Large Time Step Methods for Hyperbolic Partial Differential Equations

Transcription

1 Large Time Ste Methods for Hyerbolic Partial Differential Eqations Rolf Nygaard Master of Science in Mechanical Engineering Sbmission date: Jne 7 Servisor: Bernhard Müller, EPT Co-servisor: Tore Flåtten, Uavhengig Norwegian University of Science and Technology Deartment of Energy and Process Engineering

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5 Abstract In this thesis we consider elicit finite volme methods that are not limited by the Corant- Friedrichs-Lewy (CFL) condition, referred to as large time ste (LTS) methods. LeVeqe roosed the first LTS method in the 98 s as an etension of the Godnov method. Since then, classic concets in nmerical analysis, sch as total variation diminishing (TVD) schemes, modified eqation and higher order schemes have been etended to LTS, as well as aroimate Riemann solvers, sch as the Roe scheme, the La-Friedrichs scheme and the HLL scheme. At large time stes, LTS methods often yield entroy violating soltions, and oscillations aear de to interacting waves, esecially for systems of eqations. Becase of this redction in robstness, the maimm allowable time ste is in ractice limited for many LTS schemes. We will look at LTS methods from a new angle, by introdcing an artificial fl fnction framework. We show how the fl-difference slitting coefficients and nmerical diffsion coefficient can be evalated nmerically from the artificial fl fnction, which gives s a convenient way of eerimenting with new LTS schemes. In his master s thesis, Solberg develoed a class of LTS schemes, with an inherent mechanism for adding nmerical diffsion. As an etension of this work, we develo a new three arameter LTS scheme, denoted LTS-HLLφ, which is the main original contribtion in this thesis. We roose secial choices of arameters, which aears to give a good trade off between robstness and accracy. Nmerical simlations are erformed on the Brgers eqation and the Eler eqations, assessing the robstness and accracy of the new LTS-HLLφ scheme, comared to more established LTS schemes. i

6 Sammendrag Denne avhandlingen omhandler metoder for høye tidssteg (LTS). Dette er ekslisitte nmeriske metoder som ikke er begrenset av Corant-Friedrichs-Lewy (CFL) betingelsen. LeVeqe innførte begreet å 98-tallet, da han tvidet Godnov-metoden for høye tidssteg. Senere har klassiske begreer som totalvariasjonsforminskende (TVD) skjema, modifisert likning og høyere ordens skjema blitt tvidet til LTS. Flere Riemann løsere har også blitt tvidet til LTS, blant annet Roe-, La-Friedrichs- og HLL-skjemaet. Mange LTS-metoder gir entroi-redserende og oscillerende løsninger når tidsstegene overstiger CFL-betingelsen. Disse fenomenene er sesielt fremtredende for ligningssystemer, fordi de like bølgene overkjører hverandre. I raksis gir disse fenomenene en begrensning for størrelsen å tidsstegene. I denne avhandlingen ser vi å LTS-metoder fra en ny vinkel, når vi introdserer knstige flksfnksjoner. I dette rammeverket viser vi hvordan flksdifferanseslittingskoeffisientene og den nmeriske diffsjonskoeffisienten kan trykkes nmerisk for en gitt knstig flksfnksjon. Dette åner for enkel ekserimentering med nye LTS-skjema i fremtiden. Solberg tviklet en klasse av LTS-skjema i sin masterogave, som har en innebygd mekanisme for å jstere nmerisk diffsjon. Det viktigste originale bidraget i denne avhandlingen er tviklingen av det nye LTS-HLLφ-skjemaet, som er basert å Solbergs skjema. Vi foreslår en metode for å velge arametere for LTS-HLLφ-skjemaet som gir en god balanse mellom robsthet og nøyaktighet. Vi gjennomfører nmeriske simleringer å Brgers likning og Elerlikningene, for å stdere hvor robst og nøyaktig det nye LTS-HLLφ-skjemaet er i forhold til mer etablerte LTS-skjema. ii

7 Preface The figre on the cover of this thesis shows a simlation of the D Brgers eqation, where the seven eaks show the evoltion from the initial data to the sith time ste. We see a Gass crve moving to the left, forming a shock. Soltions were obtained sing the Solberg scheme on a [ ] grid, with a maimm Corant nmber of CFL=. This master s thesis is a contination of my roject work, concerning sorce term treatment in large time ste methods. Hence, chaters and 3 in this thesis are heavily based on content from the reviosly sbmitted reort [7]. I wold like to give a secial thanks to my servisors, Bernhard Müller and Tore Flåtten for ecellent gidance and sort throghot this whole rocess. I wold also like to thank Marin Prebeg and Anders Solberg for fritfl discssions. iii

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9 Contents Introdction. Previos work Otline of this thesis Hyerbolic conservation laws. Characteristic strctre Analytical soltion Shock formation The Riemann roblem Elicit finite volme methods 3. The fl-difference slitting formlation The Godnov method Aroimate Riemann solvers The Roe scheme The HLL scheme The CFL condition Large time ste methods 9 4. Large time ste methods Modified eqation and nmerical diffsion The total variation diminishing condition Some large time ste methods The LTS-Godnov method The LTS-Roe scheme The LTS-La-Friedrichs scheme The LTS-HLL scheme Artificial fl fnctions The artificial fl framework Some relative fl fnctions The Godnov method for relative fl fnctions v

10 4..4 Similarity soltions A new three arameter LTS scheme 9. Similarity soltion The fl-difference slitting coefficients Nmerical diffsion coefficient Parameter stdy Ceiling schemes The Solbergφ scheme The LTS-HLLEφ scheme Nmerical simlations The inviscid Brgers eqation Transonic rarefaction Sqare Plse Doble shock The Eler eqations Toro s test Toro s test Toro s test Toro s test Toro s test Woodward-Colella blast-wave roblem Conclsion Artificial fl fnction framework The new LTS scheme Nmerical simlations Ftre rosects Bibliograhy 78 vi

11 List of Figres. Analytical soltion of the inviscid Brgers eqation for the initial data in (.). Left: The conserved qantity, q is convected in sace from its initial state (solid line) ntil a shock is formed at t = (dashed line). Right: The conserved qantity is constant along characteristic lines. When t >, some oints are ambigosly defined The shock soltion of the inviscid Brgers eqation for the initial data in (.). Left: The conserved qantity, q is convected in sace from its initial state (solid line) ntil a shock is formed at t = (dashed line). The shock is simly translated in sace (dotted line). Right: The conserved qantity is constant along characteristic lines. When t >, the shock follows the dashed characteristic line The rarefaction wave soltion of a Riemann roblem for the inviscid Brgers eqation. Left: The conserved qantity is eeriencing rarefaction (dashed line). Right: The characteristic lines of a rarefaction wave sread into an eansion fan Comtational grid in the finite volme method Illstration of the HLL scheme. The discontinity is slit into two discontinities, traveling at seedss L ands R Illstration of characteristic lines when CFL> The non-dimensional wave seed of the Roe scheme The non-dimensional wave seed of the HLL scheme The non-dimensional wave seed of the Solberg scheme The non-dimensional wave seed of the LTS-HLLφ scheme The nmerical diffsion coefficient of the Solberg scheme as a fnction of Corant nmber Transonic rarefaction. t =., N = Transonic rarefaction. t =., N = vii

12 viii 6.3 Sqare lse. Soltion att =.s obtained at CFL= Sqare lse. Soltion att =.s obtained at CFL= Doble shock. N =, CFL= 4, t =.4s Nmerical soltions of Toro s test sing LTS-Roe withn = Nmerical soltions of Toro s test sing LTS-Roe withn = Nmerical soltions of Toro s test sing LTS-HLLEφ withn = Nmerical soltions of Toro s test sing LTS-HLLEφ withn =.. 6. Nmerical soltions of Toro s test sing Solberg withn = Nmerical soltions of Toro s test sing Solberg with N = Nmerical soltions of Toro s test sing LTS-HLLE withn = Nmerical soltions of Toro s test sing LTS-HLLE withn = Nmerical soltions of Toro s test sing the Solberg scheme with N = Nmerical soltions of Toro s test sing the Solberg scheme with N = Nmerical soltions of Toro s test 3 sing LTS-Roe withn = Nmerical soltions of Toro s test 3 sing LTS-Roe withn = Nmerical soltions of Toro s test 3 sing LTS-HLLEφ with N = Nmerical soltions of Toro s test 3 sing LTS-HLLEφ with N = Nmerical soltions of Toro s test 3 sing Solberg withn = Nmerical soltions of Toro s test 3 sing Solberg withn = Nmerical soltions of Toro s test 4 sing LTS-Roe withn = Nmerical soltions of Toro s test 4 sing LTS-Roe withn = Nmerical soltions of Toro s test 4 sing LTS-HLLEφ withn = Nmerical soltions of Toro s test 4 sing LTS-HLLEφ withn = Nmerical soltions of Toro s test 4 sing Solberg withn = Nmerical soltions of Toro s test 4 sing Solberg withn = Nmerical soltions of Toro s test sing LTS-Roe withn = Nmerical soltions of Toro s test sing LTS-Roe withn = Nmerical soltions of Toro s test sing LTS-HLLEφ withn = Nmerical soltions of Toro s test sing LTS-HLLEφ withn = Nmerical soltions of Toro s test sing Solberg withn = Nmerical soltions of Toro s test sing Solberg withn = Nmerical soltions of density for the Woodward-Colella blast-wave roblem, withn =

13 List of Tables. Different LTS schemes eressed as secial cases of the LTS-HLLφ scheme. A dash indicates that the variable is free. E refers to Einfeldt s choice, defined in (3.3) Smmary of test cases for the Brgers eqation Smmary of Toro s tests for the Eler eqations i

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15 Chater Introdction Hyerbolic artial differential eqations (PDEs) are imortant models for many hysical systems, sch as, gas dynamics, meteorology, traffic modelling and geohysics to mention a few [4,, 7, 3]. In this thesis we consider a class of PDEs known as hyerbolic conservation laws q t +f(q) =, whereqis the vector of conserved variables andf(q) is the fl fnction. For scientific and engineering roses, accrately solving roblems of hyerbolic conservation laws, with minimal comtational effort is highly valable. In the ast centry, different nmerical methods have been develoed for solving hyerbolic conservation laws [, 4]. Unfortnately, in most of these methods, there are trade-offs between accracy, stability and comtational effort. Elicit methods are simle to evalate and well sited for arallel comting. However, they often fall short comared to imlicit methods, since elicit methods are tyically limited by the Corant Friedrichs Lewy (CFL) condition [] CFL = t ma λ, where CFL is the largest Corant nmber inherent to the roblem, λ is the -th wave seed, is the length of one comtational cell and t is the time ste. The CFL condition is tyically necessary for stability becase most elicit methods cannot handle convecting conserved qantities more than one comtational cell length dring one time ste. Large time ste (LTS) methods are elicit finite volme methods that are also stable for CFL>. We can rela the CFL condition to a less strict CFL-like condition, CFL k.

16 by allowing the conserved qantity to convect for to k comtational cell lengths. When calclating the vale of the conserved qantity in the net time ste, we now need to consider a stencil of(k +) cells. Most of the LTS methods stdied so far, are not very robst for large time stes. Entroy violations and nhysical oscillations are tyically more freqent at larger time stes, which in ractice limit the maimm allowable time ste for many LTS methods. Althogh diffsive LTS methods are often more robst against these errors, they are also less accrate. This motivates s to search for new LTS methods that are robst, withot being nnecessarily diffsive. This thesis on LTS methods is art of an ongoing research roject, SIMCOFLOW, at SINTEF Materials and Chemistry. The ltimate goal of this roject is to develo a highly efficient oen sorce comter code for comle roblems in mltihase flow.. Previos work The first LTS methods were roosed by LeVeqe in a series of aers in the 98 s [,, 3], where he generalized the Godnov method for arbitrary Corant nmbers. Althogh reslts were romising for scalar eqations, LeVeqe observed nhysical oscillations in the soltions for systems of eqations. Harten later etended the entroy satisfying Harten scheme to LTS, and showed that this method is total variation diminishing (TVD) [7]. Harten also develoed a rocedre for creating large time ste TVD schemes, that are second order accrate away from discontinities. More recently, Lindqist et al. eressed LeVeqe s original LTS-Godnov method in closed form [6]. They defined a LTS etension of the La-Friedrichs scheme, and showed that the LTS-Roe scheme and the LTS-La-Friedrichs scheme are the least and most diffsive(k +)-oint TVD schemes, resectively. From these LTS schemes, they rodced a hybrid scheme, that combines the shar resoltion of the LTS-Roe scheme with the robstness of the LTS-La-Friedrichs scheme. A random time steing was also roosed for redcing entroy violations in the LTS-Roe scheme. The same year, Prebeg et al. develoed a LTS etension of the HLL scheme and the HLLC scheme [9]. Prebeg later showed that the LTS-HLL scheme with Einfeldt s choice of arameters [4], referred to as LTS-HLLE, yield entroy satisfying soltions for all Corant nmbers [8]. In his master s thesis, Solberg roosed a new LTS scheme, with an inherent mechanism for adding nmerical diffsion [3]. He se this method to smear ot the oscillations that LeVeqe reorted in his original articles on LTS methods. Becase of Solberg s romising reslts, a large ortion of this thesis is devoted to investigating and frther develoing this scheme.. Otline of this thesis In chaters and 3, a review of hyerbolic conservation laws and elicit finite volme methods is given. This will give the reader the necessary tools and definitions to tackle the more advanced toics of large time ste methods in chater 4. In this chater we

17 resent the (k + )-oint fl-difference slitting framework, and stdy the nmerical diffsion of a general fl-difference slitting scheme from a modified eqation aroach. In this framework, we briefly discss the total variation diminishing condition, and resent the LTS-Godnov method, the LTS-Roe scheme, the LTS-La-Friedrichs scheme and the LTS-HLL scheme. In section 4., we look at LTS from a different ersective, when we introdce the artificial fl fnction framework. We find the artificial fl fnction of the LTS-Roe scheme and the LTS-HLL scheme, as well as a generalized form of Solberg s CDˆk scheme [3], denoted here as the Solberg scheme. In chater we resent a new three arameter LTS scheme, denoted as LTS-HLLφ, based on the Solberg scheme and the LTS-HLL scheme. This new scheme is the main original contribtion in this thesis. Using the concets resented in the revios chaters, we derive the fl-difference slitting coefficients of the LTS-HLLφ, and discss how the three arameters affect the nmerical diffsion of the scheme. Nmerical simlations for the inviscid Brgers eqation and the Eler eqations are resented in chater 6. Reslts obtained from the LTS-HLLφ scheme are discssed and comared to reslts obtained from the LTS-Roe scheme. Finally, some conclding remarks and roosals for ftre work are fond in chater 7. 3

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19 Chater Hyerbolic conservation laws Physical henomena in flid dynamics, sch as, transort of mass, momentm and energy are stdied by considering the change of conserved qantities over a control volme Ω. A qantity, q, is conserved if the change of q inside Ω eqals the net fl of q throgh the bondaries of the control volme, Ω. In general, the fl of one conserved qantity can deend on other conserved qantities. A system of m conserved qantities, is then governed by the hyerbolic conservation law, t Ω qdv + Ω f(q) nds =, (.) where q(,t) = [q,...,q m ] T is the vector of conserved variables and f(q) n is the fl of q throgh Ω. We will refer to (.) as the integral form. Assming that q is differentiable, we obtain the differential form q t + f(q) =, (.) by alying the divergence theorem. In one sace dimension, the differential form redces to q t +f(q) =. (.3) In the rest of this thesis we will only consider one-dimensional hyerbolic artial differential eqations of the form (.3). However, many of the concets discssed in the thesis are etendable to mltile dimensions, e.g. throgh dimensional slitting [].. Characteristic strctre Using the chain rle, we can rewrite (.3) in the qasi-linear form q t +J(q)q =, (.4)

20 where J(q) = f q (q) is the Jacobian matri of the fl fnction f(q). A roerty of systems of hyerbolic artial differential eqations is that the Jacobian matri is diagonalizable with real eigenvales []. We can therefore writej(q) on the from J(q) = RΛR, (.) wherer = ( r,..., r m) is the matri of right eigenvectors, wherer is the-th right eigenvector ofj(q). The eigenvale matri,λ, is defined as λ λ Λ = (.6) , λ m where λ is the-th eigenvale ofj. Pre-mltilying (.4) by R gives the characteristic form w t +Λw =, where w = [w,...,w m ] T is the vector of characteristic variables, defined by w = R q. (.7) (.8) Becase the rows in Λ only have one non-zero comonent, we have now transformed the system ofmcoled eqations intomncoled, scalar eqations of the form w t +λ w =, (.9) for all =,...,m. We can se this transformation to aly theory develoed for scalar eqations on systems of eqations. Note that for linear systems, we can se (.8) to reconstrctqas the rodct of the matri of right eigenvectors,rand the vector of characteristic variables,w: m q = w r. (.). Analytical soltion Let s consider a scalar conservation law on the qasi-linear form (.4), = q t +λ(q)q =, whereλ(q) = f q. We define a characteristic line,x(t), and note that (.) d (.) dt q(x(t),t) = q t(x(t),t)+ dx(t) q (X(t),t). dt We observe that the left hand side of (.) is eqivalent to the right hand side of (.) if dx(t) dt = λ(q). Ths, on any characteristic line X(t) = + λ(q)t, the vale of q will remain constant becase d q(x(t),t) =. This means that for any given initial vale dt q(,), the soltion at time t is simly a translation in -direction at seed λ(q(,)), in other words, q(,t) = q( λ(q(,))t,). (.3) 6

21 q t t = t = Figre.: Analytical soltion of the inviscid Brgers eqation for the initial data in (.). Left: The conserved qantity, q is convected in sace from its initial state (solid line) ntil a shock is formed at t = (dashed line). Right: The conserved qantity is constant along characteristic lines. Whent >, some oints are ambigosly defined..3 Shock formation Unfortnately, we cannot always se the analytical soltion (.3) for a general nonlinear conservation law, de to the henomenon known as shock formation. We will illstrate this by an eamle. Given the inviscid Brgers eqation q t +qq =, (.4) and the initial data if <, q(,) = if, if >, (.) we can draw characteristic lines X(t) = +q(,)t, and solve the roblem sing (.3) as long as t <, as illstrated in figre.. If we naively etend the characteristic lines beyond t =, we see that some oints are defined by more than one characteristic line, ths the vale ofq(,t) is ambigos in these oints. This haens whenever the soltion form a discontinity. We get an ambigos reslt, becase the differential eqation (.4) is no longer defined for discontinos data. However, as we will see in the net section, we can find meaningfl weak soltions for the ambigos oints. A good way of nderstanding weak soltions is by stdying the Riemann roblem. 7

22 .4 The Riemann roblem A Riemann roblem is an initial vale roblem consisting of a conservation law of the form (.3) and initial data { q L if < q(,) = (.6) q R if >. where the vectorsq L and q R are constant. As illstrated in the revios section, sing the inviscid Brgers eqation (.4), differential eqations are not defined when the initial data is discontinos. In order to solve the Riemann roblem at =, we need to consider weak soltions of the conservation laws, that satisfies the corresonding integral eqation. An imortant weak soltion is the shock soltion. A shock is characterized by a discontinity moving at seed s in sace, withot changing shae (see figre.). We can find the seed of sch a discontinity by integrating (.6) over a sfficiently large interval Ω = [ l,l]. This gives l l qd+ f(q)d =, t l l (.7) d dt ((l shock)q R +( shock +l)q L )+f(q R ) f(q L ) =, where shock is the osition of the discontinity at any given time. Becase dl dt eression simly redces to s(q R q L ) = f(q R ) f(q L ), (.8) =, the (.9) where s = d shock is the seed of the discontinity. In the scalar case, we can find this dt seed by simly dividing by(q R q L ), which yields s = f(q R) f(q L ) q R q L. (.) For a linear system of m eqations, the Jacobian matri is constant, and we can write (.9) as s(q R q L ) = J(q R q L ), (.) which imly that the scalar s is an eigenvale of J. By writing q on the form (.) we can define waves as m q R q L = (w R w L )r = m = W, = 8 (.)

23 q t t = t = t = Figre.: The shock soltion of the inviscid Brgers eqation for the initial data in (.). Left: The conserved qantity, q is convected in sace from its initial state (solid line) ntil a shock is formed at t = (dashed line). The shock is simly translated in sace (dotted line). Right: The conserved qantity is constant along characteristic lines. When t >, the shock follows the dashed characteristic line. wherew is the -th wave. Using these definitions, we find that J(q R q L ) = RΛR R(w R w L ) m = λ W. = (.3) Hence, the discontinity slits intomdiscontinities, or wavesw, traveling at seeds λ. Althogh the shock soltion is a valid weak soltion of the Riemann roblem, it is not necessarily a niqe soltion. Since inviscid conservation laws often describe hysical systems where some viscosity is resent, we are interested in the niqe soltion of the corresonding viscos roblem with diminishing viscosity. For a scalar conservation law, the diminishing viscosity roblem is given by q t +f(q) = νq, (.4) where the viscosity coefficient ν. Becase soltions of (.4) ensre non-decreasing entroy for the Eler eqations, we will refer to them as entroy satisfying soltions. It can by shown that the shock soltion is only entroy satisfying when f (q L ) > s > f (q R ), (.) where we assme the fl fnction f(q) is either conve or concave, i.e. f (q) > or f (q) < resectively. This is known as the La entroy condition []. 9

24 q t Figre.3: The rarefaction wave soltion of a Riemann roblem for the inviscid Brgers eqation. Left: The conserved qantity is eeriencing rarefaction (dashed line). Right: The characteristic lines of a rarefaction wave sread into an eansion fan. When the La entroy condition (.) is not satisfied, the entroy satisfying soltion is a rarefaction wave. A rarefaction wave is, as the name sggests, a gradal redction in the density of a conserved qantity, and is associated with eansion in gas dynamics. The characteristic lines of a rarefaction wave are searating from the discontinity, forming an eansion fan in-between, as shown in figre.3. For a system of m eqations, the m new discontinities can be either shocks waves, rarefaction waves or contact waves. The entroy satisfying soltion can be decided by a more generalized version of the La entroy condition [].

25 Chater 3 Elicit finite volme methods Finding an analytical soltion of an initial vale roblem is often very hard, or even imossible. It is therefore desirable to find aroimate nmerical soltions of the roblem. Before we can calclate nmerical soltions, we need to convert the initial vale roblem into a discrete, arithmetic roblem that can be solved by simle oerations. One olar method for discretizing conservation laws on the integral form (.), is to divide the domain into small finite control volmes. Given a one-dimensional control volmeω = [ L, R ], we can integrate (.) in time over the interval[t n,t n+ ]: tn+ R t n L q t ddt+ tn+ R t n L f(q) ddt =. (3.) as By sing the fndamental theorem of calcls and Fbini s theorem, we can write (3.) R tn+ [q(,t n+ ) q(,t n )]d+ [f(q( R,t)) f(q( L,t))]dt =, L t n (3.) Let s now define average vales forqand f as R Q n = q(,t n )d, R L L tn+ F L = f(q( L,t))dt, t n+ t n t n (3.3a) (3.3b) resectively. By sing the new definitions in (3.3), we can write (3.) as Q n+ = Q n t n+ t n R L (F R F L ). (3.4) This eression is valid for any domain, or control volme. In the finite volme method (FVM), we divide the -ais into smaller sbdomains, Ω j = [ j /, j+/ ], where

26 j / j+/ Q j Q j Q j+ F j / F j+/ j j j+ Figre 3.: Comtational grid in the finite volme method. j±/ are the faces of the comtational cell j, as illstrated in figre 3.. We associate every comtational cell with the average vale of the cell, defined in (3.3). For each cell j we have Q n+ j = Q n j t (F j+/ F j / ). (3.) where the grid size = j+/ j / is assmed to be niform. Eqation (3.) imlies that the change in Q j dring one time ste is eqal to the net fl entering the comtational cell from the neighboring cells dring the time ste. We now need a way of aroimating the fles at the interfaces. Elicit finite volme methods only se vales from the revios time ste to aroimate the fles, which imlies that we can calclate Q n+ j elicitly. Many of the classic elicit schemes only consider the neighboring cells in the revios time ste, which means that the nmerical fl fnction, F n j / = F(Qn j,q n j), (3.6) only deends on the two comtational cells that are sharing the interface at j /. We will refer to sch schemes as 3-oint schemes, becase Q n+ j deends on the vales in three comtational cells. 3. The fl-difference slitting formlation Looking at figre 3., the vale of Q is iecewise constant fnction, with discontinities at the interfaces between cells. Becase of these discontinities, we need to solve a Riemann roblem on every interface. We can think of the discontinities as waves entering the cell from the interfaces. This wave descrition of the finite volme method (3.) will be referred to as the fl-difference slitting formlation. By sing the definition of waves

27 (.), (3.) becomes Q n+ j = Q n j t m = ( ) (λ j / )+ W j / +(λ j+/ ) W j+/ (3.7) where (λ j / )+ = ma(λ j /,) and (λ j+/ ) = min(λ j+/,). For scalar conservation laws, we will write this as Q n+ j = Q n j ( C + j / j / +C j+/ j+/ ), (3.8) where we have introdced the shorthand j / = Q j Q j, C ± j / = t λ± j /. (3.9a) (3.9b) We can think of the coefficients C ± as a measre of how far a wave will travel dring one time ste. IfC + j / = this means that the wave from the left interface will travel all the way to the right interface, and C j+/ = means the wave from the right interface will travel all the way to the left interface dring one time ste. 3. The Godnov method As discssed in the revios section, we need to solve Riemann roblems on every cell interface in elicit finite volme methods. In 99, Godnov roosed solving these Riemann roblems eactly [], and this method has been named after him. For scalar eqations, the eact niqe entroy satisfying soltion of the local Riemann roblem at the interface j / is obtained sing the nmerical fl fnction min F f(q) ifq j < Q j, q [Q j / = j,q j] ma f(q) ifq j > Q j, q [Q j,q j ] (3.) In the Godnov method, we se this eact nmerical fl fnction in (3.4). Althogh we can now solve any scalar conservation law, the Godnov method is not always the most ractical method, since we need to solve an otimization roblem in order to find F j /. If this cannot be done analytically, we need to se some iterative method (like Newton s method), which can be very costly. For non-linear systems, the Godnov method is even more comle and comtationally costly. 3.3 Aroimate Riemann solvers It is often too comtationally costly to solve the Riemann roblems at the interfaces eactly, and aroimate Riemann solvers are therefore often more eedient. Some relevant aroimate schemes are resented below. 3

28 3.3. The Roe scheme In the Roe scheme [], the Riemann roblems on each cell interface are linearized. At interface j /, the tre Riemann roblem is aroimated by q t +Ĵj /q = (3.) q(,) = { Q j Q j if < j / if > j / (3.) whereĵis a constant matri known as the Roe matri. In order for the linearized roblem to be consistent with the tre Riemann roblem, the Roe matri mst have the following roerties: Proerty. The linearized roblem is hyerbolic, hence,ĵj / has real eigenvales and linearly indeendent eigenvectors. Proerty. WhenQ j = Q j = Q, the Roe matri is consistent with the eact Jacobian matri in the sense, Ĵ j / (Q,Q) = J(Q). (3.3) Proerty 3. The Roe matri is conservative in the sense that, Ĵ j / (Q j Q j ) = f(q j ) f(q j ). (3.4) From roerty, we know that the Roe matri can be written in the form Ĵ j / = R j / Λj / R j /, (3.) where Λ j / is a matri with the Roe seeds, ˆλ j /, in the diagonal. Using (.3), we can write (3.4) as a sm of waves m Ĵ j / (Q j Q j ) = ˆλ j / W j /, = (3.6) giving the following fl-difference slitting coefficients for the Roe scheme: (λ j / )± = ±ma(,±ˆλ j / ), (3.7) or for a scalar eqation where we have introdced the shorthand C ± j / = ±ma(,±ĉ j /), ĉ = t ˆλ. (3.8) (3.9) 4

29 q Q j s R Q HLL j / s L Q j j / Figre 3.: Illstration of the HLL scheme. The discontinity is slit into two discontinities, traveling at seedss L and s R The HLL scheme Harten, La and van Leer [8] roosed aroimating the soltion of the Riemann roblems at the interfaces as Q j ifζ s L, Q(,t) = Q HLL j / if s L ζ s R, (3.) Q j if ζ s R. where ζ = j / t t (3.) which means that the discontinity slit into two waves, moving at seedss L and s R, i.e. s(q j Q j ) = s R (Q j Q HLL j / )+s L(Q HLL j / Q j ). (3.) This is illstrated in figre 3.. For now, let s assme s L and s R to be known. We then need to chooseq HLL j / in sch a way that (3.) is a weak soltion of the hyerbolic conservation law. Given the domain [ L, R ], we can integrate (.) in the time interval [t n,t n + t], R L q(,t n + t)d = R L q(,t n )d tn+ t t n [f( R,t) f( L,t)]dt. (3.3) If L s L t+ j / and R s R t+ j /, we can evalate the integrals on the right hand side sing the cell averages (3.3), R L q(,t n + t)d = R Q j L Q j t(f j F j ). (3.4)

30 By sing the desired soltion in (3.), R L q(,t n + t)d can be slit into three integrals, R L q(,t n + t)d = sl t L Q j d+ sr t s L t Q HLL j / d+ R s R t Q j d. (3.) Comaring this to (3.4), we can show that Q HLL j / = s RQ j s L Q j +F j F j s R s L. (3.6) For a linear system, we can insert this into (3.) and se the wave definition in (.3). We then get ( m λ j / s(q j Q j ) = s L s R + s ) R λj / s L W (3.7) s R s L s R s j /, L = which we can also aly to non-linear systems throgh Roe averaging, described in section We see that the fl-difference slitting coefficients of the HLL scheme are given by ˆλ (λ j / )± j / = ± s L s R s L ma(±s R,)± s R ˆλ j / ma(±s L,), s R s L (3.8) which for scalar conservation laws can be written as C ± j / = ±ĉj / c L c R c L ma(±c R,)± c R ĉ j / c R c L ma(±c L,), (3.9) sing the shorthand c R = t s R, (3.3) The HLLE scheme c L = t s L. (3.3) So far we have not discssed how we chooses L and s R. Einfeldt [4] roosed sing s L,j / = min(λ (Q j ),ˆλ (ˆQ j / )), s R,j / = ma(ˆλ m (ˆQ j / ),λ m (Q j )). (3.3a) (3.3b) which ensres that ˆλ (ˆQ j / ) [s L,s R ],. Einfeldt s choice also has La entroy conditions (.) bilt in, so that the scheme redces to the Roe scheme for thest andmth wave when the entroy satisfying soltion is a shock. We will refer to the HLL scheme with Einfeldt s choice of arameters as the HLLE scheme. 6

31 t t > ma j, ( λ j ) j j+ Figre 3.3: Illstration of characteristic lines when CFL>. 3.4 The CFL condition For simle elicit 3-oint schemes, the maimal allowed time ste is limited by the Corant-Friedrichs-Lewy (CFL) condition []: CFL = t ma j, λ j, (3.33) where CFL is the greatest Corant nmber inherent to the roblem. To get a more intitive nderstanding of the CFL condition, let s consider a scalar eqation whereλ j / > t. In this case, the Roe scheme will give C + j / >, meaning that the wave from the left interface of cell j shold travel throgh the entire cell, and start entering cell j +, as illstrated in figre 3.3. However, in a 3-oint scheme, this is not taken into accont when we erform calclations for cellj +. Becase of the CFL condition we are often forced to se small time stes, which is comtationally costly. Since t, we mst se even smaller time stes when we refine the grid. 7

32 8

33 Chater 4 Large time ste methods Large time ste methods (LTS) are elicit finite volme methods that are not limited by the CFL condition. In this chater we show how the fl-difference slitting formlation (section 3.) can be eanded to allow for higher time stes, and generalize the aroimate Riemann solvers to LTS. For the sake of simle elanation, we only consider scalar eqations in this chater. However, we can etend the LTS schemes to non-linear systems throgh Roe linearization, described in section Large time ste methods If a wave is allowed to travel frther than one cell length dring one time ste, this imlies the nmerical fl fnction mst deend on a wider stencil of cells F j / = F(Q j k,...,q j +k ). (4.) When k =, we have a 3-oint scheme that is limited by the CFL condition. In general, a(k +)-oint scheme is limited by a relaed CFL-like condition CFL = t ma λ j k, j (4.) where λ = df. In fl-difference slitting formlation, the change in the conserved dq qantity is now a sm of all the waves assing throgh or ending inside the cell. In total, a cell can be affected by tok waves from the left, andk waves from the right. The large time ste etension of the fl-difference slitting formlation (3.7) is therefore Q n+ j k = Q n j i= ( ) C i+ j / i j / i +C i j+/+i j+/+i, (4.3) where the coefficient C i+ j / i describes how far the wave from interface j / i has traveled into cell j. Note that for 3-oint schemes (4.3) redces to (3.7) where C ± j / = C ± j /. 9

34 4. Modified eqation and nmerical diffsion Becase a nmerical soltion is often based on aroimated fles and rojected onto a finite nmber of cells, it will generally not satisfy the hyerbolic conservation law eactly. If this is the case, we can find modified eqations that the nmerical soltion satisfies more accrately than the original conservation law. Bore showed that a (k + )-oint elicit finite volme method on the form (4.3), gives a second order accrate aroimation to the modified eqation [], q t +f(q) = t [( N ) ] (i+)(c i+ C i ) C q, i= (4.4) wherec = t f (q) is the local Corant nmber. We can also write the modified eqation in terms of a modified fl fnctionf(q) g(q), where g(q) = t σ(c)q, and σ(c) is the nmerical diffsion coefficient N σ(c) = (i+)(c i+ C i ) C. i= (4.) (4.6) We will refer to the error term on the right hand side of (4.4) as the nmerical diffsion of a scheme, becase it is roortional to q for linear eqations. A nmerical method is then a second order aroimation of a viscos artial differential eqation, where the nmerical viscosity coefficient is ν nm = t σ(c). For more detailed derivations of the modified eqation (4.4), we refer to []. 4.3 The total variation diminishing condition The CFL condition is not sfficient to ensre stability for a general non-linear conservation law. In 983, Harten [6] introdced the stricter total variation diminishing (TVD) condition, TV n+ TV n, (4.7) where TV is the total variation given by TV n = j Q n j Q n j. (4.8) The TVD condition is a strong stability condition, that garantees that a scheme will converge to a weak soltions []. Harten showed that a 3-oint scheme is nconditionally TVD if and only if C + j / C j /. (4.9)

35 This was later generalized for (k + )-oint schemes by Jameson and La [9]. A (k +)-oint scheme is TVD if C (i+)+ j / Ci+ j /, i, C + j / C j /, (4.b) C (i+) j / Ci j /, i. (4.a) (4.c) for allj. 4.4 Some large time ste methods When a LTS scheme redces to a known 3-oint scheme for k =, and they share many characteristics, we say that the LTS scheme is a LTS etension of the 3-oint scheme. A few LTS etension are given in the net sections The LTS-Godnov method The LTS-Godnov method, which is characterized by solving every Riemann roblem eactly, was the first LTS-method sed by LeVeqe in [3]. A closed form formlation of the method was derived by Solberg in his roject work [], and later blished in [6]. The fl-difference slitting coefficients for the LTS-Godnov method are given by [C i+ ] j / = Q j ( ) ( ) t t +M j / f(q) (i+)q M j / f(q) iq, [C i ] j / = Q j ( ) ( ) t t +M j / f(q)+iq M j / f(q)+(i+)q, where the oeratormis defined as min f(q) if Q j < Q j, q [Q M j / (f(q)) = j,q j] ma f(q) if Q j Q j. q [Q j,q j ] (4.a) (4.b) (4.) 4.4. The LTS-Roe scheme The Roe scheme treats the Riemann roblem at every cell interface as a linearized Riemann roblem, and waves are convected nchanged at Roe seed. If a ositive wave from the interface at j / asses throgh cell j +i, this cell is flly affected by the wave, hence C i+ j / =. If the wave ends inside cell j +i, the cell is only artially affected, and we get < C i+ j / = ĉ j / i <. We can erform the same analysis for negative waves, giving the fl-difference slitting coefficients of the LTS-Roe scheme,

36 if i ĉ C i+ = ĉ i if ĉ < i < ĉ if i ĉ if i ĉ C i = ĉ+i if ĉ < i < ĉ if i ĉ (4.3a) (4.3b) which we can write more comactly as C i± = ±ma(,min(±ĉ i,). (4.4) If we insert (4.4) into the eression for the nmerical diffsion coefficient (4.6), we can show that σ Roe (ĉ) = α( α), (4.) where α = ĉ ĉ. Note that σ Roe (ĉ) = when ĉ Z. Lindqist et al. [6] showed that the LTS-Roe scheme is the least diffsive LTS scheme that satisfies the TVD conditions (4.). Note that the 3-oint LTS-Roe scheme redces to the original Roe scheme (3.8), C ± = C ± = ±ma(,±ĉ). (4.6) The LTS-La-Friedrichs scheme The most diffsive LTS scheme that satisfies the TVD conditions (4.), is obtained when C ±i j / = C±(i+) j / for all i and j. The LTS-La-Friedrichs scheme given in fldifference slitting formlation as C i± = k (ĉ±k), (4.7) is the only consistent scheme that satisfies this condition [6]. Here, we differentiate between the local LTS-La-Friedrichs scheme, where k = ĉ, and the global LTS-La- Friedrichs scheme, wherek = CFL. When we insert (4.7) into the eression for the nmerical diffsion coefficient (4.6), we get that σ LF (ĉ) = k ĉ. (4.8) The LTS-HLL scheme The HLL scheme is characterized as a scheme that slits discontinities into two shock waves. A LTS etension of the HLL scheme was recently develoed by Prebeg et al. [9]. The fl-difference slitting coefficients of this scheme are given by:

37 C i± = ± ĉ c L c R c L ma(,min(±c R i,)± c R ĉ c R c L ma(,min(±c L i,)). (4.9) Note how similar this eression is to the corresonding eression for LTS-Roe scheme (4.4). In the limit when c L ĉ and c R ĉ the LTS-HLL scheme redce to the LTS-Roe scheme. If we insert (4.9) into the general eression for the nmerical diffsion coefficient in (4.6), Prebeg showed in [8] that σ HLL (ĉ) = (c R ĉ)(ĉ c L )+ ĉ c L c R c L α R ( α R )+ c R ĉ c R c L α L ( α L ), (4.) whereα L = c L c L and α R = c R c R. 4. Artificial fl fnctions Another way of defining an aroimate Riemann solver is by finding an aroimate eqation on the form, q t + f(q;q L ;q R ) =, (4.) that, when solved eactly by the Godnov method, yields the same soltion as if we solve the eact roblem sing the aroimate Riemann solver. We will refer to f(q;q L ;q R ) as the artificial fl fnction. 4.. The artificial fl framework In order to find schemes that are consistent to the eact roblem, f(q;ql ;q R ) mst have the following roerties: Proerty. The artificial fl fnction is Lischitz continos. Proerty. The artificial fl fnction is consistent with the tre fl fnction in the sense f(q L ;q L ;q R ) = f(q L ), f(q R ;q L ;q R ) = f(q R ). (4.a) (4.b) Proerty 3. The artificial fl fnction is conservative when, qr q L f (q;q L ;q R )dq = f(q R ) f(q L ). (4.3) In the net sections, we will mostly eress the artificial fl fnction, f(q;q L ;q R ), in terms of a relative fl fnctionsf(θ) defined by the relationshi f(θ(q;q L ;q R )) = f L + t F(θ), (4.4) 3

38 where θ = q q L q R q L, (4.) and = q R q L. This formlation is ractical, becase an artificial fl fnction f(q;q L ;q R ) always satisfies roerty and roerty 3 as long as F() =, F() = ĉ. (4.6a) (4.6b) Note that for any relative fl fnctions, the chain rle gives f (q;q L ;q R ) = t F (θ), (4.7) ths, the fnctionf (θ) is a non-dimensional wave seed, whose size determine the nmber of cells a wave will travel in one time ste. 4.. Some relative fl fnctions In (4.4) we defined the relative fl fnctionf(θ). In this section we look at some secial relative fl fnctions and show how these relate to known aroimate Riemann solvers. The Roe scheme If we choosef(θ) to be linear, F(θ) = a +a θ, (4.8) there is only one choice that satisfies (4.6), namelya = anda = ĉ. This corresonds to the artificial fl fnction f(q;q L ;q R ) = ˆλq, (4.9) which is the same linearization as in the Roe scheme. The Roe scheme has a constant non-dimensional wave seed, as illstrated in 4.. The HLL scheme Another interesting choice of F is a continos fnction consisting of two iecewise linear fnctions F(θ) = { c L θ ifθ θ c L θ +c R (θ θ ) ifθ > θ = c L θ +(c R c L )ma(,θ θ ), (4.3) where c L and c R are the sloes of the two linear fnctions and θ is the oint of intersection. When enforcing the consistency condition (4.6) we get that 4

39 F (θ) ĉ θ Figre 4.: The non-dimensional wave seed of the Roe scheme θ = c R ĉ c R c L. (4.3) which corresonds toq HLL in (3.6) since q = q L +(q R q L )θ = s Rq R s L q L +f(q L ) f(q R ) s R s L. (4.3) The non-dimensional wave seed of the HLL schemes consist of two iecewise constant vales, F (θ) = c L +(c R c L )H(θ θ ), (4.33) whereh() is the Heaviside fnction. This is illstrated in figre 4.. The Solberg scheme Solberg [3] roosed a LTS scheme in his master s thesis, which he denoted the Constant- Diffsion-ˆk (CDˆk) scheme. We will now eress a similar scheme in artificial fl formlation, denoted here as the Solberg scheme. Let the relative fl fnctionf(θ) be a second order olynomial, F(θ) = a +a θ +a θ. (4.34)

40 F (θ) c R c L θ Figre 4.: The non-dimensional wave seed of the HLL scheme θ When enforcing the consistency conditions (4.6), we find that a =, a = ĉ a and a = ˆk, where ˆk is a free arameter. With these restrictions, we get a one arameter scheme F(θ) = (ĉ ˆk)θ +ˆkθ, (4.3) with the non-dimensional wave seed F (θ) = ĉ ˆk +ˆkθ, (4.36) which is illstrated in figre 4.3. If we denote the lowest and highest non-dimensional wave seed in the Solberg scheme as F () = c L = ĉ ˆk, F () = c R = ĉ+ˆk, (4.37a) (4.37b) we can think of the Solberg scheme as a HLL-tye scheme, bt where the non-dimensional wave seed changes linearly betweenc L andc R in stead of throgh a discontinity. Note that ˆk can be any ositive nmber, while Solberg s original CDˆk scheme was restricted by ˆk Z. We will discss the Solberg scheme frther in section.4.. 6

41 F (θ) ĉ+ˆk ĉ ĉ ˆk θ Figre 4.3: The non-dimensional wave seed of the Solberg scheme 4..3 The Godnov method for relative fl fnctions We can find the fl-difference slitting coefficients for any scheme defined by a relative fl fnction, by relacing the tre fl fnction in the Godnov method (4.) by an artificial fl fnction on the form (4.4). This yields ( ) ( ) [C i+ ] = +M (F(θ) (i+)θ) M (F(θ) iθ), ( ) ( ) (4.38a) [C i ] = M (F(θ)+iθ) M (F(θ)+(i+)θ). (4.38b) Since the terms inside the oerator M are mltilied by, we can simlify the oeratormas min (z(θ) ) if >, θ [,] M(z(θ) ) = ma (z(θ)( )) if <. θ [,] = min θ [,] (z(θ)), wherez(θ) is an arbitrary fnction. Using this simlification, we can divide by in (4.38) to obtain ( ) ( ) C i+ = + min (F(θ) (i+)θ) min (F(θ) iθ), θ [,] ( ) ( θ [,] ) C i = min (F(θ)+iθ) min (F(θ)+(i+)θ). θ [,] θ [,] (4.39) (4.4a) (4.4b) 7

42 4..4 Similarity soltions A similarity soltion can be written as where q(,t) = q(ζ) ζ = t t. If the similarity soltion satisfies a scalar conservation law, we get that f ( q) q (ζ) = ζ q (ζ), (4.4) (4.4) (4.43) which for q (ζ) redces to f ( q) = ζ. (4.44) Hence, if the derivative of an artificial fl fnction f (q) is an injective fnction, we can write the similarity soltion in terms ofζ, q(ζ) = ( f ) (ζ). (4.4) 8

43 Chater A new three arameter LTS scheme As mentioned in section 4.., the Solberg scheme can be thoght of as a HLL-tye scheme, where the non-dimensional wave seeds, c L and c R, are connected by a straight line in stead of by a discontinity. Can we make a similar scheme that take in arbitraryc L and c R? Starting from the non-dimensional wave seed of the HLL scheme, we can connectc L and c R by a linear fnction in the intervalθ [θ φ,θ +φ], c L ifθ θ φ, F (θ) = a +a θ ifθ (.) φ < θ < θ +φ, if θ θ +φ, c R where c L, c R and φ are free arameters. We have illstrated this fnction in figre.. There is only one choice ofa and a that makes F (θ) continos, that is c L if θ θ φ, F c (θ) = R +c L (.) + c R c L φ (θ θ ) ifθ φ < θ < θ +φ, ifθ θ +φ, c R where we have inserted a = c R+c L c R c L φ θ and a = c R c L φ. By integrating (.) we get that the relative fl fnction is given by c ( L θ ) ifθ θ φ, F(θ) = cr +c L c R c L φ (.3) θ θ + c R c L 4φ θ ifθ φ < θ < θ +φ, c L θ +c R (θ θ ) ifθ θ φ. To ensre that the scheme is consistent, we enforce the conditions in (4.6), which yields θ = c R ĉ c R c L. (.4) 9

44 F (θ) c R c L θ φ θ θ +φ θ Figre.: The non-dimensional wave seed of the LTS-HLLφ scheme We have now created a new three arameter LTS scheme, based on the LTS-HLL scheme and the Solberg scheme, which we will denote as the LTS-HLLφ scheme.. Similarity soltion Since the non-dimensional wave seeds of the LTS-HLLφ scheme (.) is one-to-one for θ (θ φ,θ + φ), we can eress the scheme as a similarity soltion in this interval by sing (4.4). The similarity soltion of the LTS-HLLφ scheme is then q L ifζ s L, q(ζ) = φ q R q L ( s R s L ζ s R +s L ) +q HLL ifs L < ζ < s R, q R ifζ s R, which we can also eress in one line as (.) ( q(ζ) = q L +H(ζ s L ) φ q R q L s R s L ( H(ζ s R ) φ q R q L s R s L ( ζ s R +s L ( ζ s R +s L )+q HLL q L ) )+q HLL q R ) (.6) sing the Heaviside fnctionh(). 3

45 . The fl-difference slitting coefficients The fl-difference slitting coefficients of the LTS-HLLφ scheme can be written as [ C i± cr ĉ = ± c R c L +φ (c ] R +c L )±i±ma(,min(±c L i,)) ma(,min(±c L i,)) c R c L [ ĉ cl ± c R c L φ (c R +c L )±i±ma(,min(±c R i,)) c R c L ] ma(,min(±c R i,)). (.7) Proof. The LTS-HLLφ Riemann solver can be written as ( Q(ζ)=Q j +H(ζ s L ) φ Q j Q j s R s L ( H(ζ s L ) φ Q j Q j s R s L which is eqivalent to Q(ζ) = Q j + H(ζ s ( L) φ s R s L H(ζ s ( L) φ s R s L ( ζ s R +s L ( ζ s R +s L ( ζ s R +s L ( ζ s R +s L ) ) +s L ˆλ )+Q HLL Q j ) )+Q HLL Q j ) ) +s R ˆλ (Q j Q j ) ) (Q j Q j ) (.8) (.9) From [6], we know that fori we have i t (i ) t Q(ζ)dζ = t Q j t C( i) (Q j Q j ). Using the identities, H( a) d = ma( a,), H( a) d = ma( a,) +ama( a,), (.) (.a) (.b) ma(,) ma( a,) = ma(,min(,a)), (.c) ma(,) ma( a,) = [ ma(,min(,a))]ma(,min(,a)), (.d) we can solve the integral in (.), and get that 3

46 [ C i cr ĉ = c R c L +φ (c R +c L ) i ma(,min( c L i,)) c R c [ L ĉ cl c R c L φ (c R +c L ) i ma(,min( c R i,)) c R c L Using the same rocedre, we can find a similar eression forc i+. ] ma(,min( c L i,)) ] ma(,min( c R i,)). (.) By inserting the coefficients (.7) into the TVD conditions (4.), we can show that the LTS-HLLφ scheme is TVD if φ min(θ, θ ). (.3).3 Nmerical diffsion coefficient We can calclate the nmerical diffsion of the LTS-HLLφ scheme by inserting the fldifference slitting coefficients (.7) into the eression for the nmerical diffsion coefficient (4.6). For the 3-oint LTS-HLLφ scheme, this yields σ HLLφ (ĉ) = σ HLL (ĉ) φ c R c L (c R c L c R c L ), (.4) where σ HLL (ĉ) is given in (4.). Note that the nmerical diffsion is identical to the nmerical diffsion of the LTS-HLL scheme when c R and c L have eqal signs. If c R and c L have different signs, the nmerical diffsion decreases roortionally to φ, becase c R c L c R c L >. We can therefore conclde that for a 3-oint scheme σ HLLφ σ HLL..4 Parameter stdy In this section, we show how other LTS schemes can be written as secial cases of the LTS-HLLφ scheme, by selecting aroriate arameters. We also discss which arameters give the most robst scheme. A smmary of the different LTS schemes, and the corresonding arameters are listed in table...4. Ceiling schemes We define a ceiling scheme as a LTS-HLLφ scheme with arametersc R = k andc L = k, where k = ĉ. For a ceiling scheme, the fl-difference slitting coefficients simly redce to 3