Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed

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1 Well-Balaced High-Order Cetred Schemes for No-Coservative Hyperbolic Systems. Applicatios to Shallow Water Equatios with Fixed ad Mobile Bed Alberto Caestrelli a Auziato Siviglia b Michael Dumbser b Eleuterio F. Toro b a Departmet IMAGE, Uiversity of Padova Via Loreda, I Padova, Italy b Departmet of Civil ad Evirometal Egieerig, Uiversity of Treto, Via Mesiao 77, I-381 Treto, Italy Abstract This paper cocers the developmet of high-order accurate cetred schemes for the umerical solutio of oe-dimesioal hyperbolic systems cotaiig o-coservative products ad source terms. Combiig the PRICE-T method developed i [9] with the theoretical isights gaied by the recetly developed path-coservative schemes [5,6], we propose the ew PRICE-C scheme that automatically reduces to a modified coservative FORCE scheme if the uderlyig PDE system is a coservatio law. The resultig first-order accurate cetred method is the exteded to high order of accuracy i space ad time via the ADER approach together with a WENO recostructio techique. The well-balaced properties of the PRICE-C method are ivestigated for the shallow water equatios. Fially, we apply the ew scheme to the shallow water equatios with fix bottom topography ad with variable bottom solvig a additioal sedimet trasport equatio. Key words: No-coservative hyperbolic systems, cetred schemes, high-order WENO Fiite Volume methods, shallow water equatios, sedimet trasport, ADER, FORCE addresses: caestrelli@idra.uipd.it (Alberto Caestrelli), uzio.siviglia@ig.uit.it (Auziato Siviglia), michael.dumbser@ig.uit.it (Michael Dumbser), toroe@ig.uit.it (Eleuterio F. Toro). Preprit submitted to Elsevier 5 February 9

2 1 Cetred Schemes for No-Coservative Hyperbolic Systems We cosider systems of hyperbolic partial differetial equatios of the form Q t + A(Q) Q x =, (x, t) R R+, Q Ω R N, (1) i which Q = [q 1,... q N ] T is the vector of ukows ad A = A(Q) is the coefficiet matrix. We suppose that the ukow fuctio Q = Q(x, t) takes its values iside a ope covex set Ω icluded i R N ad that Q A(Q) is a smooth locally bouded map. We assume system (1) to be hyperbolic with real eigevalues λ 1, λ,... λ N ad with a full set of correspodig liearly idepedet right eigevectors r 1,r,...,r N. The umerical methods developed i this paper are of the cetred type ad will oly require a estimate for the maximum sigal speed i absolute value i order to satisfy the Courat stability coditio for the time step. The vector of ukows Q i (1) will be always chose to be the vector of physically coserved variables. So i the case that A(Q) is the Jacobia matrix A(Q) = F/ Q of some flux fuctio F = F(Q), the o-coservative system (1) ca be expressed i coservatio form Q t + F(Q) =. () x I [9] a series of primitive cetred (PRICE) umerical schemes for solvig systems of hyperbolic partial differetial equatios writte i the ocoservative form (1) has bee developed. The most promisig of these schemes, amely the PRICE-T scheme, will be the basis of the high-order cetred schemes proposed i this article. 1.1 The FORCE Scheme for Coservative Systems Sice the PRICE-T scheme [9] is the o-coservative aalogue of the coservative FORCE scheme [31,3], that is i tur a determiistic re-iterpretatio of the staggered-grid versio of the Radom Choice Method (RCM) of Glimm [15], we briefly recall here the defiitio of the FORCE scheme for coservatio laws. The FORCE scheme for the coservative system () ca be writte either i a two-step staggered grid versio as Q +1 i+ 1 = 1 (Q i + Q i+1) 1 t [ F(Q x i+1 ) F(Q i ) ], (3) Q +1 i = 1 ( Q +1 i 1 + Q +1 i+ 1 ) 1 [ t x F(Q +1 i+ 1 ] ) F(Q +1 ), (4) i 1

3 or i the more coveiet coservative o-staggered oe-step formulatio with two-poit fluxes as Q +1 i Here, the FORCE flux F FORCE i+ 1 ad the Lax-Wedroff flux, i.e. = Q i t [ F FORCE i+ x 1 ] F FORCE i. (5) 1 is the arithmetic average of the Lax-Friedrichs F FORCE i+ 1 = 1 ( F LF i+ 1 + F LW i+ 1 ), (6) with the Lax-Friedrichs flux F LF i+ 1 = 1 [ F(Q i+1 ) + F(Q i )] 1 x ( ) Q t i+1 Q i (7) ad the Lax-Wedroff flux F LW i+ 1 ( = F Q +1 i+ 1 ), (8) where Q +1 is give by (3). It is easy to prove via simple algebraic maipulatios that the two schemes (3) & (4) ad (5)-(8) are idetical. For the purpose i+ 1 of this paper the Lax-Wedroff scheme as give i (8)-(3), is ot coveiet. The mai problem is its two-step ature ad the resultig o-liearity of the umerical flux fuctio with respect to the argumets Q i, Q i+1, F(Q i ) ad F(Q i+1 ), which makes it cumbersome for further aalytic maipulatios, sice we do ot wat to make ay further assumptios o F, other tha hyperbolicity. We therefore propose the followig variat of the coservative FORCE flux: F FORCE i+ = 1 ( F LF 1 i+ 1 where the modified Lax-Wedroff-type flux is ow give by ) + F LW i+, (9) 1 F LW i+ 1 = 1 [ F(Q i+1 ) + F(Q i )] 1 t [ Â x i+ 1 F(Q i+1 ) F(Q i )]. (1) The matrix Âi+ 1 = Âi+ 1 (Q i,q i+1 ) is a fuctio of the left ad the right states ad still has to be chose appropriately. For liear systems with costat coefficiet matrix A, the fluxes give by (8) & (3) ad (1) are idetical. We poit out that the modified Lax-Wedroff-type flux (1) has to be itroduced for techical reasos, i order to be able to prove later o that the proposed o-coservative cetred schemes reduce exactly to the coservative cetred scheme (5) with the modified FORCE flux (9), if the matrix A(Q) is the Jacobia of some flux fuctio F(Q). 3

4 1. The Origial Two-Step PRICE-T Scheme The PRICE-T scheme itroduced i [9] for o-coservative systems of the form (1) is the followig two-step scheme: Q +1 i+ 1 = 1 (Q i + Q i+1 ) 1 t x  i+ 1 ( Q i+1 Q i ), (11) Q +1 i = 1 ( Q +1 i 1 + Q +1 i+ 1 ) 1 t x where the matrices are evaluated as follows:  i = A ( 1 [Q+1 i 1 ) + Q +1 ] i+ 1, Âi+ 1  i (Q +1 i+ 1 Q +1 i 1 ), (1) ( ) 1 = A [Q i + Q i+1 ]. (13) Whe applied to the liear scalar advectio equatio q t + λq x =, i [9] it was foud that the PRICE-T scheme is first-order accurate, mootoe ad obeys the stadard CFL stability coditio c = λ t x 1, (14) where c is the CFL umber. It was show i [9] that for the shallow water equatios the scheme (11) & (1) provides a reasoable approximatio of weak shock waves, but i the presece of strog shocks the scheme is uable to capture either the exact positio of the frot or the exact post-shock values. I fact, the theorem of Hou ad LeFloch [] states that o-coservative methods will coverge to the wrog solutio i the presece of shock waves. It is therefore the declared objective of this cotributio to create a modified PRICE-T scheme that automatically reduces to the modified coservative FORCE scheme (5) & (9) i the case A(Q) is the Jacobia matrix of some flux fuctio F(Q), i.e. whe A(Q) = F/ Q. The relevace of this result will be oteworthy: it has bee prove that i the coservative case that FORCE is the optimal cetred scheme resultig from a covex average of 8) ad 7) i the sese that it is the least dissipative of all three-poit cetred methods that are mootoe ad have stability coditio (14), see [3] for details. Also the coservative FORCE scheme has bee show to be coverget for the case of two particular o-liear hyperbolic systems [8]. Furthermore, we are also lookig for a scheme that preserves some particular equilibria of the goverig PDE (well-balaced property) ad that is easily extedable to high order of accuracy i space ad time. 4

5 1.3 The PRICE-R Scheme We ote that the system (1) cotais a o-coservative product which, i geeral, does ot make sese i the classical framework of the theory of distributios. With the theory developed by Dal Maso, LeFloch, ad Murat [9], a rigorous defiitio of weak solutios ca be give usig a family of paths Ψ = Ψ(Q L,Q R, s) coectig two states Q L ad Q R across a discotiuity with (s [, 1]). For all umerical test cases preseted i this paper, we always use the simple segmet path, give by Ψ(Q L,Q R, s) = Q L + s (Q R Q L ). (15) Oce a family of paths is chose, it is possible to give a sese to the ocoservative product at discotiuities as a Borel measure (see [9] for details). Moreover, based o the theoretical advaces i [9], geeralizatios of the Roe method to systems of the form (1) have bee itroduced i [33,5,6]. Give a family of paths Ψ, a matrix A Ψ is called a Roe matrix if it satisfies the followig properties: for ay Q L,Q R Ω, A Ψ (Q L,Q R ) has N real eigevalues; A Ψ (Q,Q) = A(Q), for ay Q Ω; for ay Q L,Q R Ω: A Ψ (Q L,Q R )(Q R Q L ) = 1 A(Ψ(s,Q L,Q R )) Ψ ds. (16) s I the case whe A(Q) is the Jacobia matrix of a flux F(Q), the (16) is idepedet of the choice of the path ad we have the classical Roe property: A Ψ (Q L,Q R )(Q R Q L ) = F(Q R ) F(Q L ). (17) With this isight, we ow cosider a modified versio of the PRICE-T scheme, called PRICE-R i the followig, where we evaluate the matrices A i ad A i+ 1 i equatios (11) ad (1) as ) Â i = A Ψ (Q +1,Q +1 i 1, i+ 1 Âi+ 1 = A Ψ ( Q i,q i+1). (18) Usig algebraic maipulatios ad equatio (17), it is easy to prove that the scheme (11) & (1) with (18) reduces to the origial coservative FORCE scheme (3) & (4) ad therefore to (5) with the origial FORCE flux (6) with (7) ad (8), if A(Q) is the Jacobia matrix of a flux F(Q). The choice of the matrices give by (18) has the advatage that the resultig PRICE-R method becomes exactly coservative if applied to coservatio laws. However, it has the obvious disadvatage that oe eeds to compute 5

6 the Roe matrix, which may become very cumbersome or eve impossible for complicated hyperbolic systems. Sice we are iterested i a truly cetred approach that does ot eed ay wave propagatio iformatio cotaied i the uderlyig goverig PDE, we therefore do ot wat to compute the Roe matrix explicitly. A obvious alterative to the aalytical computatio of the Roe-type matrix A Ψ is to use defiitio (16) ad the segmet path (15), which yields ( 1 A Ψ (Q L,Q R )(Q R Q L ) = ) A(Ψ(s,Q L,Q R ))ds (Q R Q L ). (19) Hece, we obtai the followig defiitio of the Roe matrix A Ψ i the case of a segmet path: A Ψ (Q L,Q R ) = 1 A(Ψ(s,Q L,Q R ))ds. () It is ow a key idea of this article to compute the Roe matrix A Ψ directly usig the itegral alog the segmet path Ψ, as give by the right had side of eq. (). The exact coservatio properties of the PRICE-R schemes described above are still valid i this case, if the itegral is computed exactly. For complicated oliear hyperbolic systems, the exact computatio of the itegral may quickly become too cumbersome, so that we propose to resort to classical high order accurate Gaussia quadrature rules to compute the right had side of eq. () umerically. Give a M-poit Gaussia quadrature rule with weights ω j ad positios s j distributed i the uit iterval [; 1], a very accurate umerical approximatio of the Roe matrix A Ψ is give by the followig cetered Roe-type matrix: M A M Ψ(Q L,Q R ) = ω j A(Ψ(s j,q L,Q R )). (1) j=1 Recall that a M-poit Gaussia quadrature rule itegrates polyomials up to degree M 1 exactly, which meas that oe Gaussia poit is eough if the system matrix A(Q) is a liear fuctio i Q. I order to study the sesitivity of the resultig PRICE-R scheme usig the approximate Roe matrix (1) we show the behavior of the method for the shallow water equatios i presece of a strog shock wave. The results are depicted i the top row Fig. 1. The computatios are carried out with differet umbers of Gaussia poits. It appears as if with three or more Gaussia poits the solutio ca ot be distiguished ay more from the solutio obtaied usig the exact Roe matrix. I the bottom row we show the behaviour of the scheme usig three Gaussia poits ad differet umbers of cells. We highlight that i this way coservatio is ot maitaied exactly ay more but it ca be preserved umerically up to ay desired order of accuracy by simply icreasig the umber of Gaussia poits. For eve more sophisticated methods i computig the path itegral 6

7 Gaussia poit approx. Gaussia poit approx. 3 Gaussia poit approx. Aalytical ROE matrix exact Gaussia poit approx. Gaussia poit approx. 3 Gaussia poit approx. Aalytical ROE matrix exact Depth h [m] 5 Depth h [m] x [m] x [m] Gaussia poit approx. N=1 3 Gaussia poit approx. N=5 3 Gaussia poit approx. N=1 3 Gaussia poit approx. N= exact Gaussia poit approx. N=1 3 Gaussia poit approx. N=5 3 Gaussia poit approx. N=1 3 Gaussia poit approx. N= exact Depth h [m] 5 Depth h [m] x [m] x [m] Fig. 1. Dam break problem geeratig a strog shock wave (iitial coditios h l = 1 m, h r =.1 m, u l = ad u r = 3 m/s). Solutio is obtaied at time t =.5 s usig the first-order PRICE-R scheme. Top row: compariso amogst results obtaied usig 1 to 3 Gaussia poits (symbols), aalytical Roe matrix A Ψ ad exact solutio (lie) ad usig 5 cells. O the right a zoom aroud the shock is show. Bottom row: behaviour of the scheme usig three Gaussia poits ad differet umbers of cells. O the right a zoom aroud the shock is show. umerically, oe eve could thik of usig adaptive ad extrapolatio strategies, such as Romberg itegratio. However, for all the test cases preseted i this article, three Gaussia quadrature poits have show to be eough. We also ote that the dam-break test cosidered i fig. 1 is ot physical at all but is just used to show the quality ad the robustess of the umerical method sice the right iitial velocity of -3 m/s ad the left iitial depth of 1 m are ever reached i real situatios. So for ay applicatios of shallow water equatios the umerical approximatio give by eq. (1) ca be cosidered a good choice. Fially the reader ca easily verify that the origial PRICE-T scheme (11) ad (1) with the matrices Âi ad Âi+ 1 give by (13) ca be reiterpreted as the PRICE-R scheme, where equatio (1) is approximated with just oe sigle Gaussia poit. This choice was show i [9] to give already reasoable shock-capturig properties i the case of weak shocks. Note that the cetered Roe-type matrix A M Ψ could also be used i the class of cetered schemes developed i [7]. We fially would like to remark that eve exactly path-coservative schemes may fail to coverge for o-coservative systems as reported recetly i [6]. 7

8 1.4 Alterative Formulatio of the PRICE-R Scheme After some algebra, the two-step PRICE-T scheme give by (11) ad (1) ca be rewritte as a oe-step scheme, as follows where ad Q +1 i = Q i t [ ] A (Q x i+ 1 i+1 Q i ) + A + i 1 (Q i Q i 1), ()  i+ 1  + i 1 = 1 4 = 1 4 [  i x t I + Âi+ 1 t ] x  iâi+ 1 (3) [  i + x t I + Âi 1 + t ] x  iâi 1, (4) with the idetity matrix I ad all matrices  computed as i (18). We emphasize the idetical form of this scheme with the path-coservative Roe scheme proposed i [6,5,4], the oly differece beig i the matrices A ad A +. i+ 1 i 1 I our case, these Roe-type matrices are cetred, that is they do ot use explicit wave properties iformatio. Moreover, they are computed umerically, whereas i [5,4] they are computed as A ± i+ 1 ( ) = A Ψ Q i,q ± i+1 = RΨ Λ ± ΨR 1 Ψ. (5) Here, the usual defiitios apply, i.e. R Ψ is the matrix of right eigevectors of the Roe matrix A Ψ ad Λ Ψ is the diagoal matrix with the eigevalues of A Ψ. The matrices Λ ± Ψ are, as usual, either the positive or the egative part of the diagoal matrix Λ Ψ. For very complicated o-coservative systems oe ( ) could still costruct a upwid method by usig A ± ± = A M i+ 1 Ψ Q i,q i+1 ad computig the eigestructure fully umerically, e.g. usig the RG subroutie of the EISPACK library. We emphasize that the use of A M Ψ istead of A Ψ i (5) still has the advatage that the Roe-averages do ot have to be computed aalytically, which may be very difficult or eve impossible for very geeral oliear systems, however the ecessary umerical computatio of the full eigestructure is very costly. So the basic idea of our ew PRICE-C scheme preseted i the followig sectio is to avoid the use of the aalytical Roe matrix ad the computatio of A ±, which requires the kowledge of wave propagatio iformatio (upwid i+ 1 philosophy), ad to use istead oly the cetered Roe-type matrix A M Ψ which is computed umerically with a umber of Gaussia poits that is adequate for the problem to be solved (cetred philosophy). 8

9 1.5 The PRICE-C Scheme The mai drawback of the scheme (),(3),(4) is that the matrices A + ad i 1 A are three-poit fuctios, i.e. each of them depeds o the three states i+ 1 Q i 1, Q i ad Q i+1. This prevets a direct extesio of the PRICE-R method to multiple space dimesios ad high order of accuracy usig a polyomial recostructio of Q. To circumvet this problem, we therefore propose to modify the matrices A + i 1 ad A, substitutig the matrix i+ 1 Âi i (3) with Âi+ 1 ad the matrix Âi i (4) with Âi 1, i order to make them oly two-poit fuctios of the two adjacet states. After these modificatio, the fial o-coservative versio of the FORCE method, called PRICE-C scheme i the followig, reads as follows: Q +1 i = Q i t [ ] A (Q x i+ 1 i+1 Q i ) + A+ (Q i 1 i Q i 1 ), (6) with A i+ 1 = 1 4 [ ( A M Ψ Q i,qi+1) x t I t ( ( ] A M x Ψ Q i,qi+1)) (7) ad A + i 1 = 1 4 [ ( ) A M Ψ Q i 1,Q x i + t I + t ( ( )) ] A M x Ψ Q i 1,Q i. (8) Now the matrices A ad A + oly deped o two adjacet states. With i+ 1 i 1 the properties () ad (17) it ca be easily prove that if the PDE (1) is a coservatio law (), the we have A i+ 1 ( ) Q i+1 Q i = F FORCE F(Q i ), (9) i+ 1 ( A + i Q 1 i Qi 1) = F(Q i ) F FORCE i. (3) 1 Therefore, the PRICE-C scheme (6)-(8) reduces to the modified coservative FORCE method (5), (9), (1) if A is the Jacobia of a flux F. We ote that, idepedetly of the preset work, a similar method has bee proposed i [7], however, with the importat differece that i our case the cetered Roe-type matrices A M Ψ are computed via a etirely umerical procedure, usig M-poit Gaussia quadrature of appropriate order to evaluate the path itegral i (1), whereas i [7] the Roe matrices are computed usig aalytical expressios for the Roe averages. 9

10 We emphasize that our formulatio has the importat advatage that a explicit computatio of the Roe averages is ot ecessary, followig the origial philosophy of cetred schemes that by defiitio do ot eed ay additioal iformatio o the PDE system. At the same time coservatio ca be practically maitaied up to ay desired precisio usig Gaussia quadrature rules of appropriate order of accuracy. For complicated oliear PDE, as they typically arise i idustrial, civil ad evirometal egieerig, closed aalytical expressios for the Roe averages may be impossible to obtai for a give PDE system. A example for this will be show later whe we cosider shallow-water-type systems with movig bed usig a complex closure relatio. High Order Extesio.1 Noliear Recostructio Techique I this sectio we briefly discuss the proposed oliear weighted essetially o-oscillatory (WENO) recostructio procedure to recostruct higher order polyomial data withi each spatial cell T i = [x i 1; x i+ 1] at time t from the give cell averages Q i. We emphasize already at this poit that the recostructio procedure is oliear ad depeds strogly o the iput data Q i. Thus, the resultig umerical scheme, eve whe applied to a completely liear PDE, will be oliear ad thus it will ot be possible to give a closed expressio of the scheme. The recostructio procedure described here for the oe-dimesioal case follows directly from the guidelies give i [1] for geeral ustructured twoad three-dimesioal meshes. It recostructs etire polyomials, as the origial ENO approach proposed by Harte et al. i [19]. However, we formally write our method like a WENO scheme [1,3] with a particularly simple choice for the liear weights. The most importat differece of our approach compared to classical WENO schemes is that stadard WENO methods recostruct poit values at the Gaussia itegratio poits istead of a etire polyomial valid iside each elemet T i. Recostructio is doe for each elemet o a recostructio stecil S s i, which is give by the followig uio of the elemet T i ad its eighbors T j, i+s+k Si s = j=i+s k T j, (31) where s is the stecil shift with respect to the cetral elemet T i ad k is the spatial extesio of the stecil to the left ad the right. A cetral recostructio stecil is give by s =, a etirely left-sided stecil is give by 1

11 s = k ad a etirely right-sided stecil is give by s = k. I our approach, we always will use the three fixed recostructio stecils Si, S k i ad Si k. Give the cell average data Q i i all elemets T i we are lookig for a spatial recostructio polyomial obtaied from Si s at time t of the form w s i (x, t ) = N l= Ψ l (x)ŵ (i,s) l (t ) := Ψ l (x)ŵ (i,s) l (t ), (3) where we use the rescaled Legedre polyomials for the spatial recostructio basis fuctios Ψ l (x) such that the Ψ l (x) form a orthogoal basis o the elemet T i. I the followig, we will use stadard tesor idex otatio, implyig summatio over idices appearig twice. The umber of polyomial coefficiets (degrees of freedom) is L = N +1, where N is the degree of the recostructio polyomial. To compute the recostructio polyomial w i (x, t ) valid for elemet T i we require itegral coservatio for all elemets T j iside the stecil S s i, i.e. 1 wi s x (x, t )dx = 1 Ψ l (x)dx ŵ (i,s) l (t ) = Q j x, T j Si s. (33) T j T j Equatio (33) yields a liear equatio system of the form B jl ŵ (i,s) l (t ) = Q j (34) for the ukow coefficiets ŵ (i,s) l (t ) of the recostructio polyomial o stecil Si s. Sice we choose k = N/ for eve N ad k = (N +1)/ for odd N, the umber of elemets i Si s may become larger tha the umber of degrees of freedom L. I this case, we use a costraied least-squares techique accordig to [1] to solve (34). To obtai the fial o-oscillatory recostructio polyomials for each elemet T i at time t, we fially costruct a data-depedet oliear combiatio of the polyomials wi (x, t ), wi k (x, t ) ad wi k(x, t ) obtaied from the cetral, left-sided ad right-sided stecils as follows: w i (x, t ) = ŵ i l(t )Ψ l (x), (35) with ŵl i (t ) = ω ŵ (i,) l (t ) + ω k ŵ (i, k) l (t ) + ω k ŵ (i,k) l (t ). (36) The oliear weights ω s are give by the relatios ω s = ω s ω + ω k + ω k, ω s = λ s (σ s + ǫ) r. (37) I our particular formulatio, the oscillatio idicators σ s are computed from σ s = Σ lm ŵ s l (t )ŵ s m(t ), (38) 11

12 with Σ lm = N α=1 1 x α 1 α Ψ l (x) x α α Ψ m (x) x α dx. (39) Here, Σ lm is the oscillatio idicator matrix for elemet T i. If all computatios are doe i a referece elemet, the this matrix does either deped o the problem or o the mesh, see [1]. The parameters ǫ ad r are costats for which we typically choose ǫ = 1 14 ad r = 8. For the liear weights λ s we choose λ k = λ k = 1 ad a very large liear weight λ o the cetral stecil, typically λ = 1 5. It has bee show previously [1,3] that the umerical results are quite isesitive to the WENO parameters ǫ ad r ad also with respect to the liear weight o the cetral stecil λ, see [1]. The proposed recostructio usually uses the accurate ad liearly stable cetral stecil recostructio i those regios of Ω where the solutio is smooth because of the large liear weight λ. However, due to the strogly oliear depedece of the weights ω s o the oscillatio idicators σ s, i the presece of discotiuities the smoother left- or right-sided stecils are preferred, as for stadard ENO ad WENO methods. For the oliear scalar case, the recostructio operator described above ca be directly applied to the cell averages Q i of the coserved quatity Q. For oliear hyperbolic systems, the recostructio should be doe i characteristic variables [19,13] i order to avoid spurious oscillatios that may appear whe applyig ENO or WENO recostructio operators compoet-wise to oliear hyperbolic systems.. High-Order Accurate Oe-Step Time Discretizatio The result of the recostructio procedure is a o-oscillatory spatial polyomial w i (x, t ) defied at time t iside each spatial elemet T i. However, we still eed to compute the temporal evolutio of these polyomials iside each space-time elemet [x i 1; x i+ 1] [t ; t +1 ] i order to be able to costruct our fial high order accurate oe-step fiite volume scheme. I order to obtai a high order accurate oe-step method we use the ADER approach of Titarev ad Toro [3]. The key idea therei is to solve high order Riema problems at the elemet boudaries, this is accomplished by a Taylor series expasio i time, use of the Cauchy-Kowalewski procedure ad solutios of classical Riema problems, the state variables ad its spatial derivatives. I this paper we adopt the followig strategy: We expad the local solutio Q i (x, t) of the PDE i each cell i a space-time Taylor series with respect to the elemet barycetre x i Q i (x, t) =Q(x i, t ) + (x x i ) Q x + (t t ) Q t + 1 (x x i) Q x + (x x i )(t t ) Q t x + 1 (t t ) Q t +... (4) 1

13 where we the use the classical Cauchy-Kovalewski procedure i order to substitute time derivatives with space derivatives, usig repeated differetiatio of the goverig PDE system (1) with respect to space ad time. I the followig, we illustrate the Cauchy-Kovalewski procedure symbolically for third order of accuracy. For a efficiet implemetatio up to ay order of accuracy i space ad time we refer the reader to [14] ad [13]. For two more geeral ad fully umerical alteratives to the semi-aalytical Cauchy-Kovalewski procedure see [11] ad [1], where local space-time fiite elemet methods are used i order to compute the polyomial Q i (x, t). The first time derivative ca be directly obtaied from (1) as Q t = A(Q) Q x. (41) The mixed space time derivative is the obtaied after a differetiatio with respect to space ad the secod time derivative of Q is Q t x = x A(Q) Q x A(Q) Q x, (4) Q t = t A(Q) Q x A(Q) Q t x. (43) The value of Q i (x i, t ) ad all purely spatial derivatives are obtaied from the WENO recostructio polyomial w i (x, t )..3 The Fully Discrete High Order Accurate Oe-Step Scheme Oce the WENO recostructio ad the Cauchy-Kovalewski procedure have bee performed for each cell, PDE (1) ca be itegrated over a space-time cotrol volume [x i 1; x i+ 1] [t ; t +1 ] (see [5] ad [6] for datails) ad our fial high-order accurate oe-step scheme ca be writte as follows: where ad Q +1 i = Q i 1 x AQ x t [ ] D + D +, (44) x i+ 1 i 1 AQ x = D ± i+ 1 t +1 t = 1 t x i+ 1 x + i 1 t +1 A(Q i (x, t)) x Q i(x, t)dxdt (45) t A ± i+ 1 ( ) Q + Q dt, (46) i+ 1 i+ 1 13

14 with Q i+ 1 = Q i (x i+ 1, t) ad Q + i+ 1 = Q i+1 (x i+ 1, t). (47) All the itegrals are approximated usig Gaussia quadrature formulae of suitable order of accuracy. Note that the term AQ x, which itegrates the smooth part of the o-coservative product withi each cell (excludig the jumps at the boudaries), vaishes for a first order scheme where we have x Q i(x, t) =. I the followig we briefly summarize the etire high-order oe-step algorithm: (1) Perform the WENO recostructio described i sectio.1 i order to obtai the recostructio polyomials w i (x, t ) for each cell. () Compute the spatial derivatives of w i (x, t ) ad isert them ito the Cauchy-Kovalewski procedure i order to get all missig space-time derivatives i the Taylor series (4). This step geerates a space-time polyomial Q i (x, t) for each cell T i. (3) Use the space-time polyomials Q i (x, t) together with Gaussia quadrature to compute the itegrals appearig i the fully discrete scheme (44) ad perform the update of the cell averages. 3 Numerical Results The PRICE-C scheme preseted i this paper is very geeral ad is applicable to ay system of hyperbolic equatios cotaiig o-coservative products. I this sectio we assess the performace of the proposed high order algorithm usig as model system the time-depedet o-liear shallow water equatios without ad with sedimet trasport. I the followig, umerical results for differet test cases are reported. The computatios are carried out usig a third order WENO versio of the proposed PRICE-C scheme. The Courat umber is set to CFL=.9. The matrix () has bee evaluated usig a three-poit Gaussia quadrature rule with the followig poits s j ad weights ω j : s 1 = 1, s,3 = 1 ± 15 1, ω 1 = 8 18, ω,3 = (48) 3.1 Shallow Water Equatios We cosider the 1D system of shallow water equatios with variable bottom topography. The bottom frictio is eglected. The system ca be writte as: 14

15 H t + q x =, q t + ( q x H b + 1 ) gh ghb + gh b =, (49) x where H = h + b is the free surface elevatio, h is the water depth, q = hu is the discharge per uit width, b represets the bottom topography ad g is the acceleratio due to gravity. I order to obtai a well balaced scheme, we follow the idea developed i [5,16,17]. Addig the trivial equatio b/ t = i system (49), the problem ca be writte i the o-coservative form (1), i which the forces due to the variable bottom topography are iterpreted as a o-coservative product. The vector Q ad the matrix A assume the followig form where h = H b ad u = q/h. H 1 Q = q, A = gh u u u, (5) b We ote here that the scheme (44) with matrices (7) ad (8) whe applied to the shallow water equatios produces a artificial motio of the bottom. I fact whe the bottom is variable, the compoet (3,3) of the idetity matrix I gives a udesirable diffusio that teds to flatte the bottom also if the water is quiescet. So i the follow we use a modified idetity matrix I m that reads: 1 I m = 1, (51) where the udesirable diffusio of the bottom is elimiated Verificatio of the C-Property Proof. It is well-kow that umerical methods for the shallow water system with variable bottom must satisfy the so-called C-property as itroduced by []. This meas that the term due to the bottom elevatio must balace the term due to hydrostatic pressure uder quiescet flow coditios over ay bottom profile, icludig discotiuous bottom. For quiescet flow, we have 15

16 H = cost., u = ad therefore H 1 gh Q = q =, A Ψ = gh, A Ψ = gh, b b (5) with h = 1 h(s)ds = 1 (h L + s(h R h L ))ds. Usig the well-balaced idetity matrix I m it follows trivially from eqs. (6)-(8) ad (51) that A ± i 1 Q = (53) ad therefore the first order scheme verifies the exact C-property. For the higher order scheme (44) we poit out that usig recostructio of the free surface elevatio H ad the bottom topography b leads to a so-called wellbalaced recostructio i the sese of [5], hece also the term AQ x =. Numerical verificatio. The aim of these simulatios is ow to verify whether also our actual implemetatio of the proposed PRICE-C scheme i computer code satisfies the exact C-property to machie precisio. I order to verify this property we perform two differet umerical experimets as proposed i [34]. We take b(x) = 5e ( 5 (x 5)) m (54) for simulatig a smooth bottom ad 4 m if 4 m x 8 m, b(x) = otherwise. (55) for the discotiuous case. The iitial data for both tests are: H = h + b = 1 m, q =. (56) To test the ability of the scheme to maitai the iitial coditio, a simulatio is carried out util t =.5s, usig a mesh of cells i a 1 m log domai. We use double precisio arithmetics. The errors betwee umerical ad exact solutio are give i Table 1, from which we ca deduce that the C-property is exactly satisfied up to machie precisio A Small Perturbatio of a Steady State Water This test was first proposed by LeVeque [] ad aims to assess the capability of the scheme to capture a small pulse propagatig over a quiescet state. The 16

17 Table 1 Verificatio of the C-property: water depth ad specific discharge orms Testcase H q L 1 L L 1 L Test 1 (smooth) 3.5e e-14.4e e-14 Test (o-smooth) 4.34e e e e-14 bottom topography cosidered is described by:.5 cos(1π(x 1.5)) + 1)) m if 1.4 m x 1.6 m, b(x) = otherwise. ad the iitial coditios are: 1 + ǫ if 1.1 m x 1. m, q(x, ) = ad H(x, ) = 1 otherwise, (57) (58) with ǫ beig a small perturbatio of the free surface that we choose to be ǫ =. m for the first test ad ǫ =.1 m for the secod oe. This is a very difficult problem ad it is reported i the literature [] that may umerical schemes fail i computig correctly the propagatio of such small perturbatios over variable bottom topography. Results for the free surface ad the velocity are give i Figs. ad Fig. 3. The solutio obtaied usig H (m) 1 q (m /s).9 4 cells (third order) 3 cells (third order) 4 cells (first order) x (m) cells (third order) 3 cells (third order) 4 cells (first order) x (m) Fig.. Small perturbatio of a steady state water: pulse (ǫ =. m) over iitial quiescet water. Results at time t =. s of the third-order PRICE-C scheme with 4 cells (symbols) ad with 3 cells (lie). Results of the first-order scheme with 4 cells are also show for compariso. the third order PRICE-C scheme with 4 cells is compared with a umerical referece solutio obtaied o a very fie mesh with 3 cells. The method produces accurate o-oscillatory solutios that are i good agreemet with the referece solutios. It is worth oticig that i the case of the small pulse ( ǫ/h << 1), theoretically the iitial disturbace should split ito two waves, propagatig to the left ad right at the characteristic speed gh. This is correctly reproduced i our umerical simulatios. 17

18 1.1 x H (m) 1 q (m /s) cells (third order) 3 cells (third order) 4 cells (first order) x (m) 1 4 cells (third order) 3 cells (third order) 4 cells (first order).5 1 x (m) 1.5 Fig. 3. Small perturbatio of a steady state water: pulse (ǫ =.1 m) over iitial quiescet water. Results at time t =. s of the third-order PRICE-C scheme with 4 cells (symbols) ad with 3 cells (lie). Results of the first-order scheme with 4 cells are also show for compariso Steady Flow Over a Smooth Hump The aim of such simulatios is to aalyze the covergece i time towards a steady flow over a smooth bump. To this ed we have used three differet tests (a, b, c) with exact solutio, proposed by the Workig group o dam break modelig [18], which are broadly used for testig umerical methods. The bottom topography is the followig:..5(x 1) m if 8 m x 1 m, b(x) = otherwise. (59) while the domai has a legth of L = 5 m, divided i cells. Steady solutios have bee obtaied by marchig i time to steady state, startig from a iitial profile (horizotal free surface profile) that is far away from the steady solutio. The iitial coditios are take as q(x, ) = ad H(x, ) =.5 m. (6) Modifyig the value of the upstream discharge q or the dowstream water surface level H results i differet steady cofiguratios ad therefore we select differet boudary coditios, which are summarized i Table. I test Table Boudary coditios for the steady flow over a smooth hump Test case q(x =,t) [m /s] H(x = L,t) [m] (a) (b) (c) 4.4. case (a) the solutio is characterized by a trascritical flow without a shock, for test (b) the solutio is characterized by a trascritical flow with a shock, while i test case (c) the solutio is give by a completely subcritical flow. The umerical ad exact solutios for all test cases are depicted i Fig. 4 18

19 at time t = s. The agreemet betwee umerical ad exact solutio for the free surface elevatio H is excellet. No spurious oscillatios are produced at the discotiuities ad the positio of the shock wave is also correct. The small errors that appear i the discharge are also preset i other high order schemes documeted i the literature, see e.g. [34]. 1.8 umerical solutio aalytical solutio umerical solutio aalytical solutio H (m).6.4. q (m /s) x (m) x (m).4 umerical solutio aalytical solutio.4 umerical solutio aalytical solutio H (m).3. q (m /s) x (m) x (m) 3.5 umerical solutio aalytical solutio umerical solutio aalytical solutio H (m) q (m /s) x (m) x (m) Fig. 4. Steady flow over a smooth hump. Top row: test case (a). Middle row: test case (b). Bottom row: test case (c). Results are show at time t = s usig the third-order PRICE-C scheme (symbols) as well as the exact solutio (lie). 3. Sedimet Trasport A system of equatios that govers the trasport of sedimets i gravel bed rivers is obtaied couplig the shallow water equatios (49) with a equatio that describes the bottom evolutio, amely the Exer equatio. It reads: b t + q s x =, (61) 19

20 where b = b(x, t) is the movable bed elevatio. q s is the bedload sedimet trasport rate for uit width. Several umerical solutios have bee proposed i literature for this problem [3,4]. Ad for the quatificatio of q s differet relatioships are available i literature. We have used a simple power law for testig the method agaist exact solutios, amely: q s = A(u u c) m, (6) (1 λ p ) where u is the velocity of the water, u c is the critical velocity below which the sedimet trasport vaishes, m is a positive expoet, while λ p is the porosity. Moreover other two empirical formulae available i literature have bee implemeted. They are of the type: q s = (s 1)gd 3 s Φ (θ), (1 λ p ) (63) with s beig the relative desity, ad the local Shields stress is give by θ = S fh (s 1)d s, (64) where d s is the mea sedimet diameter. The frictio term S f is calculated usig the usual formula of Maig that reads: S f = q f h 1/3 (65) f beig the Maig coefficiet of roughess. I this paper we make use of the sedimet discharge fuctio Φ(θ) proposed by Parker [7], which reads Φ =.18 θ 3/ G(ξ), ξ = θ θ r, θ r =.386, (66) with 5474(1.853/ξ) 4.5 ξ 1.59, G = exp [14.(ξ 1) 9.8(ξ 1) ] 1 ξ 1.59, ξ 14. ξ < 1. ad the oe proposed by Meyer-Peter ad Müller [5]: 8(θ.47) 3/, if θ >.47 Φ = otherwise (67) (68) It is worth oticig that the empirical ature of the relatioships aimig to quatify the solid discharge q s leads to the availability of a great umber of differet formulae. As a cosequece, each particular choice for the closure

21 relatio for q s leads to a differet system matrix A ad therefore to a differet formulatio of the aalytical Roe matrix A Ψ. This would result i usurmoutable problems for evirometal egieers, whose scope is to try may differet available empirical formulatios for reproducig field measuremets or laboratory experimets. The mai advatage of the proposed PRICE-C method with the fully umerical computatio of the cetered Roe-type matrix A M Ψ via Gaussia quadrature alog the path is that it completely avoids the eed for a explicit computatio of the Roe averages, beig at the same time accurate up to the prescribed order for ay choice of the solid trasport formula. The system of goverig equatios describig the coupled evolutio of the fluid ad the bed ca be writte i the form (1), see [8], with the vector Q ad matrix A beig respectively: q H s 1 + qs q s H q b Q = q, A = gh u u u. (69) b I the followig we show the results provided by the proposed PRICE-C scheme for three differet test cases. q s H q s q q s b 3..1 Propagatio of a Small Sedimet Hump Near Critical Coditios A test aimig to reproduce bed movemet ear critical coditios is carried out. Uder these coditios, the couplig betwee the shallow water equatios ad the Exer equatio withi the time step is madatory. I this rage, i fact, each of the wave propagatio speeds ca o loger be idetified solely with a surface wave or solely with a bed wave, ad a full couplig of the equatios is ecessary to correctly solve the propagatio of bed disturbaces. The iitial Froude umber is take as Fr U =.979, where U idicates the uiform uperturbed state. The iitial bed topography is described by: b(x, ) = b max e x m with 15 m x 15 m, (7) where b max = 1 5 m is the amplitude of the iitial bed perturbatio. The iitial coditio is obtaied ruig the code with a fixed bed cofiguratio. The upstream discharge is fixed accordig to the Froude umber, while dowstream a fixed water depth equal to 1 m is imposed. The domai legth L=5 m is divided i spatial steps of.5 m, leadig to 5 cells. The umerical results are compared with a exact solutio, obtaied by Ly ad Altiakar [4] liearizig the goverig system of equatios usig a small parameter ψ U 1

22 defied as follows: ψ U = 1 (1 λ p )h U q s u. (71) The adopted sedimet trasport formula is of the form (6) i which A = ad m =.65, while u c is determied solvig equatio (71) settig ψ U = I Fig. 5 the compariso betwee umerical ad aalytical solutio is give for t = s for both the bottom ad the water surface. The quatities are plotted i dimesioless form, with the scalig parameters beig: b max h ref = (1 FrU ), b ref = b max. (7) We ote two waves geerated o the bottom: the scour wave propagatig umerical solutio aalytical solutio (h h U )/h ref.4.6 umerical solutio aalytical solutio x (m) (b b U )/b ref x (m) Fig. 5. Propagatio of a small sedimet hump ear critical coditios. Results at time t = s of the third-order PRICE-C scheme (symbols) ad the exact solutio (lie). upstream ad a depositioal wave propagatig dowstream, while i correspodece of the bottom disturbaces two egative waves are geerated i the water depth h. The umerical solutio obtaied with the third-order PRICE- C scheme is i excellet agreemet with the aalytical solutio preseted i [4]. 3.. Propagatio of a Sedimet Bore I this test case we apply differet trasport formulae comparig umerical results with those obtaied experimetally i [1]. The experimet cosists i a steep-sloped, rectagular chael of fiite legth. The bed profile is i quasiequilibrium ad a costat sedimet supply is fed upstream. At referece time t =, this equilibrium situatio is perturbed by the rapid raise of a submerged weir at the dowstream ed of the flume, imposig a subcritical coditio. The water ad sedimet discharges at the upstream sectio are kept costat. This hydraulic cofiguratio gives rise to a hydraulic jump ad a sedimet bore. The aggradatioal shock frot caused by the presece of trascritical flow represets a demadig test case i which umerical schemes may fail i predictig both itesity ad propagatio velocity of the frot itself. The flume is 6.9 m log,.5 m wide ad the slope is equal to 3.%. The sedimet

23 ad water discharge are respectively q s =.136 l/s ad Q = 1 l/s. The iduced water level at the dowstream ed is H =.93 cm; uiform coarse sad with a mea diameter of 1.65 mm ad porosity of.4 are cosidered. Fially, Maig coefficiet is f =.165 m 3 s. Numerical treatmet of the frictio term is made by usig the approach of Gosse [16,17], which leads to the followig system of goverig equatios: H q Q =, A = b x q s H 1 + qs q q s b gh u u u ghs f. (73) q s q s q s H q b Numerical simulatios are coducted usig the sedimet trasport formulae (66), (68) ad a formula of the type (6) calibrated with parameters A =.4, m=3 ad u c =.3 m/s. I Fig. 6 the positio of the sedimet bore obtaied with all three differet sedimet discharge formulae is plotted as a fuctio of time ad is compared with the experimetal data. Shock positio at time t +1 is defied as the x coordiate of the barycetre of the first cell (startig from the right boudary) that satisfies: b +1 i b i > τ (74) τ is a give tolerace fixed to. m i all the computatios ad for all the differet sedimet formula. The celerity of the frot is give by the iverse of the slope of the above curves. As it is see the propagatio celerity depeds o the trasport formula used. This meas that thaks to its geerality ad simplicity, the umerical tool that we propose ca be very useful for practitioers whe they wat to reproduce real data because they ca test may differet trasport formula without havig to adapt the umerical method to each specific closure relatio for q s. 3.3 Numerical Covergece Study I the previous sectios we have show that the proposed umerical method well reproduces usteady solutios ad the results are essetially o oscillatory. Here we compute the order of accuracy of the scheme to verify that the expected theoretical order is achieved. We solve the iviscid shallow water equatios coupled with a bottom evolutio equatio, that writte with respect to the variables h, q ad b read: 3

24 t (s) 15 1 Meyer Peter ad Muller calibrated power law formula Parker experimetal data x (m) Fig. 6. Compariso of frot positio for the third-order PRICE-C scheme usig three differet formulae for the quatificatio of the solid discharge. Table 3 Covergece rates study for the sedimet trasport problem with source terms for the third order PRICE-C method, (c =.1m, h = 5m, T p = 1s, L w = 5m) variable h variable q N L 1 O(L 1 ) L O(L ) L 1 O(L 1 ) L O(L ) 5.54E E E-.14E E E E E E E E E E E E E E E E E E E E E t h + x q =, ( t q + x qu + 1 gh) = gh x b, t b + x q s =. (75) I order to validate the order of accuracy a exact solutio is costructed by prescribig three fuctios for h(x, t), q(x, t) ad b(x, t) which satisfy exactly (75). They read h(x, t) = h + c si(kx ωt), q(x, t) = ω k h ω + c si(k ωt), k b(x, t) = h(x, t), q s (x, t) = q(x, t), k = π L w, ω = π T p. (76) We uderscore that the relatio q s = q is ot-physically based, but it allows us to fid a exact solutio of system (75). Table 3 shows the errors quatified through the stadard orms L 1, L ad relative covergece rates for 4

25 variables h ad q at the time t = 1 s with c =.1m, h = 5 m, T p = 1 s, L w = 5 m. We ca see that the third of accuracy is achieved with each orm. 4 Coclusios We have preseted a first order mootoe cetred scheme, called PRICE-C, which is based o the cetred FORCE scheme for coservatio laws [31], [3]. It ca be see as a extesio of the PRICE-T method proposed i [9] usig the isights gaied by the path-coservative methods developed recetly i [5] ad [6]. We have exteded this first-order method to third-order of accuracy i space ad time via the ADER approach usig a WENO recostructio techique. Extesive umerical experimets suggest that the scheme is very geeral, though efficiet ad simple. It yields very satisfactory results compared to exact ad experimetal referece solutios. A first attractive feature of the preseted method is the simplicity due to a approximate computatio of the Roe matrix via Gaussia quadrature rules of suitable order of accuracy. I practice, we foud that for shallow-water-type PDE systems, three Gaussia poits seem to be eough to esure coservatio. This avoids the eed of a aalytical Roe matrix. A secod importat aspect cocers the future extesio of the method to multiple space dimesios: this ca be achieved sice the matrices A ± of the PRICE-C method have bee modified i such i+ 1 a way as to become two-poit fuctios of the two adjacet states, i cotrast to the origial PRICE-T method or the PRICE-R scheme show i this paper, where these matrices were depedet o three states. The high order cetred schemes preseted here are very geeral ad ca be applied to ay hyperbolic system i o-coservative form that may exhibit at the same time smooth ad discotiuous solutios. The advatage of the preseted cetred scheme over upwid-based methods is its simplicity ad efficiecy, ad will be fully realized for hyperbolic systems i which the provisio of upwid iformatio is very costly or is ot available. Ackowledgemets The first author thaks Cariveroa for fiacial support uder the project MODITE. 5

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