic expansion developed by LeFloch and Raviart. We show that for scalar problems the Toro-Titarev solver reproduces the truncated Taylor series expansion of LeFloch-Raviart exactly, whereas for nonlinear systems the Toro-Titarev solver introduces an error whose size depends on the height of the jump in the initial data. Thereby, our analysis answers open questions concerning the justification of simplifying steps in the Toro-Titarev solver. We illustrate our results by giving the full analysis for a nonlinear 2-by-2 system and numerical results for shallow water equations.. Introduction The classical Godunov method approximates the solution of a hyperbolic conservation law by a piecewise constant function and then solves local Riemann problems exactly to evolve that data. Clearly, piecewise constant approximation limits the order of accuracy, and so the natural question to ask is: Can we construct more accurate schemes by using piecewise smooth functions, e.g., polynomials of higher degree, rather than piecewise constant functions? To construct a generalized Godunov scheme we need to solve the initial value problem with piecewise smooth data. We call any Cauchy problem with piecewise smooth initial data that may be discontinuous at the origin generalized Riemann problem GRP. While classical Riemann problems RPs can be solved exactly for many relevant cases, generalized Riemann problems GRPs are much more complicated. In the case of nonlinear systems, analytical expressions for the solution of GRPs are usually not available. Toro and Titarev [36] have proposed a computational method, Toro-Titarev solver, for approximately solving the GRP. While the Toro-Titarev solver has been used quite successfully in a wide range of applications see [, 4, 24, 29, 32, 34], only very few contributions concerning the solver s theoretical properties have been provided so far. In fact, it is the demand for a more rigorous analysis on the properties of the Toro-Titarev solver that has motivated this paper. Starting with the pioneering work of Kolgan [5] and van Leer [37], piecewise linear reconstruction in space has become a commonly used tool for improving accuracy over the Godunov scheme. An early example for the use of higher order polynomials is the piecewise parabolic method PPM of Collela and Woodward [] and indeed, the numerical flux proposed by Harten, Engquist, Osher and Chakravarthy in their seminal work on ENO methods [2] can be interpreted in a generalized Godunov framework. However, the scheme that has, from a conceptual point of view, the most in common with what we discuss in this paper is the GRP scheme of Ben-Artzi and Falcovitz [3, 4]. Date: September4,23. Key words and phrases. Hyperbolic conservation laws, generalized Riemann problems, ADER methods.

2 2 CLAUS R. GOETZ AND ARMIN ISKE Astate-of-the-artvariantofthegeneralizedGodunovapproachistheADERscheme[28,33]. The ADER scheme relies on a high order WENO-reconstruction [2, 3, 22] from cell-averages and the solution of GRPs at the cell-interfaces. To solve GRPs numerically, Toro and Titarev [36] proposed to build a Taylor approximation of the solution whose coefficients are computed by solving a sequence of classical RPs. In this paper, we focus on hyperbolic systems in conservation form in one spatial dimension, but the ADER approach can be extended to a much broader set of problems, see e.g. [, 4, 24, 29, 32, 34]. Stability and the order of accuracy can be verified numerically, see [3] and references therein. However, it was reported by Castro and Toro [7] that the Toro-Titarev solver in [36] encounters difficulties for nonlinear systems with large jumps. We analyse the Toro-Titarev solver by comparing it to the local asymptotic expansion for the solution of the GRP that was constructed by LeFloch and Raviart [9]. We show analytically that both methods yield the same truncated Taylor series expansion for nonlinear scalar problems, whereas there is a difference for nonlinear systems. Both methods formally construct the same Taylor series expansion, but in the case of nonlinear systems the Toro-Titarev solver uses an approximation to spatial derivatives at the origin that differs from the values obtained in the LeFloch-Raviart expansion through the Rankine-Hugoniot conditions. Moreover, that difference becomes larger when there is a large jump in the initial data. The outline of this paper is as follows. In Section 2 we set up the analytic framework and review well-known results on the structure of solutions to classical RPs and GRPs. We explain generalized Godunov schemes in Section 3, before we discuss the Toro-Titarev solver in Section 4. Key steps for the construction of the LeFloch-Raviart expansion are presented in Section 5. In Section 6 we compare the resulting approximations to the solution of the GRP. We finally apply the two solution strategies in Section 7, by using a 2 2 system arising from two-component chromatography to illustrate the analytical techniques and provide numerical examples for shallow water equations. 2. Classical and Generalized Riemann Problems Consider a nonlinear m m system of hyperbolic conservation laws, 2. t u + x fu =, x R, t >, ux, t U Rm, where U R m is an open and convex subset and f : U R m is a smooth function. The classical Riemann problem RP is the Cauchy problem for 2. with initial data û 2.2 ux, = L, if x<, û R, if x>. The initial data is piecewise constant and given by the vectors û L, û R U. Assume 2. to be strictly hyperbolic, i.e., the Jacobian Au =Dfu has m distinct real eigenvalues λ u <λ 2 u < <λ m u for all u U. We choose bases of left and right eigenvectors of Au, i.e.,basesofr m, l u,...,l m u, and r u,...r m u, suchthatforallu U we have Aur i u =λ i ur i u, l i u T Au =λ i ul i u T, i =,...,m. The eigenvectors are normalized to 2.3 r i u =, l j u T r i u =, i = j,, i j, for all u U.

3 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 3 We restrict our analysis to systems for which every characteristic field is either genuinely nonlinear, λ i u T r i u in the sense of Lax [8], or linearly degenerate, i.e., λ i u T r i u. for all u U, Under these assumptions we have the following well-known result: Given two states û L, û R U with û R û L > sufficiently small, the classical RP 2., 2.2 has a unique entropy admissible weak solution ux, t =u x/t that is self-similar and consists of m +constant states û L = u, u,..., u m, u m =û R, separated by rarefaction waves, shock waves or contact discontinuities. For a comprehensive analysis of the classical RP and the properties of its solution, see e.g. [25]. Next, assume that the initial data ûl x, if x<, 2.4 ux, = û R x, if x>, with û L, û R : R U, is piecewise smooth but discontinuous at x =.TheCauchyproblem2., 2.4 is called generalized Riemann problem GRP. We let û L = û L and û R = û R. It is well-known see [2, 26] that for sufficiently small û R û L >, thereexistsaneighbourhood around the origin in which 2., 2.4 has a unique entropy admissible weak solution. Moreover, for sufficiently small T>, thestripr [,T can be decomposed into m +open domains of smoothness D i, i m, separated by smooth curves γ j t passing throuh the origin, or by rarefaction zones with boundaries γ j t, γ j t, whereγ j t, γ j t are smooth characteristic curves passing through the origin. More precisely: We have curves γ j t and rarefaction zones R j = x, t R [,T γ j t <x<γ j t. For γ j t, weletγ j t =γ j t =γ j t for all t [,T. Then,wecanwrite D = x, t R [,T x<γ t, D m = x, t R [,T γ m t <x, D i = x, t R [,T γ i t <x<γ i+ t, i m. The solution u is smooth inside each domain D i and inside each rarefaction zone R j. Moreover, u has a shock or contact discontinuity across each curve x = γ j t and is continuous across the characteristic curves x = γ j t, x = γ j t. The solution of the GRP and the solution of the corresponding classical RP with the initial states û L =û L and û R =û R have a similar wave structure, at least for small time t>. That is, if the j-wave in the solution of the classical RP is a shock wave, a contact discontinuity or a rarefaction wave, then the corresponding j-wave in the GRP is of the same respective type. In this paper, we focus on the case where the solution contains only shock waves and contact discontinuities. In this case, the connection between the wave structures can be described more precisely: Denote the constant states in the solution of the RP with initial data û L, û R by u i,for i =,...,m, and the wave speeds by σ j, j =,...,m.then,thecurvesγ j satisfy γ j = and lim t γ j t =σ j, for j =,...,m,

4 4 CLAUS R. GOETZ AND ARMIN ISKE and the solution u of the GRP satisfies within each domain of smoothness D i the convergence lim t x,t D i ux, t =u i for i =,...,m. We remark that the solution of GRPs is a popular subject of ongoing research. Quite recently, special emphasis has been placed on the global existence and the structural stability of solutions. We refer to [8, 9, 6, 7] and references therein for an up-to-date account on the solution of GRPs. For the following analysis in this paper, we can rely on available results on the local existence and on the local structural stability. To solve the Cauchy problem 3. Generalized Godunov Schemes 3. t u + fu =, x R, t >, ux, = ûx, x numerically, we extend the classical Godunov finite volume scheme, with assuming a uniform grid x i+/2 =i +/2 x, t n = n t i Z, n N, for simplicity, where x >, t >. AgeneralizedGodunovschemeconsistsofthefollowingsteps:Startingwithcellaverages ū i = x xi+/2 x i /2 ûx dx, i Z, for any time step n =,,...,giventhevaluesū n i i Z, performthefollowingsteps: Find a piecewise smooth, conservative reconstruction. That is, compute a function û n : R U such that for all i Z we have: û n i =û n [xi /2,x i+/2 ] is smooth, and x xi+/2 x i /2 û n x dx =ū n i. For ADER schemes this is usually done by a WENO-reconstruction [2, 3, 22], such that each û n i is a polynomial of degree r, wherer> is a given integer. 2 Use the function û n as initial data, i.e., pose the Cauchy problem t u + fu =, x x R, t >, ux, = û n i x, x [x i /2,x i+/2 ], i Z. Solve this problem exactly and evolve the data for one time step. Denote by E the exact entropy evolution operator associated with 3. and by t the limit t t, t < t. We compute E t û n. 3 Update the cell averages by ū n+ i = x ū n+ i xi+/2 x i /2 =ū n i t x E t û n x dx. In a finite volume frame work we can perform evolution and averaging in one step by the update f n i+/2 f i /2 n.

5 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 5 Here, f n i+/2 is the exact averaged flux through the cell interface x i+/2 during one time step: 3.2 f n i+/2 = t t f Eτû n x i+/2 dτ, and τ = t t n is the local time. However, computing the exact integral 3.2 may be exceedingly complicated, if not impossible. Therefore, we are looking for an approximation to 3.2 based on an approximate solution of the GRP. 4. The Toro-Titarev Solver for the Generalized Riemann Problem We describe the method of solution for the GRP proposed by Toro and Titarev [36], based on a state expansion approach. That is, we use a Taylor series expansion of order r in time of the solution around τ =right at the cell-interface x i+/2, r k u 4. ux i+/2,τ ux i+/2, + + t k x i+/2, + τ k k!. Note that while the solution u may be discontinuous, the function ux i+/2, for a fixed point x i+/2 in space as a function of time is smooth, provided that τ> is small enough. If we can solve the GRP and give a meaning to the time derivatives in 4., the easiest way to define a numerical flux is to approximate the time-integral in 3.2 by a Gaussian quadrature, k= N fi+/2 n = ω γ fux i+/2,τ γ, γ= where ω γ,τ γ are suitable weights and nodes and N is the number of nodes, which is chosen according to the desired accuracy. The values ux i+/2,τ γ are computed by 4.. For a discussion of more refined numerical fluxes in the ADER context, see [3, 35]. The key idea in the method of Toro and Titarev [36] is to reduce the solution of the GRP to a sequence of classical RPs. To find the sought value ux i+/2, + we take the extrapolated values û L =û Lx i /2 and û R =û Rx i+/2 + to solve the classical RP t u + fu =, x R, t >, x û ux, = L if x<x i+/2, û R if x>x i+/2. As described in Section 2, this problem has a self-similar entropy solution that we denote by u x x i+/2 /τ. Theleadingtermoftheexpansion4.isthengivenbyux i+/2, + = u. We call this the Godunov state of 4.2, 4.3. For nonlinear systems of conservation laws, computing the solution of the RP may be difficult, so we might need to employ a numerical approximative Riemann solver see [3] to compute the leading term. However, in this paper we are mainly interested in the analytical aspects of the scheme, so we assume the Godunov states of 4.2, 4.3 can be computed exactly. For higher order terms we formally perform a Cauchy-Kowalewskaya procedure to express all time derivatives as functions of lower order spatial derivatives. That is, we use a recursive mapping k u t k =Φk u, u u x,..., k x k, k =,...,r, with Φ u =u.

6 6 CLAUS R. GOETZ AND ARMIN ISKE Since the classical Cauchy-Kowalewskaya theorem assumes analytical initial data, it does not apply to the case of piecewise smooth data. But to illustrate the basic ideas, assume u was smooth. In that case, the equations in the following can be obtained by simple manipulations of derivatives. We can compute the expansion 4. via Φ k,providedthatwecanfindthespatialderivatives u k = To do so, we use the one-sided derivatives û k L = lim x x i+/2 t + k u x, t. xk k û L lim x x i+/2, x k x, k û R ûk R = lim x x i+/2,+ x k x. These values can be used as initial conditions for classical RPs. As regards the evolution equations for the spatial derivatives, we can rely on the following result: Let u be a smooth solution of 3., and for k, denotethek-th spatial derivative of u by u k. Then, all u k satisfy a semi-linear hyperbolic equation of the form 4.4 t uk + Au x uk = H k u, u,...,u k, where Au =Dfu is the Jacobian of the flux and the function H k depends only on u, u,...,u k. We remark that for smooth u, 4.4canbeobtainedbystraightforwardcomputation. Forthe sake of brevity, however, we omit the details here. Moreover, it is easy to see that in the linear case, where Au A is a constant matrix, the function H k vanishes identically. Although we can derive 4.4 for smooth u, we do not have a rigorous analysis yet to see whether these equations can also be used for discontinuous solutions. We simplify the problem 4.4 in two ways: Firstly, we neglect the source terms and secondly, we linearise the equations: t uk + A LR x uk =, for k =,...,r, x R, t > u k ûk x, = L, if x<x i+/2, û k R, if x>x i+/2. Here A LR = Aux i+/2, +. Then, the self-similar solutions u k of these linear problems can be computed easily. Note that for all k we have the same A LR. Finally, we approximate the solution u along the t-axis by the truncated Taylor expansion r ux i+/2,τ u + Φ k u,u,...,u k τ k k!. k= 5. Asymptotic Expansion of the Solution to the Generalized Riemann Problem 5.. Basics. We describe how to construct a local power series expansion for the solution of the GRP. The main source for our work is the expansion constructed by LeFloch and Raviart [9]. See [5] for an application of this techniques to the Euler equations of gas dynamics. Related approaches are discussed in [] and [2]. Another somewhat different approach to asymptotic expansion for

7 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 7 the Euler equations is given in [23]. We discuss the local properties of the solution of the GRP t u + fu =, x x ûl x, if x<, ux, = û R x, if x>. Our goal is to construct an asymptotic expansion of the form 5. ux, t = k t k q k ξ R,t>, with ξ = x/t. This is possible in any domain of smoothness D i,bysimplytakingataylorseries expansion of the solution u in that domain. So every q k is a polynomial of degree k. Wereturnto this point in Section 6. It can be shown that such a series expansion can also be constructed inside a rarefaction zone R. However, for our numerical scheme we only need detailed information about the solution along the line segment x = [, t] in local coordinates. We assume that the solution does not contain a transonic rarefaction wave. In that case, the solution along that line segment is given by some function u i inside a domain of smoothness D i, i m, and we do not need the explicit construction of the expansion inside a rarefaction zone. Roughly speaking, the construction can be summarized as follows: Take a Taylor expansion wherever the solution u is smooth and then investigate the jump conditions at the boundaries of the domains of smoothness. As we are looking for an expansion in terms of self-similar functions, it is useful to change the variables and work with ξ = x/t. Wesetũξ,t =uξt,t and check that 5.2 x = t ξ, u t = ũ t ξ ũ t ξ. For illustration we will give more details for scalar problems and 2 2 systems of conservation laws, v u = U R 2, f : U R 2 f v, w, fu =, w f 2 v, w in which case we denote the expansion by vx, t = k t k v k ξ, wx, t = k t k w k ξ, q k ξ = v k ξ w k ξ We give the full detail for the construction of the expansion up to quadratic terms. First order terms were presented in [5] for the Euler equations, but there seems to be no explicit computation of higher order terms available in the literature Step I: Derivation of the Differential Equations. At first we derive a series of ordinary differential equations satisfied by the functions u k in 5.. We change the variables according to 5.2 and the conservation law becomes t tũ ξ ξũ + fũξ,t =. ξ. Observe that the expansion ũξ,t = k t k q k ξ

8 8 CLAUS R. GOETZ AND ARMIN ISKE gives 5.3 t ũ t ξ ũ dq = ξ ξ dξ + t k kq k ξ dqk. dξ k Inserting this expansion into the flux function f yields 5.4 fũξ,t = fq + k t k Aq u k + f k Q k. Here, the function f k depends only on the previous terms Q k =q,...,q k. We get f k by a Taylor expansion of the flux in powers of t, suchthatf k accounts for all terms in that expansion belonging to t k that do not depend on q k,i.e.,allbutaq q k.thiswillbeourstandardtrickin the following analysis, so we discuss the method in somewhat more detail. At first, consider the expansion of the flux around t =for the scalar case: f t k q k ξ = fq +t t f t k q k ξ + t2 2 k k 2 t 2 f t k q k ξ +... k t= t= = fq +tf t k q k ξ kt k q k ξ k k So we have t= + t2 2 f t k q k ξ k + t2 2 f t k q k ξ k t= t= t= 2 kt k q k ξ k t= kk t k 2 q k ξ k t= +... fũξ,t = fq +tf q q + t 2 f q q f q q 2 + Ot 3 for t, and we see that f q =and f 2 q,q = 2 f q q 2.Fora2 2 system, we have 2 f l u 2 f l q t 2 = v 2 f l q +2 t= v 2 v f l q v v w v w + 2 f l q w 2 f l q +2 w 2 w for l =, 2, and thus where fux, t = fq +taq q + t 2 Aq q 2 + f 2 q,q + Ot 3 for t, 2 f l v 2 q v 2 2 f l +2 v w q v w + 2 f l w 2 q w 2 f 2 q,q =. 2 l=,2 By that Taylor expansion of f in powers of t it is easy to see that f k is a polynomial of degree at most k, ifeveryq l is a polynomial in ξ ofdegreeatmostl, forall l k. Next, we combine 5.3 and 5.4 to find ξ dq dξ + d dξ fq + t k kq k ξ dqk dξ + d Aq q k + f k =, dξ k w2,

9 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 9 which yields for k =: 5.5 ξ dq dξ + d dξ fq =, and for k : kq k ξ dqk dξ + d Aq q k + f k =. dξ Letting h k ξ = d dξ f k q,...,q k, this becomes 5.6 kq k ξ dqk dξ + d Aq q k = h k. dξ We remark that in 5.6 the coefficient Aq depends on q but not on q k. Thus, 5.6 is a semilinear equation. Moreover, recall that f k is a polynomial in ξ of degree at most k, soh k is a polynomial of degree at most k Step II: Jump Conditions. The above construction is valid wherever u is smooth. So next we need to investigate the jump conditions satisfied by q k at the boundaries of the domains of smoothness of u. Takeacurvex = γt that separates two domains of smoothness of u. Sincethese curves are all smooth, we can use a Taylor expansion to write γt =σ t + σ t σ k t k +... In fact, the solution u is smooth, not only in D i,butalsointheclosured i,see[2]. Sowecan use, again, a Taylor expansion in powers of t around the origin to obtain from 5. that uγt,t= γt t k q k = t k q k t l σ l t k k l = q σ +t q σ +σ dq dξ σ + t q 2 2 σ +σ 2 dq dξ σ +z 2 Σ,Q t k q k σ +σ k dq dξ σ +z k Σ k,q k +..., where the functions z k depend only on Σ k =σ,...,σ k and Q k =q,...,q k.similar to the f k in 5.4, we plug all terms belonging to t k that do not depend on σ k or q k in a Taylor expansion into this z k.inparticular,z =and 5.8 z 2 Σ,Q = 2 σ 2 d2 q dξ 2 σ +σ dq dξ σ. We denote the jump of a function u at a point ξ by [[u]]ξ =uξ + uξ. Therefore, if u is continuous across the curve x = γt, wesimplyget 5.9 [[q ]]σ = from 5.7 for k =.Moreover,fork we get ]] 5. [[q k + σ k dq dξ + zk Σ k,q k σ =. We see that q is continuous at the point σ,whereasq k is in general discontinuous at σ for k.

10 CLAUS R. GOETZ AND ARMIN ISKE Now let u have a jump across the curve x = γt. Then, by the Rankine-Hugoniot conditions, γt[[u]]x =[[fu]]x, x = γt. To derive the correct jump conditions satisfied by the functions q k,wewilltaketheexpansionsfor fu and γu along x = γt, respectively.westartwiththefluxalongthatcurveofdiscontinuity: By a Taylor expansion around t =we get fuγt,t = f t k q k σ +σ k d dq ξ σ +z k Σ k,q k k = f q σ + ta q σ q σ +σ dq dξ σ + t 2 A q σ q 2 σ +σ 2 dq dξ σ + a 2 Σ,Q t k A q σ q k σ +σ k dq dξ σ + a k Σ k,q k +..., where for the 2 2 system we can express a 2 explicitly as 5. a 2 Σ,Q = fl 2 v q σ σ 2 d2 v dξ 2 σ +σ dv dξ σ + f l w q σ σ 2 d2 w dξ 2 σ +σ dw dξ σ f l v 2 q σ v σ +σ dv dξ σ + 2 fl w 2 q σ w σ +σ dw dξ σ +2 2 f l v w q σ v σ +σ dv dξ σ w σ +σ dw dξ σ Further, we have with 5.2. l=,2 γtuγt,t = σ q σ +t σ q σ +σ dq dξ σ +2σ q σ + t 2 σ q 2 σ +σ 2 dq dξ σ +3σ 2 q σ +b 2 Σ,Q t k σ q k σ +σ k dq dξ σ +k +σ k q σ +b k Σ k,q k +... b 2 Σ,Q =2σ q σ +σ dq dξ σ + σ z 2 Σ,Q. In summary, at ξ = σ the jump conditions are 5.3 σ [[q ]] = [[fq ]] at σ, 2

11 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS for k =,andfork we get 5.4 [[ Aq σ q k]] + σ k [[Aq σ dq dξ with a function c k Σ k,q k =a k Σ k,q k b k Σ k,q k. Finally, we remark that for ξ large enough, say ξ ξ, 5.5 q û ξ = R, ξ > ξ, û L, ξ < ξ. We now can summarize the above construction: ]] σ k [[ k +q ]] + [[ c k]] = at σ, Lemma 5.. The function q satisfies the relations 5.5,5.9,5.3,5.5, whichcharacterizethe piecewise continuous self-similar entropy solution q x, t =q ξ of the Riemann problem t q x, t+ x f q x, t =, x R, t > û L, if x<, q x, = û R, if x>. Therefore, we can conclude that the Toro-Titarev solver sets up "the right problem", when it comes to computing the leading term of the expansion Step III: Higher Order Terms. Assume that the solution of 5.6,5.7 contains no transonic rarefaction wave. Then line segment x = [, t] is contained in a domain of smoothness, say in D i.sincewedonotexplicitlyneedtheexpansioninsidetherarefactionzones, we only consider the simplified case that the solution q of contains only shock waves or contact discontinuities. The full problem requires similar techniques, although some of the details are more involved again, see [9] for the full construction. When we only have shocks and contact discontinuities, the solution q of 5.6,5.7 has the form q =û L, for ξ,σ, q ξ = qi, qm =û R, for ξ σ i,σ i+, i m, m,. Now consider the domains in which q takes the constant value q i, D i = x, t σ i <ξ<σ i+, i =,...,m. As a convention, we let σ =, σ m+ =+. Then equation 5.6 in D i 5.8 kq k + Aq i ξ d dξ qk = h k. becomes Recall that h k is a polynomial of degree at most k. Itisthenstraightforwardtoshowthatthe general solution of 5.8 is given by 5.9 q k ξ = ξ Aq i k q k i + p k i ξ, where q k i R m is an arbitrary vector and p k i : R R m is a polynomial of degree at most k with coefficients that depend only on q,...,q k.

12 2 CLAUS R. GOETZ AND ARMIN ISKE More precisely, ξ Aqi k qi k is a solution of the homogeneous part of 5.8 and pk i is a particular solution of 5.8. Since f =,wehaveh =, and therefore, p =.Forthe2 2 system, this means that v ξ ξq w = i, f v q i q i, f w q i q i,2 ξ ξqi,2 f2 v q i q i,, f2 w q i q,2 where we denote qi =q i,, q i,2 T. Then we get h 2 ξ = v 2 f l ξ v 2 q q, + 2 f l v w q q,2 In general, writing w ξ l=,2 k k h k i ξ = βi l ξ l and p k i ξ = θi l ξ l, l= the coefficients θ l i of the polynomial pk i 2 f l v w q q, + 2 f l w 2 q q,2. l=,2 l= can be obtained as follows cf. [9, Lemma 2]. θ k i = β k i l +Aq i θ l+ i +k lθ l i = β l i for l k 2. Moreover, since the function q is piecewise constant, this allows us to simplify some of the above expressions. Let u have a jump across the curve x = γ i t, thenwehave q σ i =q i, q σ i + = q i, dq dξ σ i = dq dξ σ i + =, and thus, using the representation 5.9, we get from 5.8 for the 2 2 system z 2 Σ,Q =σi dq dξ σ =σi qi. This gives b 2 Σ,Q =σi 2q σi +σ i q i. Moreover 5., reduces to a 2 Σ,Q = σ fl 2 v q σ dv dξ σ + f l w q σ dw dξ σ + 2 f l v 2 q σ v σ 2 2 f l + w 2 q σ w σ f l v w q σ v σ w σ. l=,2 6. Connecting the Toro-Titarev Solver with the LeFloch-Raviart Expansion Let us take a look at the Taylor expansion that we used to define the functions q k :Weconsider the domains D i = ξ γ i t <ξ< γ it. t t Since we have γ i =, γ i = σ i, the domains remain close to the domains D i in which u is constant, for small t >. Inside each domain of smoothness D i we may take a Taylor

13 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 3 expansion around some x,t D i and define the Taylor expansion at the origin by the limit x,t, + D i.inthatsensethetaylorexpansionaroundtheorigingives ux, t = k k= l= l k l u, + x l t k l l!k l! xl t k l = u, + + k= t k k l= l k l u, + x l x l t k l l!k l! t Thus, the vector qi k in 5.9, which gives the leading coefficient of this polynomial, defines the value k u, +. x k To determine the vectors qi k,wefirstdescribeqk and qm, k as in [9, Lemma 6]. Using the notation from before, we can write for the initial data r û L x =û L + k= In D,thesolutionisgivenbythefunctions û k r L k! xk, û R x =û R + q k ξ =ξ Aq k q k + p k ξ. Since p k is a polynomial of degree at most k, wefind t k q k x = x k q k. t Hence, it follows that Therefore, we have ux, = lim t x<γ t lim t x<γ t r ux, t =q + k= k= q k x k. û k R k! xk. q k = ûk L, for k =,...,r. k! Analogously, we get qm k =û k R /k!, k =,...,r. Now consider the scalar case. For a strictly convex flux, f >, weonlyhavetwodomainsof smoothness. In that case, all coefficients qi k,i =,, and k =,...,r are uniquely determined by the initial data and its derivatives. Assuming that there is no transonic wave, solving linear RPs merely means picking the left or the right side, depending on the sign of the coefficient in the evolution equation. Thus, to build the expansion, we first have to solve one nonlinear RP to determine which domain of smoothness contains the line segment x = [, t]. Then,usethe data from that side, which is equivalent to solving linear RPs. So the solver of Toro and Titarev reproduces the first r terms of the expansion of LeFloch and Raviart exactly. In summary, we can state one main result of this paper. Theorem 6.. Consider the generalized Riemann problem for a scalar, non-linear hyperbolic conservation law in one spatial dimension with strictly convex flux. Let the initial data consist of piecewise polynomials of degree r. Assume the solution does not contain a transonic wave. Then, the Toro-Titarev solver and the LeFloch-Raviart expansion yield the same truncated Taylor expansion in time at x =, r Φ k u,...,u k τ k r k! = q k τ k = Eτû + O t r k= for <τ< t as t +. k=

14 4 CLAUS R. GOETZ AND ARMIN ISKE Indeed, both methods are computing the same truncated Taylor expansion. The difference is, however, that the Toro-Titarev solver uses an approximation to the spatial derivatives at the origin. For illustration, we check that for a scalar problem both methods formally construct the expansion up to quadratic terms. Assume that σi < <σ i+ and consider the Taylor approximation inside D i, u, + x u, + ux, t u, + + t + + t 2 2 u, + 2 x 2 x x t t t 2 2 u, + + x t x t u, + t 2, where function evaluations and derivatives at, + are regarded as limits D i x, t, +. The Cauchy-Kowalewskaya procedure now gives: u t = f u u x, 2 2 u u x t = f u f u 2 u x x 2, 2 u t 2 =2f uf u u x 2 +f u 2 2 u x 2. Inserting this into the above Taylor approximation yields x u ux, t u, + + t t f u, +, + x x 2 + t 2 t f 2 u u, + 2 x 2, + f u, + 2 u 6. +f u, + f u, +, +. x u, + x Now let us compute us the terms up to q 2 in the LeFloch-Raviart expansion. We have For q 2,wefirstcompute h 2 ξ = d dξ f 2 qi ξ,qi ξ = d 2 dξ q i ξ = ξ f q i q i. 2 x f q i q i ξ 2 = f q i ξ f q i q i 2 = β i ξ + β i, where βi = f qi qi 2 and β i = f qi f qi 2.Withlettingp qi 2 i ξ =θi ξ + θ i,weget θi = βi, θi = β 2 i f qi θi = f qi f qi qi 2. Thus, we have q 2 i ξ = ξ f q i 2 q 2 i f q i q i 2 ξ + f q i f q i q i 2. Noting that qi k = k u k!, +, weseethatq + tq + t 2 q 2 agrees with 6.. x k Naturally, the question arises whether this result can be extended to the case of systems. What we compare are the coefficient qi k and the Godunov state for the k-th spatial derivative in the Toro-Titarev solver. To do so, at first we write each coefficient qi k in the form m qi k = αijr k j qi. j= t

15 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 5 Note that the coefficients α,j k,αk m,j,j=,...,m are known from the initial data. To characterize the coefficients αij k, i =,...,m, weneedthefollowingtworesults. Lemma 6.2. cf. [9, Lemma 4] Assumethatthei-th wave is a shock wave or a contact discontinuity. Then for all k, thereexistsavectors k i Rm, depending only on q,...,q k,σi,...,σk i such that Aq i σi k+ q k i = Aqi σi k+ q k 6.2 i + k k +σi k qi qi +s k i. More precisely, 6.2 holds with s k i = k+ Aq i σ i p k i σ i Aq i σ i p k i σ i + c k i σ i + c k i σ i. Lemma 6.3. Corollary from Theorem in [9] Assumethei-th wave is a shock wave or a contact discontinuity. Then, we have for i j: m λp qi σi k+ lj qi r p qi αip k λ j qi σi k+ α k i,j = 6.3 p= l j qi q i q i l i qi q i q i m p= λp qi σi k+ li qi r p qi αip k λ i q i σ i k+ α k i,i l i q i s k i + l j q i s k i. Both statements are derived from the jump relation 5.4. LeFloch and Raviart show that this leads to a uniquely solvable system of linear equations for the coefficients αij k cf. [9, Theorem ]. The question whether this gives coefficients qi k that agree with the intermediate states in the linear RPs of the Toro-Titarev solver is answered by the following theorem, which is the other main result of this paper. Theorem 6.4. Consider the generalized Riemann problem for a strictly hyperbolic m m system of conservation laws, such that every characteristic field is either genuinely nonlinear or linearly degenerate. Let the initial data consist of polynomials û L, û R of degree r with sufficiently small û L û R >. Assumethatthesolutioncontainsonlyshockwavesandcontactdiscontinuities. Then, for k the coefficients qi k in the LeFloch-Raviart expansion and the states u k i in the linear Riemann problems of the Toro-Titarev solver satisfy the relation q k i = k! uk i for i =and for i = m. This does, in general, not hold for i m. Proof. The statement that q k = k! uk, q k m = k! uk m, k =,...,r, was already shown in our discussion of the scalar case. Now take 5.4 for k =,inwhichcasewe have c =and s i =,so5.4becomes [[ 6.4 Aq σi q ]] [[ ]] Aq σi σi d dξ q 2q = at σi.

16 6 CLAUS R. GOETZ AND ARMIN ISKE We note that q σ i = q i, q σ i +=q i, and thus the jump condition 6.4 becomes 6.5 Aq i σi q i Aqi σi q i 2σi d dξ q σ i ± =, q i qi =. Now consider the Toro-Titarev solver. We denote the solution of the linearised RP 4.5 by u k and let u k i,i=,...,n be the constant states in that solution. If u i is the Godunov state for u,then solving the RPs linearised around u i is equivalent to imposing the jump conditions 6.6 Au i λ i u i u i u i =, i =,...,m. Clearly, 6.5 and 6.6 do not have the same solution. We remark, however, that when all states qi are close, the solutions of 6.5 and 6.6 are close. This depends only on the leading term q, but not on higher order terms. Thus, when the jump in the initial data û L û R is small we expect 6.6 to give a good approximation to Applications and Numerical Examples 7.. Two-Component Chromatography. Consider the system v 7. + v + v + w t w x w + v + w, v,w > and denote u =v, w T, u U=,, R 2. This example is inspired by the analysis of two-component chromatography, as described by Temple [27]. A discussion on the RP for 7. can be found in [6]. The Jacobian of the flux is given as with eigenvalues Av, w = λ u = + v + w 2 +w v w +v + v + w 2, λ 2u = +v + w. The corresponding normalized right eigenvectors are r u = v2 + w 2 v w, r 2 u = 2 and the left eigenvectors, normalized to l j v, w r i v, w =δ ij,are v2 + w l v, w = 2 2, l v + w 2 v, w = v + w, w v The first characteristic field is genuinely nonlinear, while the second is linearly degenerate. For this system, shock and rarefaction curves coincide in the sense that each point in the i- Hugoniot set i =, 2 of a given point u lies on the integral curve of r i through u. Due to the simple nature of the eigenvectors, the integral curves here are straight lines in the space of conserved variables..

17 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS 7 Now consider the RP with initial data u L =v L,w L T, u R =v R,w R T. Then the Riemann solution contains the states u = u L, u =v,w T,andu 2 = u R,sothat v v ε v 7.2 = +, w w v 2 + w 2 w v2 v = + ε 2 7.3, 2 w 2 w for some ε,ε 2.Sincethesecondfieldislinearlydegenerate,wehaveλ 2 u =λ 2 u 2 and therefore 7.4 v + w = v 2 + w 2. Further, it follows from 7.2 that 7.5 v w = v w 2. Combing the conditions 7.4 and 7.5, we can explicitly compute v = v v 2 + w 2, w = w v 2 + w 2, v + w v + w and the wave strength ε is ε = v 2 + w 2 v 2 v + w + w2. Recall that the type of wave associated with the first characteristic family depends on the sign of ε :Wegeta-shock for ε and a -rarefaction for ε > the second wave is always a contact discontinuity, independent of the sign of ε 2. Thus, if v 2 + w 2 v + w,thesolutioncontainsa-shock and a 2-contact discontinuity. The shock speed σ can be computed from the Rankine-Hugoniot conditions: σ = λ θu + θu dθ = For the contact discontinuity we have σ 2 = λ 2 u =λ 2 u 2. Now consider the GRP with piecewise linear initial data, û a x = ˆva x ŵ a x = ˆv a ŵ a + v + w + v + w. + x ˆv a ŵ a for a = L, R. Denote û k a =va, k wa k T for a = L, R, and k =, and let u be the solution of the classical RP for 7. with initial data û L, û R.Denotetheintermediatestateinthatsolution by u.then,thesimplifiedprobleminthetoro-titarevsolverforthespatialderivativesis t u + A LR x u =, u x, =, û L, if x<, û R, if x>. Here, A LR = Au. We express the vectors û L, û R in terms of the basis r u,r 2 u, i.e., û L = β r u +β 2 r 2 u, û R = θ r u +θ 2 r 2 u. Then, the intermediate state u can be computed as v u = θ r u +β 2 r 2 u = 2 + w2 v + v v +w + w w w v +w. v 2 + w2 v w w + w v +w

18 8 CLAUS R. GOETZ AND ARMIN ISKE Next, we compute the first two terms of the expansion ux, t =q ξ+tq ξ+.... As above, we find q by solving the classical RP with initial states û L, û R and we denote the constant states in that solution by û L = q, q, q2 = û R. The function q ξ is given in each domain Di by q ξ = ξ Aqi qi, i 2, where we express the unknown vectors q i as q i = α i, r q i +α i,2 r 2q i. The coefficients α,j,α 2,j,forj =, 2, aredeterminedbytheinitialdataû L and û R respectively, while the remaining coefficients α,, α 2,2 are found by solving a linear 2 2 system of algebraic equations, as given by Lemma 6.3. We arrive at the system λ q σ 2l2 q r qα, + λ 2 q σ 2 l2 q r 2 qα,2 λ 2 q σ 2 α,2 7.6 = l 2q q q λ l q q q q σ 2 l q r qα,+ λ2 q σ 2 l q r 2 qα,2 + λ q σ 2 α, λ q2 σ2 2l q r q2α 2, + λ 2 q2 σ2 2 l q r 2 q2α 2,2 λ q σ2 2 α, 7.7 = l q q2 q λ l 2 q q 2 q q2 σ2 2 l2 q r q2α 2,+ λ2 q2 σ2 2 l2 q r 2 q2α 2,2 + λ 2 q σ2 2 α,2 Now recall that q,q,q 2 are the constant states in the solution of a classical RP. By for the intermediate state q,wefindq q = ε r q and q 2 q = ε 2 r 2 q. Therefore,weget from condition 2.3 l 2 q q q =, l q q q =ε, l q q 2 q =, l 2 q q 2 q =ε 2. Then we can solve 7.6, 7.7 to find α, = v2 + w2 v 2 +w 2 v2 +, α w,2 = v w v + v w + w w v + w w. Thus, the coefficient q is given by q = α,r q +α,2r 2 q = v 2 + w2 v v +w v 2 + w2 v v +w v 2 + +w 2 2 v v +w + w v 2 v 2 +w 2 v +w + w w v +w w w w v +w The only difference between u and q is the factor v 2 + w 2/v + w 2 whose size only depends on the distance of the states û L and û R..

19 APPROXIMATE SOLUTIONS OF GENERALIZED RIEMANN PROBLEMS Shallow Water Equations. We consider the GRP for the shallow water equations, h + hu t hu x hu 2 + =, 2 gh2 where g is a constant, with initial data ĥl = ĥr =and û L x =a L x 2 + b L x + c L, û R x =a R x 2 + b R x + c R. When c R < <c L,thisdataleadstoasolutionwithtwoshockwaves. We compare the resulting approximations up to terms of second order obtained by the LeFloch- Raviart expansion and the Toro-Titarev solver, respectively. Reference solutions are obtained by a random choice method RCM on a very fine grid using an exact Riemann solver and van der Corput pseudo random numbers see [3, Chapter 7] for details. We perform two series of tests: v along = v ref RCM TT t = to.5 w along = w ref RCM.66 TT t = to.5 a c L =2, û L û R = v along = v ref RCM TT t = to.5 w along = w ref RCM TT2 b c L =4, û L û R = t = to v along = v ref RCM TT t = to.5 w along = w ref RCM TT t = to.5 c c L =6, û L û R =7 Figure. Jumps in the initial states i Large jumps in the initial data, fixed derivatives. We fix a L =.2,a R =. and b L =.4, b R =.2. We solve the GRP for c R = and c L =, 2, 4, respectively. Resultsare shown in Figure. Denoting v = h and w = hu, theplotsshowthereferencesolutionthickblack

20 2 CLAUS R. GOETZ AND ARMIN ISKE line, the LeFloch-Raviart approximation circles and the Toro-Titarev approximation crosses along the line ξ =for times t.5. The difference in the two approximations increases with the size of the jump. We observe that the LeFloch-Raviart approximation is almost identical to the reference solution. ii Large jumps in the derivatives, fixed jump in states. We fix a L =.2, a R =. and c L =.2, c R =.2, sowehaveafixedjump û L û R =.4. We let b R = and test for different values of b L,seeFigure2. Foralltestcasesbothapproximationsareveryclosetothe reference solution v along = v ref RCM TT t = to.5 w along =..5.2 w ref RCM TT t = to.5 a b L =2, û L û R =3.8 v along = v ref RCM TT t = to.5 w along =..2.3 w ref RCM TT t = to.5 b b L =4, û L û R =5. v along = v ref RCM.98 TT t = to.5 w along = w ref RCM TT t = to.5 c b L =6, û L û R =7 Figure 2. Jumps in the derivatives We conclude that the Toro-Titarev solver gives a very good approximation when the jump in the initial data is small independent of the jumps in derivatives, but it introduces a larger error when the jump in the states is large. Note that this behaviour is consistent with our analysis, see the remark after Theorem 6.4.

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22 22 CLAUS R. GOETZ AND ARMIN ISKE [28] V. A. Titarev and E. F. Toro, ADER: Arbitrary high order Godunov approach, J.Sci.Comput.7 22, [29], ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys , no. 2, [3], Analysis of ADER and ADER-WAF schemes, IMAJ. Numer. Anal , [3] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 3rded., Springer, Berlin, 29. [32] E. F. Toro and A. Hidalgo, ADER finite volume schemes for nonlinear reaction-diffusion equations, Appl. Numer. Math , 73. [33] E. F. Toro, R. C. Millington, and L. A. M. Nejad, Towards very high order Godunov schemes, GodunovMethods. Theory and Applications E. F. Toro, ed., Kluwer/Plenum Academic Publishers, 2, pp [34] E. F. Toro and V. A. Titarev, ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions, J. Comput. Phys , no., [35], TVD fluxes for the high-order ADER schemes, J. Sci. Comput , no. 3, [36], Derivative Riemann solvers for systems of conservation laws and ADER methods, J.Comput.Phys , [37] B. van Leer, Towards the ultimate conservative difference scheme. IV: a second order sequel to Godunov s method, J. Comput. Phys , 36. Department of Mathematics, University of Hamburg, Bundesstr. 55, D-246 Hamburg, Germany address: claus.goetz,armin.iske@uni-hamburg.de