Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes

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1 Postvty-Preservng Well-Balanced Dscontnuous Galerkn Methods for the Shallow Water Equatons on Unstructured Trangular Meshes The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Xng, Yulong, and Xangxong Zhang. Postvty-Preservng Well- Balanced Dscontnuous Galerkn Methods for the Shallow Water Equatons on Unstructured Trangular Meshes. Journal of Scentfc Computng 57, no. 1 (October 213): Sprnger-Verlag Verson Author's fnal manuscrpt Accessed Fr Jan 4 19:29:7 EST 219 Ctable Lnk Terms of Use Detaled Terms Artcle s made avalable n accordance wth the publsher's polcy and may be subject to US copyrght law. Please refer to the publsher's ste for terms of use.

2 Postvty-preservng well-balanced dscontnuous Galerkn methods for the shallow water equatons on unstructured trangular meshes Yulong Xng 1 and Xangxong Zhang 2 Abstract The shallow water equatons model flows n rvers and coastal areas and have wde applcatons n ocean, hydraulc engneerng, and atmospherc modelng. In [36], the authors constructed hgh order dscontnuous Galerkn methods for the shallow water equatons whch can mantan the stll water steady state exactly, and at the same tme can preserve the nonnegatvty of the water heght wthout loss of mass conservaton. In ths paper, we explore the extenson of these methods on unstructured trangular meshes. The smple postvtypreservng lmter s reformulated, and we prove that the resultng scheme guarantees the postvty of the water depth. Extensve numercal examples are provded to verfy the postvty-preservng property, well-balanced property, hgh-order accuracy, and good resoluton for smooth and dscontnuous solutons. eywords: shallow water equatons; dscontnuous Galerkn method; hgh order accuracy; well-balanced; postvty-preservng methods; wettng and dryng treatment 1 Computer Scence and Mathematcs Dvson, Oak Rdge Natonal Laboratory, Oak Rdge, TN and Department of Mathematcs, Unversty of Tennessee, noxvlle, TN E-mal: xngy@math.utk.edu. Fax: (865) Research s sponsored by the Natonal Scence Foundaton grant DMS , ORNL s Laboratory Drected Research and Development funds, and the U. S. Department of Energy, Offce of Advanced Scentfc Computng Research. The work was partally performed at ORNL, whch s managed by UT-Battelle, LLC, under Contract No. DE-AC5-OR Department of Mathematcs, Massachusetts Insttute of Technology, Cambrdge, MA E-mal: zhangxx@math.mt.edu. 1

3 1 Introducton The man goal of ths paper s to present hgh order accurate dscontnuous Galerkn (DG) methods for the shallow water equatons on unstructured trangular meshes, whch are not only well-balanced for the stll water steady state solutons, but also preserve the nonnegatvty of the water depth. The shallow water equatons wth a non-flat bottom topography play a crtcal role n the modelng and smulaton of flows n rvers, lakes and coastal areas. They have wde applcatons n ocean, hydraulc engneerng and atmospherc modelng. The two dmensonal shallow water equatons take the form h t + (hu) x + (hv) y = (hu) t + (hu ) gh2 + (huv) = ghb x x (hv) t + (huv) x + (hv ) gh2 = ghb y, (1.1) where h denotes the water heght, (u, v) T s the velocty vector, b represents the bottom topography and g s the gravtatonal constant. Other terms, such as a frcton term, could also be added n (1.1). Due to the large scentfc and engneerng applcatons of the shallow water equatons, research on effectve and accurate numercal methods for ther solutons has attracted great attenton n the past two decades. One dffculty encountered s the treatment of the source terms. An essental part for the shallow water equatons and other conservaton laws wth source terms s that they often admt steady state solutons n whch the flux gradents are exactly balanced by the source term. For the shallow water equatons, people are partcularly nterested n the stll water steady-state soluton, whch represents a stll flat water surface, and often referred as lake at rest soluton: u = and h + b = const. (1.2) Tradtonal numercal schemes wth a straghtforward handlng of the source term cannot balance the effect of the source term and the flux, and usually fal to capture the steady state well. They wll ntroduce spurous oscllatons near the steady state. The well-balanced 2

4 schemes are specally desgned to preserve exactly these steady-state solutons up to machne error wth relatvely coarse meshes, and therefore t s desrable to desgn numercal methods whch have the well-balanced property. The other major dffculty often encountered n the smulatons of the shallow water equatons s the appearance of dry areas n many engneerng applcatons ncludng the dam break problem and flood waves etc. Specal attenton needs to be pad near the dry/wet front, otherwse they may produce non-physcal negatve water heght, whch becomes problematc when calculatng the egenvalues u ± gh to determne the tme step sze t, and renders the system not hyperbolc and not well posed. In the past decade, many well-balanced numercal methods have been developed for the shallow water equatons, see, e.g. [3, 1, 22, 2, 26, 25] and the references theren. There are also a number of postvty-preservng schemes [4, 14, 17, 7, 9, 6] proposed for (1.1), and a few of them [21, 1, 14, 8] can resolve both dffcultes at the same tme. Recently, hgh-order accurate DG methods have attracted ncreasng attenton n many computatonal felds, ncludng the geophyscal flud dynamcs. DG method s a class of fnte element methods usng dscontnuous pecewse polynomal space as the soluton and test functon spaces (see [1] for a hstorc revew). Several advantages of the DG method, ncludng ts accuracy, hgh parallel effcency, flexblty for hp-adaptvty and arbtrary geometry and meshes, make t partcularly suted for the shallow water equatons, see the frst work by Schwaneberg and ongeter [29], followed by [15, 13, 18, 23] and others. Recently, several well-balanced DG methods have been proposed, by Xng and Shu [33, 34, 36], Ern et al. [14], Rhebergen et al. [27] and other researchers [2]. Also, some dscussons on DG methods nvolvng wettng and dryng treatments for the shallow water equatons can be found n [5, 14, 9]. In [36], hgh order accurate DG methods, whch can mantan the stll water steady state exactly, and at the same tme can preserve the non-negatvty of the water heght, are developed for the shallow water equatons on one-dmensonal and two-dmensonal rectangular meshes. Due to the complex geometry of the computatonal domans n many real-world applcatons, trangular meshes are often used. In ths paper, we are nterested n the ex- 3

5 tenson of the postvty-preservng well-balanced methods developed n [36] on unstructured trangular meshes. A smple source term dscretzaton wll be presented, and shown to be balanced wth the numercal fluxes at the steady state soluton. Wth the ntroducton of a specal desgned Gaussan quadrature rule, we wll demonstrate that the smple postvtypreservng lmter used n [39] s stll plausble on trangular meshes, and does not affect the hgh order accuracy, as well as the mass conservaton. Ths paper s organzed as follows. In Secton 2, we present the well-balanced DG methods for the shallow water equatons on trangular meshes, followng the technque proposed n [34]. The postvty-preservng well-balanced DG methods are presented n Secton 3, whch nvolves a smple postvty-preservng lmter. Secton 4 contans extensve numercal smulaton results to demonstrate the behavor of our DG methods for two-dmensonal shallow water equatons on trangular meshes, verfyng hgh order accuracy, the well-balanced property, postvty-preservng property, and good resoluton for smooth and dscontnuous solutons. Concludng remarks are gven n Secton 5. 2 Well-balanced DG methods In ths secton, we frst renstate the classcal Runge-utta dscontnuous Galerkn (RDG) methods appled for the shallow water equatons. A few well-balanced DG methods have been developed recently, see the recent book chapter [25] for a revew and the references theren. In ths paper, we consder the approach developed by one of the authors n [34], where we observed that the classcal RDG methods are well-balanced for the stll water soluton (1.2), f a hydrostatc reconstructon s employed on the flux. The same technque s also used n [14, 2, 36] to derve well-balanced postvty-preservng methods. Ths s one of the smplest approaches to obtan a hgh order well-balanced scheme, and the extra computatonal cost due to the well-balanced property s neglgble. Its extenson to a trangulaton wll be ntroduced n ths secton, and ths scheme wll serve as the bass for the postvty-preservng technque presented n Secton 3. 4

6 Let T τ be a famly of parttons of the computatonal doman Ω parameterzed by τ >. For any trangle T τ, we defne τ := dam() and τ := max T τ τ. For each edge e ( = 1, 2, 3) of, we denote ts length by l, and outward unt normal vector by ν. Let () be the neghborng trangle along the edge e and be the area of the trangle. For the ease of presentaton, we denote the shallow water equatons (1.1) by U t + f(u) x + g(u) y = s(h, b), or U t + F(U) = s(h, b), where U = (h, hu, hv) T wth the superscrpt T denotng the transpose, f(u), g(u) or F(U) = (f(u), g(u)) are the flux and s(h, b) s the source term. In a hgh order DG method, we seek an approxmaton, stll denoted by U wth an abuse of notaton, whch belongs to the fnte dmensonal space V τ = V k τ {w L 2 (Ω); w P k () T τ }, (2.1) where P k () denotes the space of polynomals on of degree at most k. We project the bottom functon b nto the same space V τ, to obtan an approxmaton whch s stll denoted by b, agan wth an abuse of notaton. Let x denote (x, y), then the numercal scheme s gven by t Uw dx F(U) w dx + 3 =1 e F e νw ds = s(h, b)w dx, (2.2) where w(x) s a test functon from the test space V τ. The numercal flux F s defned by F e ν = F(U nt(), U ext(), ν). (2.3) where U nt() and U ext() are the approxmatons to the values on the edge e obtaned from the nteror and the exteror of. We could, for example, use the smple global Lax-Fredrchs flux F(a 1, a 2, ν) = 1 2 [F(a 1) ν + F(a 2 ) ν α(a 2 a 1 )], ( α = max ( u + gh, v + ) gh) ν, (2.4) 5

7 where the maxmum s taken over the whole regon. It satsfes the conservatvty and consstency F(a 1, a 2, ν) = F(a 2, a 1, ν), F(a 1, a 1, ν) = F(a 1 ) ν. (2.5) A smple Euler forward tme dscretzaton of (2.2) gves the fully dscretzed scheme U n+1 U n t w dx F(U) w dx + 3 =1 e F e νw ds = s(h, b)w dx. (2.6) Total varaton dmnshng (TVD) hgh order Runge-utta tme dscretzaton [31] s used n practce for stablty and to ncrease temporal accuracy. For example, the thrd order TVD Runge-utta method s used n the smulaton n ths paper: U (1) = U n + tl(u n ) (2.7) U (2) = 3 4 U n + 1 ( U (1) + tl(u (1) ) ) 4 U n+1 = 1 3 U n + 2 ( U (2) + tl(u (2) ) ), 3 where L(U) s the spatal operator. In order to acheve the well-balanced property, we are nterested n preservng the stll water statonary soluton (1.2) exactly. Followng the technque presented n [34], our wellbalanced numercal scheme, modfed from the classcal verson (2.6), takes the form: U n+1 U n t or equvalently, = w dx F(U) w dx+ 3 =1 e F e νw ds = s(h, b)w dx, (2.8) U n+1 U n 3 w dx F(U) w dx + F t e =1 e νw ds 3 s(h, b)w dx + ( F F ) e νw ds (2.9) =1 e The left sde of (2.9) s the classcal RDG scheme, and the rght sde s our approxmaton to the source term. The flux F s computed based on the hydrostatc reconstructon technque [1] and wll be explaned later. However we pont out here that the dfference 6

8 F F s a hgh order correcton term at the level of O(h k+1 ) regardless of the smoothness of the soluton U. Therefore, the scheme (2.8) s a spatally (k + 1)-th order conservatve scheme and wll converge to the weak soluton. After computng boundary values U nt() ( h,nt() = max h,ext() = max, h nt() and U ext() + b nt() (, h ext() + b ext() on the edge e, we set ) max(b nt(), b ext() ) max(b nt(), b ext() ) ) (2.1) and redefne the nteror and exteror values of U as: Introducng the notatons U,nt() = U,ext() = h,nt() h,nt() h,nt() u nt() v nt() h,ext() h,ext() h,ext() u ext() v ext() ( δ,x =, g 2 (hnt() ) 2 g ) T 2 (h,nt() ) 2,, δ on the edge e, the flux F s then gven by: = h,nt() h nt() = h,ext(),y = ( h ext() U nt(),,, g 2 (hnt() U ext(), (2.11) ) 2 g ) T 2 (h,nt() ) 2 F e ν = F(U,nt(), U,ext(), ν) + δ,x, δ,y ν. (2.12) We also requre that all the ntegrals n formula (2.8) should be calculated exactly at the stll water state. Ths can be easly acheved by usng sutable Gauss-quadrature rules snce the numercal solutons h, b and w are polynomals at the stll water state n each trangle, hence F(U) and s(h, b) are both polynomals. We can prove that the above methods (2.8), combned wth the choce of fluxes (2.12), are actually well-balanced for the stll water steady state of the shallow water equatons. The key dea s to show that, at the stll water steady state (1.2), the numercal fluxes F becomes F(U nt() ) on the edge e. We refer to [34] for the techncal detals of the proof. When appled to problems whch contan dscontnuous soluton, RDG methods may generate oscllaton and even nonlnear nstablty. We often apply nonlnear lmters to 7

9 control these oscllatons. Many lmters have been studed n the lterature. In ths paper, we use the classcal characterstc-wse total varaton bounded (TVB) lmter n [12, 3], wth a corrected mnmod functon defned by { a1, f a m(a 1,, a n ) = 1 M x 2, m(a 1,, a n ), otherwse, (2.13) where M s the TVB parameter to be chosen adequately [11] and the mnmod functon m s gven by { s mn a m(a 1,, a n ) =, f s = sgn(a 1 ) = = sgn(a n ),, otherwse. Usually, the lmter s appled on the functon U after each nner stage n the Runge-utta tme steppng. For the shallow water system, we perform the lmtng n the local characterstc varables. However, ths lmter procedure mght destroy the preservaton of the stll water steady state h + b = const, snce f the lmter s enacted, the resultng modfed soluton h may no longer satsfy ths steady state relaton. Therefore, followng the dea presented n [33, 36], we present the followng strategy to perform the lmter, whch works well wth the well-balance property. As explaned n [36], we note that the TVB lmter procedure actually nvolves two steps: the frst one s to check whether any lmtng s needed n a specfc cell; and, f the answer s yes, the second step s to apply the TVB lmter on the varables n ths cell. We frst check f the lmtng s needed, based on (h + b, hu, hv) T. If a certan cell s flagged by ths procedure needng lmtng, then the actual TVB lmter s mplemented on the varables (h, hu, hv) T. Note that f n a steady state regon where h + b = const and u = v =, we frst check f the lmtng s needed based on (h + b, hu, hv) T = (const,, ) T, whch demonstrates that lmtng s not needed n ths cell. Therefore the flat surface h + b = const wll not be affected by the lmter procedure and the well-balanced property s mantaned. Also, we observe that ths procedure wll not destroy the conservatvty of h, whch wll be mantaned durng the lmter process. When the lmtng procedure s mplemented ths way, numercal results show that ths choce of the TVB lmter does not destroy the well-balanced property, and also t works well wth the 8

10 postvty-preservng lmter presented n the next secton. 3 Postvty-preservng lmter In ths secton, we present a smple postvty-preservng lmter on trangular meshes, and couple t wth the well-balanced DG methods developed for the shallow water equatons n Secton 2. We wll start by showng the postvty of a frst order scheme wth the well-balanced flux, and later generalze the dea to hgh order schemes. For the ease of presentaton, Euler forward tme dscretzaton (2.8) wll be dscussed, but all the results hold for the TVD hgh order Runge-utta and mult-step tme dscretzatons. 3.1 Prelmnares For convenence, let F 1 and F 1 e ν denote the frst components of F and F e ν respectvely. Then F 1 e ν = F 1(U,nt(), U,ext(), ν ) by (2.12). Takng the test functon as w 1 n (2.8), we get the the scheme satsfed by the cell averages for the water heght h: h n+1 = h n t 3 F 1 (U,nt(), U,ext(), ν ) ds, (3.1) where h n stands for the average of h over the trangle at tme level n. =1 e Suppose we use L-pont Gaussan quadrature for the lne ntegral n (2.8) and (3.1), and the subscrpt (, β) wll denote the pont value at the β-th quadrature pont of the -th edge. Let w β denote the Gauss quadrature weght on the nterval [ 1/2, 1/2]. Then (3.1) becomes h n+1 = h n t 3 L F 1 (U,nt(),β, U,ext(),β, ν )w β l. (3.2) 3.2 Frst order schemes =1 β=1 To nvestgate the postvty of a hgh order scheme (3.2), we need to study ts frst order counterpart. Gven pecewse constants U n for the soluton and b for the bottom on each trangle at tme level n, consder a frst order scheme for the water heght, h n+1 = hn t 3 F 1 (U,nt(), U,ext(), ν )l, (3.3) =1 9

11 where wth U,nt() = h, U h, n n U,ext() = h, () U h (), n n () h, = max (, h n + b max(b, b () ) ), h, () = max (, h n () + b () max(b, b () ) ), Lemma 3.1: Under the CFL condton t α 3 l =1 1, wth ( α = max ( u + gh, v + gh) ν ), (3.4) f h n s non-negatve for any, then hn+1 s non-negatve n the frst order scheme (3.3). Proof: By (2.4), the flux n (3.3) s F 1 (U,nt(), U,ext(), ν) [ = 1 h, (hu) n 2 h, (hv) n ν n + h, And the scheme (3.3) can be wrtten as [ h n+1 = 1 1 t 3 2 =1 + 1 t 3 l 2 =1 () h n () l ] (hu) n (), (hv) n () ν α(h, () h, ). h, h n h, () h n () ( ) ] α + u n, v n ν h n ( ) α u n (), v() n ν h n () (3.5) Notce that h, /hn, h, () /hn () [, 1]. And we have un, vn ν α for any by (3.4). Therefore, (3.5) s a lnear combnaton of h n and hn () wth non-negatve coeffcents. Thus, h n+1 s non-negatve f hn and hn () are non-negatve. 3.3 Hgh order schemes Followng the approach n [39], the frst step s to decompose the cell average h n as a convex combnaton of pont values of the DG polynomal h (x, y) by a quadrature satsfyng: The quadrature rule s exact for ntegraton of h (x, y) on. 1

12 The quadrature ponts nclude all L-pont Gauss quadrature ponts for each edge e. All the quadrature weghts should be postve. Ths partcular quadrature rule can be constructed by a transformaton of the tensor product of M-pont Gauss-Lobatto and L-pont Gauss quadrature, whch s summarzed below (see [39] for detals). Let {v β : β = 1,, L} denote the Gauss quadrature ponts on [ 1, 1 ] wth weghts 2 2 w β, and {û α : α = 1,, M} denote the Gauss-Lobatto quadrature ponts on [ 1 2, 1 2 ] wth weghts ŵ α. In the barycentrc coordnates, the set of quadrature ponts S can be wrtten as S = {( vβ, ( ûα )( 1 2 vβ ), ( 1 2 ûα )( 1 ) 2 vβ ), ( ( 1 2 ûα )( 1 2 vβ ), vβ, ( ûα )( 1 ) 2 vβ ), (( 12 + ûα )( 12 vβ ), ( 12 ûα )( 12 vβ ), 12 ) } + vβ : α = 1,, M; β = 1,, L. (3.6) In partcular, for the P 2 -DG method used n numercal tests of ths paper, 4-pont Gauss quadrature rule s needed so that the lne ntegral n (2.8) s exactly calculated for the stll water state. And the 3-pont Gauss-Lobatto quadrature s suffcent to construct S. See Fg 3.1 for the quadrature ponts. Let h (x, y) denote the DG polynomal for the water heght at tme level n and w x denote the quadrature weght for the pont x S of the quadrature rule (3.6). Let h nt(),β denote the pont value of h (x) at the β-th Gauss quadrature pont of the -th edge of. Then the quadrature weght for h nt(),β s 2w β ŵ 1 /3, see [39] for the detal. The cell average h n can now be wrtten as a convex combnaton of pont values of h (x) va the quadrature rule S, h n = h (x) dx = h (x)w x = x S 3 L =1 β=1 2 3 w βŵ 1 h nt(),β + h (x)w x, (3.7) x S 11

13 Fgure 3.1: The quadrature ponts on a trangle for P 2 polynomals. There are 24 dstnct ponts. Three ponts near the centrod of the trangle are very close to one another. where S s the set of the ponts n S that le n the nteror of the trangle. Theorem 3.2: For the scheme (3.2) to be postvty preservng,.e., h n+1 condton s that h (x), x S for all, under the CFL condton α t, a suffcent 3 l 2 (3.8) 3ŵ1. =1 Here h (x) denotes the polynomal for water heght at tme level n, ŵ 1 s the quadrature weght of the M-pont Gauss-Lobatto rule on [ 1/2, 1/2] for the frst quadrature pont. For k = 2, 3, ŵ 1 = 1/6 and for k = 4, 5, ŵ 1 = 1/12. Proof: Rewrte the scheme (3.2) as h n+1 = h n t = h n t 3 L =1 β=1 L β=1 w β ( 3 =1 F 1 (U,nt(),β F 1 (U,nt(),β, U,ext(),β, ν)w β l, U,ext(), ν)l,β ). (3.9) 12

14 Then decompose the flux term nsde the bracket. Let 3 =1 F 1 (U,nt(),β, U,ext(), ν)l,β = F 1 (U,nt() 1,β, U,ext() 1,β, ν 1 )l 1 + F 1 (U,nt() 2,β, U,ext() 2,β, ν 2 )l 2 + F 1 (U,nt() 3,β, U,ext() 3,β, ν 3 )l 3 = F 1 (U,nt() 1,β, U,ext() 1,β, ν 1 )l 1 + F 1 (U,nt() 1,β, U,nt() 2,β, ν 1 )l 1 +F 1 (U,nt() 2,β, U,nt() 1,β, ν 1 )l 1 + F 1 (U,nt() 2,β, U,ext() 2,β, ν 2 )l 2 + F 1 (U,nt() 2,β, U,nt() 3,β, ν 3 )l 3 +F 1 (U,nt() 3,β, U,nt() 2,β, ν 3 )l 3 + F 1 (U,nt() 3,β, U,ext() 3,β, ν 3 )l 3, (3.1) where we have used the conservatvty of the flux (2.5). Pluggng (3.7) and (3.1) nto (3.9), we get the monotone form h n+1 = 3 L =1 β=1 = 2 3 w βŵ 1 h nt(),β + h (x)w x t x S L x S h (x)w x + β=1 where H 1,β, H 2,β, and H 3,β are L β=1 w β ( 3 =1 F 1 (U,nt(),β, U,ext(), ν)l 2 3 w βŵ 1 [H 1,β + H 2,β + H 3,β ], (3.11) H 1,β = h nt() 1,β 3 t [ ] F 1 (U,nt() 1,β, U,ext() 1,β, ν 1 2ŵ 1 )l 1 + F 1 (U,nt() 1,β, U,nt() 2,β, ν)l 1 1, H 2,β = h nt() 2,β 3 t 2ŵ 1 + F 1 (U,nt() 2,β, U,nt() 3,β, ν 3 )l 3 H 3,β = h nt() 3,β 3 t 2ŵ 1 [ F 1 (U,nt() 2,β, U,nt() 1,β, ν)l F 1 (U,nt() 2,β, U,ext() ],β 2,β, ν 2 )l 2, [ F 1 (U,nt() 3,β, U,nt() 2,β, ν 3 )l 3 + F 1 (U,nt() 3,β, U,ext() 3,β, ν 3 )l 3 Followng Lemma 3.1, under the CFL condton (3.8), H 2,β f h nt(),β, h ext(),β. The postvty of H 1,β and H 3,β follows the analyss of one-dmensonal frst order postvtypreservng methods presented n [36, Lemma3.1]. Therefore, f all the pont values nvolved n (3.11), h nt(),β, h ext(),β and h (x) for x S are non-negatve, whch s equvalent to h (x), x S for all, then we have the postvty h n+1 n (3.11). Remark 3.3 As mentoned n [35, 38], for those ponts n S, nstead of requrng h (x), x S, t suffces to requre x S h (x)w x to have postvty of h n+1 13 ]. n (3.11). No- )

15 tce that h (x)w x / w x s a convex combnaton of pont values of h (x), thus by the x S x S Mean Value Theorem, there exsts some pont x such that h (x 1 ) = w x h (x)w x. x S x S By (3.7), we have ( ) h (x L ) = h (x)w x = h n ŵ 1 3 w βŵ 1 h nt(),β, (3.12) x S =1 β=1 x S w x where we use the fact w x = 1 3 L x S =1 β=1 ntutve suffcent condton for (3.2) to satsfy h n+1 wth the CFL condton (3.8), where h (x ) s defned n (3.12). 3.4 The lmter 2 w 3 βŵ 1 = 1 2ŵ 1. So a more relaxed but less s, h n, h nt(),β, h ext(),β, h (x ) At tme level n, gven the water heght DG polynomal h (x) wth ts cell average h n, to enforce the suffcent condton h (x), x S, the lmter n [39] can be used drectly,.e., replacng h (x) by a lnear scalng around the cell average: h (x) = θ (h (x) h n ) + h n, (3.13) where θ [, 1] s determned by θ = mn x S θ x, θ x = mn { 1, h n h n h (x) }. (3.14) Ths lmter s conservatve (the cell average of p s stll h n ), postvty-preservng ( h (x), x S ) and hgh order accurate. See [37, 39, 36] for the dscusson of the lmter. An alternatve lmter s to enforce the relaxed condton n Remark 3.3. Let S denote the ponts n S whch le on the edges of, then we can use (3.13) wth { } { } h n θ = mn θ x, mn θ x, θ x = mn 1, x S h n. (3.15) h (x) Compared to (3.14), evaluatng h (x), x S s avoded n (3.15) snce h (x ) can be obtaned by (3.12), whch s preferred snce these pont values are not Gaussan quadrature ponts on a trangle thus not used n the DG scheme (2.8). 14

16 Notce that the postvty-preservng lmter (3.13) or (3.14) wll not take any effect f the DG polynomals satsfy (1.2). So the postvty-preservng lmter does not affect the well-balanced property. 3.5 The algorthm for Runge-utta tme dscretzatons At the end, we present the algorthm flowchart of our postvty-preservng well-balanced methods, when coupled wth thrd order TVD Runge-utta methods. Frst of all, one notces that, for Euler forward tme dscretzaton, the CFL constrant (3.8) s suffcent rather than necessary for preservng postvty. Second, for a Runge-utta tme dscretzaton, to enforce the CFL condton rgorously, we need to obtan an accurate estmaton of (3.4) for all the stages of Runge-utta based only on the numercal soluton at tme level n, whch s very dffcult n most of test examples. So an effcent mplementaton s, f a prelmnary calculaton to the next tme step produces negatve water heght, we restart the computaton from the tme step n wth half of the tme step sze. The algorthm of postvty-preservng well-balanced dscontnuous Galerkn method wth the thrd order TVD Runge-utta tme dscretzaton on trangular meshes can be summarzed as below: 1. Gven the DG polynomals U (x) at tme step n satsfyng the cell average of h s non-negatve and h (x) >, x S, calculate α = max ( ( u + gh, v + gh) ν ), where the maxmum of u, v, h s taken over S and the maxmum of ν s taken over ν for all. Set the tme step t = mn 2 ŵ 1 3 α 3. =1 l 2. Calculate the frst stage wth U (x) based on (2.8) wth the numercal fluxes (2.12). Let U 1 (x) denote the soluton of the frst stage. Modfy t by frst the TVB lmter then the postvty lmter (3.13) or (3.14) nto Ũ 1 (x). 15

17 3. Calculate the second stage wth Ũ 1 (x). Let U 2 (x) denote the soluton of the second stage. If ts cell average of water heght s negatve (by Theorem 3.2, ths means that α calculated based on U (x) s smaller than the one of Ũ 1 (x)), then go back to step two and restart wth half tme step; otherwse, modfy t by the lmters nto Ũ 2 (x). 4. Calculate the thrd stage wth Ũ 2 (x). Let U 3 (x) denote the soluton of the thrd stage. If ts cell average of water heght s negatve (by Theorem 3.2, ths means that α calculated based on U (x) s smaller than the one of Ũ 2 (x)), then go back to step two and restart wth half tme step; otherwse, modfy t by the lmters nto Ũ 3 (x), whch s the soluton at tme step n Numercal examples In ths secton we present numercal results of our postvty-preservng well-balanced DG methods when appled to the two-dmensonal shallow water equatons on unstructured trangular meshes. The thrd order fnte element DG method (.e. k = 2), coupled wth the thrd order TVD Runge-utta tme dscretzaton (2.7), s mplemented n the examples. Global Lax-Fredrchs numercal flux s used, and the gravtaton constant g s fxed as m/s 2. The tme step s taken as ndcated by the CFL condton (3.8). All the meshes are unstructured, and generated by EasyMesh [24]. 4.1 Well-balanced test The purpose of the frst test problem s to verfy the well-balanced property of our algorthm towards the steady-state soluton. We consder a rectangular computatonal doman [, 1] [, 1]. The bottom functon s chosen as: b(x, y) = max (, 1 (1x 5) 2 (1y 5) 2), (4.1) and the ntal data s the statonary soluton: h(x, y, ) = 2 b(x, y), hu(x, y, ) = hv(x, y, ) =. 16

18 A perodc boundary condton s used. Ths steady state should be exactly preserved, and the surface should reman flat. We compute the soluton untl t =.5 on the trangular meshes wth the mesh sze τ =.25. In order to demonstrate that the stll water soluton s ndeed mantaned up to round-off error, we use the double-precson to perform the computaton, and show the L 1 and L errors for the water heght h (note: h n ths case s not a constant functon!) and the dscharges hu, hv n Table 4.1. We can clearly see that all errors are at the level of round-off errors, whch verfes the well-balanced property. Table 4.1: L 1 and L errors for the statonary soluton n Secton 4.1. L 1 error L error h hu hv h hu hv 4.72E E E E E E Accuracy test In ths example we wll test the hgh order accuracy of our schemes when appled to the followng two dmensonal problem. The computatonal doman s set as a unt square [, 1] 2. The bottom topography and the ntal data are gven by: ( ) 2π b(x, y) = sn 2 (x + y), h(x, y, ) = 5 + e cos( 2π(x+y)), 2 (hu)(x, y, ) = (hv)(x, y, ) = 2 ( ( )) 2 sn cos 2π(x + y), whch are obtaned by rotatng the setup of the one-dmensonal accuracy test n [32, 35] by an angle of 45 degrees. Perodc boundary condtons are consdered here for smplcty. The fnal tme s set as t =.5 to avod the appearance of shocks n the soluton. Snce the exact soluton s also not known explctly for ths case, we use the one-dmensonal well-balanced DG methods presented n [34], wth a refned 12, 8 unform cells, to compute a reference soluton. After rotatng that soluton by an angle of 45 degrees, ths reference soluton s treated as the exact soluton n computng the numercal errors. The TVB constant M n the 17

19 TVB lmter (2.13) s taken as 1 here. The computatonal doman, and the unstructured trangular mesh wth τ =.1,.25, are shown n Fgure 4.1. Table 4.2 contans the L 1 errors and orders of accuracy for the cell averages. We can clearly see that, n ths two dmensonal test case, thrd order accuracy s acheved for the RDG scheme Fgure 4.1: The computatonal doman and unstructured trangular meshes of the accuracy test problem n Secton 4.2. Left: wth mesh sze τ =.1; Rght: wth mesh sze τ =.25. Table 4.2: FV scheme: L 1 errors and numercal orders of accuracy for the example n Secton 4.2. Meshsze h hu hv τ L 1 error order L 1 error order L 1 error order E E E E E E E E E E E E E E E-2 1.3E E E E E E E

20 4.3 A small perturbaton of a steady-state soluton Ths s a classcal example to show the capablty of the proposed scheme for the perturbaton of the statonary state. Ths test was gven by LeVeque [22], and has also been used n [32, 34, 8]. We solve the system n the rectangular doman [, 2] [, 1]. The bottom topography s an solated ellptcal shaped hump: b(x, y) =.8 exp( 5(x.9) 2 5(y.5) 2 ). (4.2) The ntal condton s gven by: h(x, y, ) = hu(x, y, ) = hv(x, y, ) =. { 1 b(x, y) +.1, f.5 x.15, 1 b(x, y), otherwse, (4.3) Intally, the surface s almost flat except for.5 x.15, where h s perturbed upward by a small magntude of.1. Theoretcally, ths dsturbance should splt nto two waves, propagatng left and rght at the characterstc speeds ± gh. We use the outlet boundary condton on the left and rght, and reflecton boundary condtons on the top and bottom sdes. TVB constant M s taken as 1 n the test. Fgure 4.2, left, dsplays the rght-gong dsturbance as t propagates past the hump on the trangular meshes wth τ =.625. The surface level h + b s presented at dfferent tmes. The results ndcate that our schemes can resolve the complex small features of the flow very well. As a comparson, we refer to [8, Fg 5] for the output f a non-well-balanced method s used. Next, we ncrease the heght of the bottom topography to reach the water surface. The modfed bottom topography takes the form: b(x, y) = exp( 5(x.9) 2 5(y.5) 2 ). (4.4) The other set up remans the same. We repeat the smulaton and plot the surface level at dfferent tmes n Fgure 4.2, rght. As one can see, the general structure of the soluton s 19

21 well resolved. For small tme when the rght-gong wave does not reach the bottom hump, the surface stays the same for these two smulatons wth dfferent bottom topography. Obvous dfference can be observed as the wave reaches and passes the hump. 4.4 Crcular dam-break problem Ths s a classcal test case for testng the complete break of a crcular dam separatng a basn of water and dry bed. It has been prevously tested n [6, 16]. We consder a square computatnal doman [ 1, 1] [ 1, 1] wth a flat bottom topography (.e. b = ). The dam s located at r = x 2 + y 2 = 6, and the water heght h s ntally set as 1 nsde the dam and outsde. Both components of the velocty u and v are set to zero ntally. At tme t =, the crcular wall formng the dam collapses. We dscretze the doman wth the trangular meshes and τ s set as 1. A 3D vew and contour lnes of the water heght at tme t = 1.75 are shown n Fgure 4.3. We can observe an almost perfectly symmetrc soluton, and there s no oscllaton (max(h) = 1) n the numercal results. 4.5 Water drop problem Next, we apply our methods to a numercal test case whch smulates the water drop problem. Followng the setup n [28], we consder the 2D Gaussan shaped peak ntal condton gven by: h(x, y, ) = exp ( 1((x.5) 2 + (y.5) 2 ) ), (4.5) hu(x, y, ) = hv(x, y, ) =, n the computatonal doman [, 1] 2. The reflectve boundary condtons are employed. The ntal Gaussan shaped water drop generates a wave that reflects off the boundary. We have provded the evoluton of water surface at varous tmes n Fgure 4.4, whch shows that the wave s well smulated by our methods. As a comparson, we also repeat the test wth a 2

22 1 Surface level at tme t=.12 1 Surface level at tme t= Surface level at tme t=.24 1 Surface level at tme t= Surface level at tme t=.36 1 Surface level at tme t= Surface level at tme t=.48 1 Surface level at tme t= Surface level at tme t=.6 1 Surface level at tme t= Fgure 4.2: The contours of the surface level h+b for the problem n Secton unformly spaced contour lnes. From top to bottom: at tme t =.12 from to 1.7; at tme t =.24 from.9999 to 1.1; at tme t =.36 from.9998 to 1.2; at tme t =.48 from to 1.5; and at tme t =.6 from.9999 to 1.1. Left: results wth the bottom (4.2). Rght: results wth the bottom (4.4). 21

23 1 5 Y X Fgure 4.3: Numercal results at tme t = 1.75 of the crcular dam-break problem n Secton 4.4. Left: 3D vew of the surface level; Rght: the contours of the surface level wth 3 unformly spaced contour lnes between and 1. non-zero bottom topography: b(x, y) =.5 exp ( 1((x.75) 2 + (y.5) 2 ) ). (4.6) The results are shown n Fgure 4.5, where we can observe the effect of the bottom on the propagaton of the wave. 4.6 Floodng on a channel wth three mounds In ths test example, we consder the smulaton of a flow through a channel whch contans three mounds on ts bottom [7, 17]. The length of the channel s 75 and wdth s 3. The bottom topography takes the form of where b(x, y) = max(, m 1 (x, y), m 2 (x, y), m 3 (x, y)), (4.7) m 1 (x, y) = 1.1 (x 3) 2 + (y 22.5) 2, m 2 (x, y) = 1.1 (x 3) 2 + (y 7.5) 2, m 1 (x, y) = (x 47.5) 2 + (y 15) 2. 22

24 Fgure 4.4: The water surface level n the water drop problem wth the flat bottom topography at dfferent tmes. 23

25 Fgure 4.5: The water surface level n the water drop problem wth the bottom topography (4.6) at dfferent tmes. 24

26 Intally, the doman s set as dry,.e. h = hu = hv =. We mpose the reflectng boundary condton on the upper and lower boundares n y-drecton. The rght x-boundary s a open-wall outflow boundary. At the left x-boundary, we mpose an nflow of the form: u = 1, v = and the water heght h =.5 for the tme t 3, h = 1 when t 3. We test our well-balanced postvty-preservng methods on ths problem wth trangulaton of mesh sze τ =.5. The numercal results obtaned at dfferent tmes are shown n Fgure Flows n convergng-dvergng channels In the last example, we consder the water flow n an open convergng-dvergng channel. The test s frst dscussed n [19] and recently used n [8]. The gravtaton constant g s taken as 1 n ths test. The computatonal doman s defned on the convergng-dvergng channel of length 3 wth a half-cosne constrcton centered at x = 1.5. It takes the form of [, 3] [ y b (x), y b (x)], where y b (x) = {.5.5(1 d) cos 2 (π(x 1.5)), f x 1.5.5,.5, otherwse, (4.8) and d s the mnmum channel breadth. Two values of the channel breadth d,.9 and.6, are tested n ths example. The computatonal doman wth d =.6 s shown n Fgure 4.7, left. We consder a bottom topography whch conssts of two ellptc Gausssan mounds: b(x, y) = B ( exp( 1(x 1.9) 2 5(y.2) 2 ) + exp( 2(x 2.2) 2 5(y +.2) 2 ) ), (4.9) where B wll be specfed later. Ths topography wth B = 1 s shown n Fgure 4.7, rght. The ntal condtons are gven by: h(x, y, ) = max(, 1 b(x, y)), hu(x, y, ) = 2, hv(x, y, ) =. 25

27 Fgure 4.6: The water surface level n the floodng problem on a channel wth three mounds at dfferent tmes t = 8, 3, 3 and 54 (from top to bottom). Left: 3D vew, color spannng between and 1.2; Rght: the 2D contours wth 3 unformly spaced contour lnes between and

28 We mpose the reflectng boundary condton on the upper and lower boundares n y- drecton. The left x-boundary s set as an nflow boundary wth u = 2 and the rght x-boundary s a zeroth-order outflow boundary Fgure 4.7: Intal setup of the convergng-dvergng channels problem n Secton 4.7. Left: the computatonal doman wth d =.6 and the unstructured trangular meshes wth τ =.25; Rght: the contours of the bottom topography (4.9) wth B = 1 and d =.6. We frst generate the trangular meshes on the rectangle doman [, 3] [.5,.5]. Followng the dea n [8], the trangulaton on the convergng-dvergng channel s obtaned through the mappng { (x, y (1 d) cos 2 (π(x 1.5))y), f x 1.5.5, (x, y) (x, y), otherwse. The resultng unstructured trangular mesh wth τ =.25 s shown n Fgure 4.7, left. Our well-balanced postvty-preservng methods are tested on ths problem. In all tests, the smulatons are carred out on two trangulaton wth mesh szes τ =.25 and 6.25E 3 respectvely. The stoppng tme s set as t = 2. We start wth the parameters d =.9 and B =,.e., a flat bottom. The numercal results are shown n Fgure 4.8, whch agree well wth the solutons n [19, 8]. Next, we keep the computatonal doman (d =.9) and set the bottom topography as B = 1 n (4.8). The two mounds are set to reach the water surface. We repeat our smulatons and show the results n Fgure 4.9. They are n good agreement wth the results shown n [8], and we can conclude that our well-balanced methods capture the complcated soluton well. 27

29 We also modfy the wdth of the channel by settng d =.6. The bottom topography s kept as B = 1. The numercal results are shown n Fgure 4.1, whch also agree well wth the results n [8]. At the end, we ncrease the heght of the mound to B = 2, to smulate the flows through two slands. Our postvty-preservng methods demonstrate to be robust and provde nce results, presented n Fgure Fgure 4.8: The contours of the water surface level of the convergng-dvergng channels problem n Secton 4.7 wth the parameters d =.9 and B =. Results are based on the trangulaton wth mesh szes τ =.25 (left) and 6.25E 3 (rght), respectvely Fgure 4.9: The contours of the water surface level of the convergng-dvergng channels problem n Secton 4.7 wth the parameters d =.9 and B = 1. Results are based on the trangulaton wth mesh szes τ =.25 (left) and 6.25E 3 (rght), respectvely. 5 Concludng remarks Postvty-preservng well-balanced dscontnuous Galerkn methods have been presented n [36] for the shallow water equatons on one-dmensonal and two dmensonal problems wth rectangular meshes. In ths paper, we showed that such methods can be naturally extended to unstructured trangular meshes, wth the ntroducton of a specal quadrature rule from [39]. We have demonstrated that ths postvty-preservng lmter can keep the water heght 28

30 Fgure 4.1: The contours of the water surface level of the convergng-dvergng channels problem n Secton 4.7 wth the parameters d =.6 and B = 1. Results are based on the trangulaton wth mesh szes τ =.25 (left) and 6.25E 3 (rght), respectvely Fgure 4.11: The contours of the water surface level of the convergng-dvergng channels problem n Secton 4.7 wth the parameters d =.9 and B = 2. Results are based on the trangulaton wth mesh szes τ =.25 (left) and 6.25E 3 (rght), respectvely. 29

31 non-negatve under sutable CFL condton, can preserve the mass conservaton, s easy to mplement, and at the same tme does not affect the hgh order accuracy for the general solutons. Extensve numercal examples are provded at the end to demonstrate the wellbalanced property, accuracy, postvty-preservng property, and non-oscllatory shock resoluton of the proposed numercal methods. The proposed methods are hghly parallelzable and our ongong work s to nvestgate ther performance on hgh performance computers. References [1] E. Audusse, F. Bouchut, M.-O. Brsteau, R. len, and B. Perthame. A fast and stable well-balanced scheme wth hydrostatc reconstructon for shallow water flows. SIAM Journal on Scentfc Computng, 25:25 265, 24. [2] D. S. Bale, R. J. LeVeque, S. Mtran, and J. A. Rossmanth. A wave propagaton method for conservaton laws and balance laws wth spatally varyng flux functons. SIAM Journal on Scentfc Computng, 24: , 22. [3] A. Bermudez and M. E. Vazquez. Upwnd methods for hyperbolc conservaton laws wth source terms. Computers and Fluds, 23: , [4] C. Berthon and F. Marche. A postve preservng hgh order VFRoe scheme for shallow water equatons: A class of relaxaton schemes. SIAM Journal on Scentfc Computng, 3: , 28. [5] O. Bokhove. Floodng and dryng n dscontnuous Galerkn fnte-element dscretzatons of shallow-water equatons. Part 1: one dmenson. Journal of Scentfc Computng, 22:47 82, 25. [6] A. Bollermann, S. Noelle, and M. Lukácová-Medvová. Fnte volume evoluton Galerkn methods for the shallow water equatons wth dry beds. Communcatons n Computatonal Physcs, 1:371 44, 21. 3

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