Running head: Weighted ENO Schemes Prfs sent t: Xu-Dng Liu Curant Institute f Mathematical Sciences 5 Mercer Street New Yrk, NY 00 AMS(MOS) subject cl

Weighted Essentially Nn-Oscillatry Schemes Xu-Dng Liu Stanley Osher y Tny Chan z Department f Mathematics, UCLA, Ls Angeles, Califrnia, 9004. Research supprted by NSF grant DMS 9-043. y Department f Mathematics, UCLA, Ls Angeles, Califrnia, 9004. Research supprted by NSF DMS 9-0304 and ONR NOO04-9-J-890 z Department f Mathematics,UCLA, Ls Angeles, Califrnia, 9004. Research supprted by NSF ASC90-9-0300, ARO DAAL03-9-G-50 and ONR NOO04-90-J-695

Running head: Weighted ENO Schemes Prfs sent t: Xu-Dng Liu Curant Institute f Mathematical Sciences 5 Mercer Street New Yrk, NY 00 AMS(MOS) subject classicatins: Primary 65M0; Secndary 65M05 Keywrds: Hyperblic Cnservatin Laws, ENO

Abstract In this paper we intrduce a new versin f ENO (Essentially Nn- Oscillatry) shck-capturing schemes which we call Weighted ENO. The main new idea is that, instead f chsing the \smthest" stencil t pick ne interplating plynmial fr the ENO recnstructin, we use a cnvex cmbinatin f all candidates t achieve the essentially nn-scillatry prperty, while additinally btaining ne rder f imprvement in accuracy. The resulting Weighted ENO schemes are based n cell-averages and a TVD Runge-Kutta time discretizatin. Preliminary encuraging numerical experiments are given. 3

Intrductin In this paper we present a new versin f ENO (Essentially Nn-Oscillatry) schemes. The cell-average versin f ENO schemes riginally was intrduced and develped by Harten and Osher in [] and Harten, Engquist, Osher and Chakravarthy in []. Later Shu and Osher develped the ux versin f ENO schemes and intrduced the TVD Runge-Kutta time discretizatin in [3] and [4]. The ENO schemes wrk well in many numerical experiments. The new ENO schemes which we call the Weighted ENO schemes are based n cell-averages and the TVD Runge-Kutta time discretizatin. The nly dierence between these schemes and the standard cell-average versin f ENO is hw we dene a recnstructin prcedure which prduces a high-rder accurate glbal apprximatin t the slutin frm its given cellaverages. The cell-average versin f ENO schemes attempts t avid grwth f spurius scillatins by an adaptive-stencil apprach, in which each cell is assigned its wn stencil f cells fr the purpses f recnstructin. Fr each cell the cell-average versin f ENO schemes selects an interplating stencil in which the slutin is smthest in sme sense. Thus a cell near a discntinuity is assigned a stencil frm the smth part f the slutin and a Gibbs-like phenmenn is s avided (see [5]). The Weighted ENO schemes develped here fllw this basic idea by using a cnvex cmbinatin apprach, in which each cell is assigned all crrespnding stencils and a cnvex cmbinatin f all crrespnding interplating plynmials n the stencils is cmputed t be the apprximating plynmial. This is dne by assigning prper weights t the cnvex cmbinatin. T achieve the essentially nn-scillatry prperty as the cell-average versin f ENO, the Weighted ENO schemes require that the cnvex cmbinatin be essentially a cnvex cmbinatin f the interplating plynmials n the smth stencils and that the interplating plynmials n the discntinuus stencils have essentially n cntributin t the cnvex cmbinatin. Thus, as in the cell-average versin f ENO schemes, a cell near a discntinuity is essentially assigned stencils frm the smth part f the slutin and a Gibbs-like phenmenn is als avided. In additin t this, the cnvex cmbinatin apprach results in cancellatin f truncatin errrs f crrespnding interplating plynmials and imprves the rder f accuracy by ne. Anther pssible advantage f Weighted ENO is smther dependence n data which may lessen sme f ENO's scillatry behavir near cnvergence and may help in getting a cnvergence prf. 4

In x we intrduce sme ntatins and basic ntins and give the TVD Runge-Kutta time discretizatin. In x3 we describe the prcedure f recnstructin frm given cell averages. In x4 we present sme preliminary numerical experiments. Basic Frmulatin and TVD Runge-Kutta Time Discretizatin We cnsider a hyperblic cnservatin law u t + f(u) x = 0; u(x; 0) = u (x): (:) Let fi j g be a partitin f R, where I j = [x j? ; x j+ ] is the j-th cell, x j+? = h. Dente fu j (; t)g t be the sliding averages f the weak slutin x j? u(x; t) f (.) i.e. R u j (; t) = h I j u(x; t) dx: (:) Integrating (.) ver each cell I j, we btain that the sliding averages fu j (; t)g satisfy @ u @t j(; t) =? [f(u(x h j+ ; t))? f(u(x j? ; t))]: (:3) T evaluate each @ u @t j(; t), we need t evaluate f(u(x; t)) at each interface x j+. First f all, frm given cell-averages u = fu j g in which u j apprximates u j (; t), we recnstruct the slutin t btain R(x) = fr j (x)g which is a piecewise plynmial with unifrm plynmial degree r?, and in which each R j (x) is a plynmial apprximating u(x; t) n I j. We shall shw hw t btain R(x) frm u = fu j g in x3. Next at each interface x j+ have tw apprximating values R j (x j+ ) and R j+ (x j+ ) fr u(x j+, R(x) may ; t). We need a tw-pint Lipschitz mntne ux ~ h(; ) which is nndecreasing fr the rst argument and nnincreasing fr the secnd argument. Sme pssible chices are (i) Engquist-Osher h EO R (a; b) = b 0 R min(f 0 (s); 0) ds + a max(f 0 (s); 0) ds + f(0); (:4) 0 5

(ii) Gdunv h G (a; b) = ( minaub f(u) if a b; max aub f(u) if a > b; (:5) (iii) Re with entrpy x 8 >< f(a) if f 0 (u) 0 fr u [min(a; b); max(a; b)]; h RF (a; b) = f(b) if f 0 (u) 0 fr u [min(a; b); max(a; b)]; >: h LLF (a; b) therwise; (:6a) where h LLF (a; b) is dened as h LLF (a; b) = [f(a) + f(b)? (b? a)]; = max j f 0 (u) j : min(a;b)umax(a;b) (:6b) We apprximate f(u(x j+ and f(u(x j? where ; t)) by ~ h(r j? (x j? ; t)) by ~ h(r j (x j+ ); R j (x j? @ u @t j(; t) L j (u); ); R j+ (x j+ )) )). Therefre (:7a) L j (u) =? h [~ h(r j (x j+ ); R j+ (x j+ ))? ~ h(r j? (x j? ); R j (x j? ))]: (:7b) In sectin x3, in which we intrduce the recnstructin prcedure, we shall btain that, in each cell I j, u(x; t) = R j (x) + O(h r ) 8x I j ; (:8) and at ne chsen pint f tw end pints f I j, u(x j; t) = R j (x j) + O(h r+ ) x j = x j? r x j = x j+ : (:9) Here and belw we always cnsider smth slutins when we discuss accuracy. Fr general upwind schemes, away frm snic pints (where f 0 (u) = 0), ( 0 f(a) in the regins f f > 0 ~h(a; b) = f(b) in the regins f f 0 < 0 6

In the regins f f 0 > 0, frm (.7b), and if we chse x j = x j+ hence L j (u) =? [f(r h j(x j+ ))? f(r j? (x j? ))]; u(x j+ u(x j? and x j? = x j? ; t) = R j (x j+ ; t) = R j? (x j? in (.9) i.e. ) + O(h r+ ) ) + O(h r+ ); @ u @t j(; t) = L j (u) + O(h r+ ): (:0) Similarly, we shall have the abut frmula (.0) in the regins f f 0 < 0 by chsing x j = x j? and x j+ = x j+ in (.9). This will be detailed in x3:4. As usual, in the regins arund f 0 = 0 (snic pints), we btain @ u @t j(; t) = L j (u) + O(h r ): Fr high rder time discretizatin, because f (.0), we need (r + )-th rder TVD Runge-Kutta time discretizatins intrduced by Shu and Osher in [3]. We need nly t spell ut the 3rd and 4th rder methds, which will be implemented in ur numerical experiments. Fr 3rd rder, 8j, Fr 4th rder, 8j, u (0) j = u n j ; u () j = u (0) u () j u (3) j u n+ j = u(0) j = 9 u(0) j = 3 u() j u (0) j = u n j ; u () j = u (0) u () j u n+ j j + L j (u (0) ) + u() + 9 u() j + 3 u() j = 3 4 u(0) j = 3 u(0) j j + L j (u (0) ) + 4 u() + 3 u() j + L 4 j(u () ) j + L 3 j(u () ): j? 4 L j(u (0) ) + L j(u () ) + 3 u() + 3 u(3) j? L 9 j(u (0) )? L 3 j(u () ) + L j (u () ) j + L 6 j(u () ) + L 6 j(u (3) ): T cmplete the cnstructin f ur schemes we frm ur nvel recnstructin prcedure. 7

3 Recnstructin Prcedure 3. Purpses f Recnstructin In this sectin we present the recnstructin prcedure. The R(x) is required t satisfy (i) In each cell I j, 8x I j and ne chsen pint x j I j, we have and R j (x) = u(x; t) + O(h r ); R j (x j) = u(x j; t) + O(h r+ ); (3:a) (3:b) where (3.b) will lead t ne rder f imprvement in accuracy, see x3:4 in this paper. (ii) R(x) has cnservatin frm i.e. 8j h R I j R j (x) dx = u j : (3:) (iii) Every R j (x) achieves the \ENO prperty" which will be specied later. 3. Interplatin n Each Stencil Fllwing the recnstructin prcedure in [], given the cell averages fu j g, we can immediately evaluate the pint values f the slutin's primitive functin W (x) at interfaces fw (x j+ )g, where the primitive functin is dened as W (x) = R x x u(x; t) dx; 0 j? (3:3) where x j 0? and bviusly culd be any interface, hence u(x; t) = W 0 (x) = d W (x); (3:4) dx P W (x j+ ) = j u i h: (3:5) i=j 0 T recnstruct the slutin, we interplate W (x) n each stencil S j = (x j?r+ ; x j?r+ 3 ; ; x j+ ) t btain a plynmial p j (x) i.e. p j (x l+ ) = W (x l+ ); l = j? r; ; j: 8

Obviusly the crrespnding plynmial p 0 j(x) (with degree r? ) apprximates the slutin u(x; t) i.e. see []. u(x; t) = p 0 j(x) + O(h r ) 8x (x j?r+ ; x j+ ); Als fr each stencil S j = (x j?r+ ; x j?r+ 3 ; ; x j+ ), we dene an indi- catr f the smthness IS j f u(x; t) n S j as fllwing: First we cmpute a table f dierences f fu j g n S j, where [u j?r+ ]; [u j?r+ ]; ; [u j? ]; [u j?r+ ]; [u j?r+ ]; ; [u j? ];. r? [u j?r+ ]; [u l ] = u l+? u l k [u l ] = k? [u l+ ]? k? [u l ]: Next we dene IS j t be the summatin f all averages f square values f the same rder dierences, That is, fr r =, and, fr r = 3, IS j = r? P P ( l ( r?l [u j?r+k ]) )=l: l= k= IS j = ([u j? ]) ; IS j = (([u j? ]) + ([u j? ]) )= + ( [u j? ]) : We bserve that if u(x; t) is discntinuus n S j, IS j O(), and if u(x; t) is cntinuus n S j, IS j O(h ). Hence fr each stencil S j, we btain p 0 j(x) apprximating u(x; t) n S j and IS j indicating the smthness f u(x; t) n S j. In the fllwing subsectin, t recnstruct the slutin in I j, we shall use r interplating plynmials fp 0 j+k(x)g r? k=0 n the stencils fs j+k g r? k=0, in which all S j+k cver the I j, t btain a cnvex cmbinatin f them, and we shall explre fis j+k g r? k=0 t assign a prper weight fr each f fp 0 j+k(x)g r? k=0 in the cnvex cmbinatin fr the purpses f recnstructin. 9

0 3.3 Cnvex Cmbinatin f fp (x)gr? j+k k=0 fr Each Cell I j Fr each cell I j we have r stencils fs j+k g r? k=0 = f(x j+k?r+ ; x j+k?r+ 3 ; ; x j+k+ )g r? k=0 which all include tw end pints x j? and x j+ f I j. We als have r interplating plynmials fp 0 j+k(x)g r? k=0 n the crrespnding stencils fs j+k g r? k=0. The main idea f the cell-average versin f ENO is t chse the \smthest" ne frm these r interplating plynmials. Fr Weighted ENO, instead f chsing ne, we use all r interplating plynmials and cmpute a cnvex cmbinatin f them t btain a plynmial R j (x) as fllws R j (x) = r? P j k r? P k=0 j l l=0 p 0 j+k(x); (3:6) where the j k > 0 (k = 0; ; ; ; r? ). Obviusly u(x; t) = R j (x) + O(h r ) in the smth regins f u(x; t) which is the purpse f (3.a). Nte that fr any k = 0; ; ; r?, p j+k (x j? ) = W (x j? ) and p j+k (x j+ ) = W (x j+ ), hence we achieve the purpse f (3.) R h I j R j (x) dx = r? P j k h r? P k=0 j l l=0 (p j+k (x j+ = h fw (x j+ )? W (x j? )? p j+k (x j? )) r? P )g k=0 j k r? P j l l=0 = u j : (3:7) Nte that n matter hw we dene f j k gr? k=0, R j (x) satises the purpses f (3.a) and (3.). We specify the \ENO prperty" f R j (x) by the crrespnding f j k gr? k=0. Denitin : The R j (x) has the \ENO prperty" if the crrespnding f j k gr? k=0 satisfy that (i) If the stencil S j+k is in the smth regins, the crrespnding j k satisfy j k P r? l=0 j l = O(): (3:8a) (ii) If the stencil S j+k is in a discntinuus regin f the slutin u(x; t), the crrespnding j k satisfy j k P r? l=0 j l O(h r ): (3:8b) 0

r? P Nte that, if f j k gr? k=0 satisfy the \ENO prperty" (3.8), the R j (x) = j k r? P k=0 j l l=0 p 0 j+k(x) will be a cnvex cmbinatin f the interplating plynmials n the smth stencils (3.8a), and the interplating plynmials n the discntinuus stencils have essentially n cntributin t R j (x) (3.8b). Dene, j k = C j k=( + IS j+k ) r ; k = 0; ; r? ; (3:9) where C j k = O() and C j k > 0 will be dened later fr imprvement f accuracy. Nte that because IS j+k culd be zer and =x is t sensitive as x is near zer, we add a small psitive number = 0?5 in the denminatr. Nte that if the stencil S j+k is in the smth regins j k P r? l=0 j l = O(); and if the stencil S j+k is in the discntinuus regins f u(x; t) j k P r? l=0 j l max(o( r ); O(h r )): Hence these f j k gr? k=0 (3.9) satisfy the \ENO prperty" (3.8)(O( r ) O(0?0 )). Here we assume there is at least ne stencil f fs j+k g r? k=0 in the smth regins. N matter hw we dene the cnstants fc j k gr? k=0, we have achieved the purpses f (3.a), (3.) and the \ENO prperty" (3.8). Hwever we shall specify fc j k gr? k=0 fr (3.b) which will lead ut ne rder imprvement in accuracy in sectin x3.4, ur last purpse f the recnstructin. Fr analysis we assume that in [x j?r+ ; x j+r+ ]. u(x; t) C r+ ; (3:0)

Fr each p 0 j+k(x), we express its truncatin errr as e j+k (x) where a j k(x) = r P = u(x; t)? p 0 j+k(x) = W 0 (x)? p 0 j+k(x) = d fw [x; x rq dx j+k?r+ ; ; x j+k+ ] (x? x j+k?l+ )g l=0 = d W [x; x rq dx j+k?r+ ; ; x j+k+ ] (x? x j+k?l+ ) l=0 rp rq +W [x; x j+k?r+ ; ; x j+k+ ] f (x? x j+k?l+ s=0 l=0;l6=s = W [x; x j+k?r+ ; ; x j+k+ ] a j k(x) + O(h r+ ); s=0 rq f l=0;l6=s (x? x j+k?l+ )g. We express the truncatin errr fr R j (x) E j (x) = u(x; t n )? R j (x) = W 0 (x)? R j (x) = r? P j k r? P (W 0 (x)? p 0 j+k(x)) = r? P P k=0 j l l=0 j k r? k=0 j l l=0 Because f the assumptin (3.0), 8k = 0; ; ; r?, j IS j+k j O(h ) j IS j+k? IS j j O(h ) j a j k(x) j O(h r ) j W [x; x j+k?r+ ; ; x j+k+ We have, frm (3.a) and (3.b), E j (x) = r? P j k r? P k=0 j l l=0 = r? P j k r? P k=0 j l l=0 r? C j k r? P = f P k=0 C j k l=0 e j+k (x) W [x; x j+k?r+ ]? W [x; x j?r+ a j k(x)g W [x; x j?r+ e j+k (x): ; ; x j+ ] j O(h): ; ; x j+k+ ] a j k(x) + O(h r+ ) ; ; x j+ ] + O(h r+ ): )g (3:a) (3:b) (3:c) The idea is that fr ne chsen pint x j [x j? ; x j+ ], we dene C j k t make the rst term in (3.c) equal t zer and btain E j (x j) = O(h r+ ):

Fr x j [x j? ; x j+ ], we dente p be the number f psitive terms in fa j k(x j)g r? and k=0 n be the number f negative terms in fa j k(x j)g r?, then we k=0 dene 8 if a j >< k(x j) = 0, C j h r k = if a j pja j k (x)j k (x j) > 0, (3:) j h >: r if a j nja k(x j) < 0. j k (x)j j Obviusly the C j k are independent f grid size h. E j (x j) r? P = f k=0 = f C j k r? P C j l l=0 P a j k(x j)gw [x; x j?r+ ; ; x j+ ] + O(h r+ ) p r? P a j k (x)>0 C j j l l=0 = 0 + O(h r+ ) = O(h r+ ):? P a j k (x j )<0 n r? P C j l l=0 gw [x; x j?r+ ; ; x j+ ] + O(h r+ ) (3:3) Remark : We have t have p and n t guarantee (3.3). Thus we btain that, fr ne chsen pint x j and any ther pint x [x j? ; x j+ ], dening C j k by (3.) gives us E j (x) = O(h r ); (3:4a) and E j (x j) = O(h r+ ): (3:4b) Up t nw, we have achieved all purpses f recnstructin (3.a), (3.b), (3.) and (3.8). 3.4 One Order Imprvement in Accuracy using (3.b) In this subsectin, we shall see hw (3.b) r (3.4b) gives us ne rder f imprvement in accuracy by chsing x j prperly in each cell. Let us cnsider the numerical spatial apprximatin (.7b) L j (u) =? h [ ~ h(rj (x j+ ~h(r j? (x j? 3 ); R j+ (x j+ ))? ))]: ); R j (x j?

Cnsider three cells in a smth regin, say cells I j?, I j and I j+, which are away frm snic pints. If f 0 (R(x)) > 0 in the cells, we have In (3.b), we chse x j = x j+ that Thus L j (u) =? [f(r h j(x j+ ))? f(r j? (x j? ))]: R j (x j+ R j? (x j? )? u(x j+ )? u(x j? L j (u) =? h [f(u(x j+ If f 0 (R(x)) < 0 in the cells, we have In (3.b), we chse x j = x j? that and x j? = x j?, then by (3.4b) we btain ; t) = E j (x j+ ) = O(h r+ ); ; t) = E j? (x j? ) = O(h r+ ): ; t))? f(u(x j? ; t))] + O(h r+ ): L j (u) =? [f(r h j+(x j+ ))? f(r j (x j? ))]: R j+ (x j+ R j (x j? )? u(x j+ )? u(x j? and x j+ = x j+, then by (3.4b) we btain ; t) = E j+ (x j+ ) = O(h r+ ); ; t) = E j (x j? ) = O(h r+ ): Thus L j (u) =? [f(u(x h j+ ; t))? f(u(x j? ; t))] + O(h r+ ): Hence in the smth regins and away frm snic pints, the numerical spatial peratrs fl j (u)g apprximate f @ u @t j(; t)g t the rder O(h r+ ). We specify x j in each I j in the fllwing way: First we cmpute f 0 (u j ). Then (i) if f 0 (u j ) > 0 we chse x j = x j+, (ii) if f 0 (u j ) < 0 we chse x j = x j?, (iii) if f 0 (u j ) = 0 we chse x j = x j+ r x j = x j?. If the cell I j is in the smth regins and away frm snic pints, then in general f 0 (R(x)) f 0 (u j ) > 0 arund the cell I j, hence accrding t the abve analysis @ @t u j(; t) = L j (u) + O(h r+ ): (3:5) 4

Because snic pints are islated, in general, we btain (3.5) in mst f the cells and btain @ @t u j(; t) = L j (u) + O(h r ) in a bunded, in fact small, number f cells near which there are snic pints as h decreases t zer. Remark 3: We have achieved ne rder imprvement in accuracy. Fr r = and r = 3, the cst f cmputing f the Weighted ENO schemes is cmparable t (f curse a little mre expensive than) that f standard ENO schemes (with the same rder accuracy) n sequential cmputers. Hwever n parallel cmputers, t achieve the same rder accuracy, the frmer schemes are much less expensive than the latter because the latter need mre expensive data transprt between cells. 3.5 Schemes fr r = The purpse f the fllwing tw subsectins x3:5 and x3:6 is t spell ut the details f the general schemes fr tw specic values f r, perhaps t aid the reader in implementin. In this subsectin, we cnsider ur schemes when r =. In this case we use linear interplatin t achieve the \ENO prperty" and 3rd rder accuracy (in ur numerical experiments, we achieved 4th rder accuracy) with cnservatin frm. Here we give the recnstructin prcedure fr r =. Fr each cell I j, we have tw stencils S j = (x j? 3 crrespnding t I j = [x j? linear interplatins and ; x j+ ; x j? ; x j+ ) and S j+ = (x j? ; x j+ ; x j+ 3 ) ]. On these tw stencils, we btain tw p 0 j(x) = u j + u j?u j? h (x? x j ) p 0 j+(x) = u j + u j+?u j h (x? x j ); and tw indicatrs f smthness IS j = (u j?u j? ) and IS j+ = (u j+?u j ). The recnstructed slutin R j (x) will be a cnvex cmbinatin f p 0 j(x) and p 0 j+(x) i.e. R j (x) = j 0 p 0 j(x) + j p 0 j 0 +j j+(x); (3:6) j 0 +j 5

where j 0 = C0=( j + IS j ), j = C=( j + IS j+ ). We shall specify C j 0 and C j in the fllwing tw cases. Case : If f 0 (u j ) > 0, we chse x j = x j+. We cmpute a j 0(x j+ ) = h and a j (x j+ ) =?h, and btain p = and n =, hence C j 0 = = and C j =. Thus in (3.6). j 0 = (+IS j ) j = (+IS j+ ) Case : If f 0 (u j ) 0, we chse x j = x j? and a j (x j? C j = =. Thus in (3.6). (3:7a). We cmpute a j 0(x j? ) =?h ) = h, and btain p = and n =, hence C j 0 = and j 0 = (+IS j ) j = (+IS j+ ) (3:7b) 3.6 Schemes fr r = 3 In this subsectin, we cnsider ur schemes when r = 3. In this case we use quadratic interplatin t achieve the \ENO prperty" and 4th rder accuracy (in ut numerical experiments, we achieved 5th rder accuracy) with cnservatin frm. Here we give ut the recnstructin prcedure fr r = 3. Fr each I j, we have three stencils S j = (x j? 5 ; x j? 3 ; x j? ; x j+ ), S j+ = (x j? 3 ; x j? ; x j+ ; x j+ 3 ), and S j+ = (x j? ; x j+ ; x 3 j+ ; x 5 j+ ) crrespnding t I j = [x j? three stencils, we btain three quadratic interplatins and p 0 j(x) = u j?u j? +u j? h (x? x j? ) + u j?u j? h (x? x j? )+ u j?? u j?u j? +u j? 4 p 0 j+(x) = u j+?u j +u j? h (x? x j ) + u j+?u j? h (x? x j )+ u j? u j+?u j +u j? 4 p 0 j+(x) = u j+?u j+ +u j h (x? x j+ ) + u j+?u j (x? x h j+ )+ u j+? u j+?u j+ +u j 4 ; 6 ; x j+ ]. On these

and three indicatrs f smthness IS j = ((u j?? u j? ) + (u j? u j? ) )= + (u j?u j? +u j? ), IS j+ = ((u j?u j? ) +(u j+?u j ) )=+(u j+?u j +u j? ) and IS j+ = ((u j+? u j ) + (u j+? u j+ ) )= + (u j+? u j+ + u j ). The recnstructed slutin R j (x) will be a cnvex cmbinatin f p 0 j(x), p 0 j+(x) and p 0 j+(x) i.e. R j (x) = j 0 j 0 +j +j p 0 j(x) + j j 0 +j +j p 0 j+(x) + j j 0 +j +j p 0 j+(x); (3:8) where j 0 = C0=( j + IS j ) 3, j = C=( j + IS j+ ) 3, j = C=( j + IS j+ ) 3. We shall specify C0, j C j and C j in the fllwing tw cases. Case : If f 0 (u j ) > 0, we chse x j = x j+. We cmpute a j 0(x j+ ) = 6h 3, a j (x j+ ) =?h 3 and a j (x j+ ) = h 3, and btain p = and n =, hence C j 0 = =, C j = = and C j = =4. Thus j 0 = (+IS j ) 3 j = (+IS j+ ) 3 j = 4(+IS j+ ) 3 (3:9a) in (3.8). Case : If f 0 (u j ) 0, we chse x j = x j??h 3,a j (x j? ) = h 3, and a j (x j? hence C j 0 = =4, C j = = and C j = =. Thus. We cmpute a j 0(x j? ) = ) =?6h 3 and btain p = and n =, j 0 = 4(+IS j ) 3 j = (+IS j+ ) 3 j = (+IS j+ ) 3 (3:9b) in (3.8). 4 Numerical Experiments 4. Scalar Cnservatin Laws In this subsectin we use sme mdel prblems t numerically test ur schemes. We use the Re ux with entrpy x as numerical ux and chse 7

r = which means we use a linear plynmial t recnstruct the slutin, and/r r = 3 which means we use a quadratic plynmial t recnstruct the slutin, and we expect t achieve 3rd and 4th rder accuracy respectively (at least away frm snic pints) accrding t ur analysis in the previus sectin. Example. We slve the mdel equatin u t + u x = 0? x u(x; 0) = u 0 (x) u 0 (x) peridic with perid : (4:) Five dierent initial data u 0 (x) are used. The rst ne is u 0 (x) = sin(x) and we list the errrs at time t = in Table. The secnd ne is u 0 (x) = sin 4 (x) and we list the errrs at time t = in Table. TABLE (=h = 0:8; t = ) l L errr L rder L errr L rder r = 80.77D-03.D-0 60.98D-04 3.8.D-03 3.45 30.06D-05 4. 4.30D-05 4.70 r = 3 80.8D-05.03D-04 60 9.65D-07 4.56 7.85D-06 3.7 30.7D-08 5.8.4D-07 5.80 640 6.07D-0 4.8.33D-09 6.73 TABLE (=h = 0:8; t = ) 8

l L errr L rder L errr L rder r = 80.77D-0 7.3D-0 60 3.08D-03.5.86D-0.94 30.46D-04 3.65.04D-03 3.9 640.4D-05 4. 9.8D-05 4.46 r = 3 80.7D-03 6.87D-03 60.3D-04 4.6 3.93D-04 3.9 30 3.7D-06 4.93 3.5D-05 5. 640 6.39D-08 5.86 6.63D-07 5.6 Here and belw l is the ttal number f cells and the step size h = =l in all scalar examples. Fr the rst tw initial data, we btain abut 4th (fr r = ) and 5th (fr r = 3) rder f accuracy respectively in the smth regin in bth L and L nrms which is surprisingly better than the 3rd and 4-th rder, the theretical results. We nte that standard ENO schemes applied t the example with the secnd initial data experienced an (easily xed) lss f accuracy, see [6], [7]. N such degeneracy was fund with ur present methds. The third initial functin is (? u 0 (x) = 5 5 0 therwise, the furth is and the fth is u 0 (x) = ( (? ( 0 3 x) )? 3 0 x 3 0 ; 0 therwise; u 0 (x) = e?300x : We see the gd reslutin f the slutins in Figures -3 which are btained by ur scheme with r = 3. Linear discntinuities are smeared a bit. We expect t x this in the future using either the subcell reslutin technique f Harten [0] r the articial cmpressin technique f Yang [] tgether with the present technique. Figure (=h = 0:8) 9

The slutin by WENO at T = 0.5 0.9 0.8 0.7 Number f Pints = 80 0.6 0.5 0.4 0.3 0. 0. 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu ++ apprx. slu r = 3 Figure (=h = 0:8) The slutin by WENO at T = 0.5 0.9 0.8 0.7 Number f Pints = 80 0.6 0.5 0.4 0.3 0. 0. 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu ++ apprx. slu r = 3 0

Figure 3 (=h = 0:8) The slutin by WENO at T = 0.5 0.9 0.8 0.7 Number f Pints = 80 0.6 0.5 0.4 0.3 0. 0. 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu ++ apprx. slu r = 3 Example. We slve Burgers' equatin with a peridic bundary cnditin u t + ( u ) x = 0? x (4:) u(x; 0) = u 0 (x) u 0 (x) peridic with perid : Fr the initial data u 0 (x) = + sin(x), the exact slutin is smth up t t =, then it develps a mving shck which interacts with a rarefactin wave. Observe that there is a snic pint. At t = 0:5 the slutin is still smth. We list the errrs in Table 3. Nte we als have abut 5th (fr r = 3) rder f accuracy respectively bth in L and L nrms.

TABLE 3 (=h = 0:6; t = 0:5) l L errr L rder L errr L rder r = 3 80.63D-05.84D-04 60.50D-06 4.3.68D-05 3.4 30 5.44D-08 4.79 3.63D-07 6. At t = the shck just begins t frm, at t = 0:55 the interactin between the shck and the rarefactin waves is ver, and the slutin becmes mntne between shcks. In Figures 4-5 which are btain by ur scheme with r = 3 we can see the excellent behavir f the schemes in bth cases. The errrs at a distance 0: away frm the shck (i.e. j x? shck lcatin j 0:) are listed in Table 4 at t = 0:55. These errrs are f same magnitude as the nes in the smth case f Table 3 and shw abut 5th (fr r = 3) rder f accuracy respectively bth in L and L in the smth regins 0: away frm the shck. This shws that the errr prpagatin f the scheme is still very lcal. TABLE 4 (=h = 0:6; t = 0:55) l L errr L rder L errr L rder r = 3 80.9D-05 7.63D-04 60 7.7D-07 4.89 3.83D-05 4.3 30 7.4D-09 6.70 7.7D-07 5.7

Figure 4 (=h = 0:6).5 The slutin by WENO at T = 0.383 Number f Pints = 80 0.5 0-0.5 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu ++ apprx. slu r = 3 Figure 5 (=h = 0:6).5 The slutin by WENO at T = 0.55 Number f Pints = 80 0.5 0-0.5 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu ++ apprx. slu r = 3 Example 3. we use tw nncnvex uxes t test the cnvergence t the physically crrect slutins. The true slutins are btained frm the Lax- Friedrichs scheme n a very ne grid. We use ur scheme with r = 3 in this example. The rst ne is a Riemann prblem with the ux f(u) = 3

4 (u? )(u? 4), and the initial data ( ul x < 0 u 0 (x) = u r x > 0: The tw cases we test are (i) u l =, u r =?, Figure 6; (ii) u l =?3, u r = 3, Figure 7. Fr mre details cncluding this prblem see [] Figure 6 (=h = 0:3) The slutin by WENO at T =.5 h = 0.05 0.5 0-0.5 - -.5 - - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 3 -- true slu apprx. slu l=80 Figure 7 (=h = 0:04) The slutin by WENO at T = 0.04 h = 0.05 0 - - -3 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu apprx. slu l=80 4

The secnd ux is the Buckley-Leverett ux used t mdel il recvery [], f(u) = 4u =(4u + (? u) ), with initial data u = in [? ; 0] and u = 0 elsewhere. The result is displayed in Figure 8. Figure 8 (=h = 0:4) 0.9 0.8 0.7 The slutin by WENO at T = 0.4 0.6 h = 0.05 0.5 0.4 0.3 0. 0. 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 -- true slu apprx. slu l=80 In this example, we bserve cnvergence with gd reslutin t the entrpy slutins in bth cases. In all the examples that we have illustrated abve, we bserve that the schemes are f abut 4th (fr r = ) and 5th (fr r = 3) rder f accuracy respectively and cnvergent with gd reslutin t the entrpy slutins. 4. Euler Equatins f Gas Dynamics In this subsectin we apply ur schemes t the Euler equatin f gas dynamics fr a plytrpic gas, u t + f(u) x = 0 u = (; m; E) T f(u) = qu + (0; P; qp ) T P = (? )(E? q ) m = q; (4:) where = :4 in the fllwing cmputatin. Fr details f the Jacbian, its eigenvalues, eigenvectrs, etc., see []. 5

Example 4. We cnsider the fllwing Riemann prblems: ( ul x < 0 u 0 (x) = u r x > 0: Tw sets f initial data are used. One is prpsed by Sd in [8]: ( l ; q l ; P l ) = (; 0; ); ( r ; q r ; P r ) = (0:5; 0; 0:): The ther is used by Lax [9]: ( l ; q l ; P l ) = (0:445; 0:698; 3:58); ( r ; q r ; P r ) = (0:5; 0; 0:57): We test ur schemes with r = 3. We use the characteristic recnstructin and Re ux with entrpy x frmed by Re's average as numerical ux. Fr details see []. The results are displayed in Figure 9-0. Figure 9a (=h = 0:4; t = ) DENSITY at time T = 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. -6-4 - 0 4 6 The Number f Pints = 00 Figure 9b (=h = 0:4; t = ) VELOCITY at time T = 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0-6 -4-0 4 6 The Number f Pints = 00 6

Figure 9c (=h = 0:4; t = ) PRESSURE at time T = 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. -6-4 - 0 4 6 The Number f Pints = 00 Figure 0a (=h = 0:; t = :5).4 DENSITY at time T =.5. 0.8 0.6 0.4 0. -6-4 - 0 4 6 The Number f Pints = 00 Figure 0b (=h = 0:; t = :5).6 VELOCITY at time T =.5.4. 0.8 0.6 0.4 0. 0-6 -4-0 4 6 The Number f Pints = 00 7

Figure 0c (=h = 0:; t = :5) 4 PRESSURE at time T =.5 3.5 3.5.5 0.5-6 -4-0 4 6 The Number f Pints = 00 Example 5. In this example we shall test the accuracy f ur schemes (r = 3) fr the Euler equatin f gas dynamics fr a plytrpic gas. We chse initial data as = +sin(x), m = +sin(x) and E = +sin(x), and peridic bundary cnditin. The true slutin was btained by applying the schemes t a very ne grid. Fr time t = when shcks haven't frmed, ur schemes achieve 5th (r = 3) rder accuracy in all three cmpnents, see Table 5. We can als see the slutin fr time t = in Figure. 8

TABLE 5 (=h = 0:6; t = ) l L errr L rder L errr L rder DENSITY 80.99D-04.9D-03 60.D-05 4.03.5D-04 3.49 30.74D-07 6..07D-06 5.80 640.9D-09 5.90 3.0D-08 6.0 MOMENTUM 80.7D-04.76D-03 60.9D-05 4.07.50D-04 3.55 30.85D-07 6..43D-06 5.95 640 3.09D-09 5.90 3.0D-08 6.5 ENERGY 80.0D-04.9D-03 60.9D-05 4.4.60D-04 3.59 30.55D-07 6.6.57D-06 5.96 640.75D-09 5.8 3.0D-08 6.37 9

Figure a (=h = 0:6; t = ).3 Density...9.8.7 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 The Number f Pints = 00 Figure b (=h = 0:6; t = ).6 Mmentum.4..8.6.4 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 The Number f Pints = 00 Figure c (=h = 0:6; t = ).4 Energy.3...9.8.7.6.5.4 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 The Number f Pints = 00 Acknwledgment We are grateful t Prfessr Chi-Wang Shu fr his 30

suggestin f the smth indicatr functin i.e. instead f IS j = r? P P ( l ( r?l [u j?r+k ]) )=l; l= l= k= IS j = r? P P ( l j r?l [u j?r+k ] j)=l; k= which we used riginally. Bth functins wrk well, hwever the latter ne leads t a smther (C vs. Lipschitz) numerical ux which may be helpful fr steady state cnvergence r cnvergence prf. References [] A. Harten and S. Osher,\Unifrmly High-Order Accurate Nn- Oscillatry Schemes I," SIAM Jurnal n Numerical Analysis, V4, pp. 79-309, 987; als MRC Technical Summary Reprt N. 83, May 985. [] A. Harten, B. Engquist, S. Osher and S. Chakravarthy, \Unifrmly High Order Accurate Essentially Nn-Oscillatry Schemes III," Jurnal f Cmputatinal Physics, V7, pp. 3-303, 987; als ICASE Reprt N. 86-, April 986. [3] C.-W. Shu and S. Osher, \Ecient Implementatin f Essentially Nn-scillatry Shck-Capturing Schemes," Jurnal f Cmputatinal Physics, V77, 988, pp. 439-47. [4] Chi-Wang Shu, Stanley Osher, \Ecient Implementatin f Essentially Nn-scillatry Shck-Capturing Schemes, II," J. Cmput. Phys., V83, 989, pp. 3-78. [5] A. Harten and S. Chakraverthy, \Multi-Dimensinal ENO Schemes fr General Gemetries," UCLA CAM reprt N. 9-6, August 99. [6] A. Rgersn and E. Meiburg, \A Numerical Study f the Cnvergence Prperties f ENO Schemes," J. Scientic Cmputing, V5, N., 990, pp.5-67. 3

[7] Chi-Wang Shu, \Numerical Experiments n the Accuracy f ENO and Mdied ENO Schemes," J. Scientic Cmputing, V5, N., 990, pp.7-49. [8] G. Sd, J. Cmput. Phys. 7, (978). [9] P. Lax. Cmmun. Pure Appl. Math. 46, (986). [0] A. Harten, \ENO Schemes with Subcell Reslutin," J. Cmput. Phys., V83 (989), pp.48-84. [] H. Yang, \ An Articial Cmpressin Methd fr ENO Schemes, the Slpe Mdicatin Methd," J. Cmput. Phys., V89 (990) pp.5-60. 3