1 Introduction We are interested in numerical solutions to the conservation laws: u t + f(u) x =; u(x; ) = u (x);» x» 1: (1.1) Here we have written (1

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1 ON THE CONSERVATION AND CONVERGENCE TO WEAK SOLUTIONS OF GLOBAL SCHEMES 1 Mark H. Carenter Comutational Methods and Simulation Branch (CMSB) NASA Langley Research Center Hamton, VA and David Gottlieb and Chi-Wang Shu Division of Alied Mathematics Brown University Providence, RI 91 ABSTRACT In this aer we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used comact schemes and sectral collocation schemes, for solving hyerbolic conservation laws. It is shown that such schemes, if convergent boundedly almost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendroff theorem concerning conservative schemes. Key Words: conservation laws, conservation, weak solutions, convergence. AMS(MOS) subject classication: 65M1, 35L65 1 Research of the second and third authors is suorted by AFOSR grant F , ARO grant DAAD , DOE grant DE-FG-98ER5346, NSF grant DMS , NASA grant NAG and NASA contract NAS while in residence at ICASE, NASA Langley Research Center, Hamton, VA

2 1 Introduction We are interested in numerical solutions to the conservation laws: u t + f(u) x =; u(x; ) = u (x);» x» 1: (1.1) Here we have written (1.1) in the one dimensional form, but the results of this aer are also valid for multi dimensions. The urose of this aer is not to study the issue of convergence. We actually assume that the numerical solution converges boundedly a.e. (almost everywhere), to a certain function u(x; t). More recisely, for a numerical scheme dened at the (uniform or nonuniform) grid oints x j ;» j» N, with x = max(x j+ j )andv j (t) asthenumerical solution at x = x j,we dene the function v x (x; t) by v x (x; t) =v j (t); x j» x<x j+1 ; (1.) and assume that v x (x; t) is uniformly bounded with resect to x, t,and x,and,as x, v x (x; t) converges ointwise a.e. to u(x; t). See, e.g., [5, 16, 17, ] for discussions, in the scalar case, of convergence of some of the schemes studied in this aer, under the L 1 boundedness assumtion. We will concentrate on the issue of whether the limit function u(x; t) is a weak solution to (1.1), that is whether it satises Z T (u(x; t)f t (x; t)+f(u(x; t))f x (x; t)) dx dt u (x)f(x; ) dx = (1.3) for any smooth function f(x; t) which is comactly suorted. Also, in this aer we only consider semidiscrete method-of-lines schemes, i.e. schemes which are discretized in the satial variable(s) only. The classical result in this area is the famous Lax-Wendroff Theorem [11]: Theorem 1.1. (Lax and Wendroff) If the numerical solution of a conservative scheme: (v j ) t + 1 ^fj+ 1 x ^f j 1 = (1.4)

3 where the numerical flux ^f j+ 1 = ^f (v j ;:::;v j+q ) (1.5) is local (i.e. and q are constants indeendent of x), Lischitz continuous in every argument, and consistent with the hysical flux ^f(v; :::; v) = f(v), converges boundedly a.e. (almost everywhere) to a function u(x; t), then u(x; t) is a weak solution to (1.1). The roof follows easily from a summation by arts and an alication of the dominant convergence theorem. See LeVeque [13] for a slightly different version of this theorem and its roof. The Lax-Wendroff Theorem, however, does not cover global schemes, i.e. schemes which can not be written in the form (1.4) with a local flux ^f j+ 1. Examles of global schemes include the comact schemes [1,,3,5],and various sectral Galerkin or collocation schemes (Fourier, Legendre, Chebyshev) [9, 1, 6]. We will extend the Lax Wendroff Theorem to these global schemes in this aer. We remark that there are discussions in the literature about schemes which are not of the conservative form (1.1) but nevertheless still converge to weak solutions. One such examle is the class of schemes for general curvilinear coordinates, see [19] for a roof that such schemes actually do converge to weak solutions. In [8] the authors discussed conservation issues of Chebyshev methods. However, they only considered the mean of the solution, that is, they veried that the limit solution satises (1.3) with f(x; t) =1. The organization of the aer is as follows. In Section we discuss comact schemes. Section 3 contains the Legendre collocation method, while Section 4 discusses the Chebyshev method. Section 5 discusses the Legendre aroximation in the multi-domain case. To close this section we mention that in this aer C (or c) is a generic constant. 3

4 Comact Schemes Comact schemes are methods where the derivatives are aroximated by rational function oerators on the discrete solutions. We consider comact schemes dened on a uniform grid x j ;» j» N. For examle, a fourth order central comact aroximation to the derivative is [1]: 1 6 ((v x) j +4(v x ) j +(v x ) j+1 ) = 1 (v j+1 v j ) (.1) and a third order uwind comact aroximation to the derivative is [5]: 1 3 ( (v x) j +5(v x ) j (v x ) j+1 ) = 1 (3v j 4v j + v j ) : (.) Adequate boundary conditions must be used for the comact schemes, to retain accuracy and stability, see [], [3] for details. Together with boundary conditions, a comact scheme for (1.1) can be written as Pv t + Qf(v) = (v B g B ) (.3) where v =(v ; :::v N ) Λ is the numerical solution, is a constant, v B =(v ; ;:::;;v N ) Λ is the boundary art of the numerical solution, and g B = (g ; ;:::;;g N ) Λ is the given boundary data. Deending on the wind direction, one or both of the rst and last comonents of v B and g B may also be zero(s). The matrices P and Q satisfy the following conditions [3]: ffl P is symmetric, and satises Pf f = O( x) (.4) Here and below, f = (f(x );:::;f(x N )) Λ, and f(x) is an arbitrary smooth (C 1 or smoother) function. O( x) for a vector means that each comonent is bounded by a constant times x, and the constant deends only on the derivatives of f(x); ffl Q is almost" anti-symmetric, that is: Q + Q Λ = R + S (.5) 4

5 where R = (r ij ), and r ij = excet for r and r NN. S is either identically for the central comact schemes, or satises Sf = O( x) (.6) for the uwind comact schemes, where f is dened as before. Also, Q is at least a rst order aroximation to the derivative: Qf f x = O( x): (.7) We can easily verify that all the comact schemes in [1,, 3, 5] satisfy the above conditions for P and Q. For such comact schemes we can state the following roosition: Proosition.1. If the solution of the comact scheme (.3) converges almost everywhere to a function u(x; t), then u(x; t) isaweak solution to (1.1). Proof: For any comactly suorted, C function f(x; t), we denote by f =(f(x ;t); :::; f(x N ;t)) Λ, left multily (.3) by f Λ, and integrate over [,T] to obtain: Z T f Λ (Pv t + Qf(v)) dt = due to the zero boundary conditions of f. Now integrating by arts in t for the rst term, taking a transose of the equation (which is a scalar), and using the symmetry of P and condition (.5) of Q, we obtain: Z T Z T (v Λ Pf t + f(v) Λ Qf) dt (v Λ Pf) j t= = f(v) Λ Sf dt Or, considering (.4), (.6), (.7), and the uniform boundedness (with resect to the mesh size x) of v, Z T (v Λ f t + f(v) Λ f x ) dt (v Λ f) j t= = O(1) (.8) where the constant term O(1) results from a summation of N = 1 O( x) terms of O( x) quantities; the constant deends only on the derivatives of f(x). 5

6 Recalling the denition of the function v x (x; t) in (1.), we can multily (.8) by x to obtain Z T where v x (x; t)f x t (x; t)+f(v x (x; t))f x (x; t) dx dt x v x (x; )f x (x; ) dx = O( x) (.9) f x (x; t) =f(x j ;t); f x x (x; t) =f x (x j ;t); f x t (x; t) =f t (x j ;t); x j» x<x j+1 ; (.1) By assumtion, v x (x; t) converges to u(x; t) boundedly a.e. There is no roblem about the uniform convergence of f x (x; t), f x x (x; t) andf x (x; t) due to the smoothness of f. By the dominant convergence theorem, taking the limit as x in (.9), we obtain (1.3). This roves that u(x; t) is a weak solution of (1.1). 3 Legendre Sectral Collocation Schemes t The Legendre collocation method can be written in the following N (x; t) N f(u N (x; t)) = SV (u N (x; t)) + Bu N (x; t) (3.1) where u N (x; t) is the numerical solution which is a olynomial of degree at most N in x, I N is the Legendre interolation oerator, i.e. for any function g(x), I N g(x) is the unique olynomial of degree at most N satisfying I N g(x j ) = g(x j ) at the N +1Legendre Gauss- Lobatto oints x j,which are the zeros of the olynomial (1 x )P N, where P N is the Legendre olynomial of degree N. The term SV is the sectral viscosity term needed to stabilize the scheme and in order for the assumtion v x (x; t) converges boundedly a.e. to a function u(x; t)" to be realistic. We consider here the suerviscosity term SV (u N ) = ffl()s N s " ( s u N (x; t) (3.) 6

7 ffl is the suerviscosity coefcient, s is an integer growing with N [1, 1, 14, 15]. We remark that this suerviscosity term is equivalent in ractice to a low ass lter. Finally, theboundary term Bu N (x; t) could be either, or (u N (1;t) g 1 )(1 + x)p N(x); or (u N (;t) g )()PN(x); or a combination, deending on the wind directions at the boundary oints. Here is a constant chosen for stability andg 1 and g are functions of the time only. Let f(x; t) be a test function in C 1. Take f N (x; t) = I N f(x; t), then clearly f N are olynomials of degree at most N 1andvanish at both boundary oints x = ±1. Also f N (x; t) f(x; t), (f N ) x (x; t) f x (x; t), and (f N ) t (x; t) f t (x; t) uniformly. We denote now by (f; g) = f(x)g(x)dx and by (f; g) N = NX f(x j )g(x j ) j where j > are the weights in the Gauss-Lobatto formula. We note that (f; g) N =(f; g) if fg is a olynomial of degree at most N 1. We rst show that the boundary terms do not cause a roblem: Lemma 3.1. (f N ;Bu N ) = : (3.3) Proof: We start by observing that (f N ;Bu N )=(f N ;Bu N ) N : Bu N vanishes for the inner Gauss-Lobatto oints and f N vanishes at the boundaries and therefore the Lemma is roven. 7

8 With (3.3), we multily (3.1) by f N (x; t), integrate over x, and integrate by arts for the second term to obtain f N (x; N(x; t) = ffl()s N s We now estimate the right hand side of (3.4): Lemma 3.. lim N1 ffl() s N N (x; t) dx I N f(u N (x; t)) dx " f N (x; t) ( s u N (x; t) dx: (3.4) " f N (x; t) ( s u N (x; t) dx = : (3.5) Also, the quantity under the limit sign is uniformly bounded with resect to t. Proof: Since f N is a olynomial of degree N 1, f N (x; t) = N X ^f k;n (t)p k (x) where ^f k;n (t) are the collocation Legendre coefcients of the test function f. Note that " ( s P k (x) =() s k s (k +1) s P k (x); and therefore (f N ;SV(u N )) = () s ffl NX N s k s (k +1) s ^fk;n (t)^u k;n (t)(p k ;P k ); Here ^u k;n are the Legendre collocation coefcients of u N. We note that as a consequence of the uniform boundedness of u N (x j ) and the fact that f is in C 1, j ^f k;n (t)^u k;n (t)j» C k 3 : (3.6) This imlies (3.5) and the uniform boundedness of the quantity under the limit sign with resect to t. 8

9 We thus only have to deal with the left hand side of (3.4). We integrate (3.4) in t, integrate by arts for the rst term, and use Lemma 3. to obtain: Z T u N (x; N(x; t) + I N f(u N (x; N(x; t) dx dt u N (x; )f N (x; ) dx = o(1): (3.7) It looks like we can immediately take the limit as in the Lax-Wendroff Theorem. The trouble is that, we have only assumed the uniform boundedness of u N (x j ;t), hence of f(u N (x j ;t)), but this does not imly the uniform boundedness of either u N (x; t) or I N f(u N (x; t)) due to the lack of regularity. We need the following Lemma: Lemma 3.3. Let v x be the iecewise linear olynomial taking the values u N (x j ;t), then u N (x; N(x; t) = v x N(x; t) + o(1); (3.8) I N f(u N (x; N(x; t) = I N f(v x N(x; t) + o(1): (3.9) Proof: We will switch back and forth between integrals and quadrature summations: where u N (x; N(x; t) (x) = = = = u N (x; N(x; t) NX j= N u N (x j N(x j ;t) j (x)v x (x; t)(f ) x N t (x; t)dx j x j+ j ; x j» x<x j+1 : In [18] it has been established that (x) is uniformly bounded and converges a.e to 1 as x. This roves (3.8). The roof for (3.9) is similar. 9

10 We can state now Theorem 3.4. If the function v x (x; t) dened in (1.), obtained from the solution of the Legendre collocation scheme (3.1) at the Legendre Gauss-Lobatto oints x j, converges almost everywhere to a function u(x; t), then u(x; t) is a weak solution to (1.1). Proof: By assumtion, v x (x; t)converges to u(x; t) boundedly a.e. Also, there is no roblem about the uniform convergence of f x N(x; t), (f ) x N x (x; t) and (f ) x N t (x; t) due to the smoothness of f. Using the dominant convergence theorem, taking the limit as x in (3.8), (3.9) and (3.7), we obtain (1.3). This roves that u(x; t) isaweak solution of (1.1). We close this section by commenting on other sectral viscosity terms in (3.1) that stabilize the Legendre method. One such term is ffl Q@u N where the sectral viscosity oerator Q is dened by where and Qf = f = NX NX ^Q k ^fk P k (x) ^f k P k (x) 1 ^Q k 1 with m N growing with N. ^Q k =; mn 4 k k» m N ; k>m N We can establish also for this viscosity term that (f N ;SV(u N )) and therefore the result above holds also for this kind of sectral viscosity. 1

11 4 Chebyshev Sectral Collocation Schemes In this section we consider the Chebyshev collocation schemes. These are more difcult to analyze than the Legendre method because of the weight function 1 1 x. The Chebyshev collocation method can be written in the following N (x; t) N f(u N (x; t)) = ffl()s N s " 1 # s u N (x; t)+bu N (x; t) (4.1) where again u N (x; t) is the numerical solution which is a olynomial of degree at most N in x, J N is the Chebyshev interolation oerator, i.e. for any function g(x), J N g(x) is the unique olynomial of degree at most N satisfying J N g(x j )=g(x j )atthen + 1 Chebyshev Gauss-Lobatto oints x j. ffl is the suerviscosity coefcient, s is an integer growing with N [1, 14, 15]. We remark again that this suerviscosity term, which in ractice is equivalent to a low ass lter, or a similar vanishing viscosity term [16, 17], is needed in order for the assumtion v x (x; t) converges boundedly a.e. to a function u(x; t)" to be realistic. Finally, the boundary term Bu N (x; t) could be either, or (u N (1;t) g 1 )(1 + x)tn(x); or (u N (;t) g )()TN(x); or a combination, deending on the wind directions at the boundary oints. Here is a constant chosen for stability andg 1 and g are functions of the time only. Let f(x; t) be a test function in C, 5 that is, all x derivatives of f(x; t) u to order 5 vanish at the boundary oints x = ±1. Such test functions are, of course, dense in C. 1 It follows that ( ) 3= f(x; t) is in C. 3 We denote the (N 5)-th degree Chebyshev interolation olynomial of the function ( ) 3= f(x; t) by ο N 5 (x; t) =J N 5 (( ) 3= f(x; t)); (4.) and note that ο N 5 (x; t) ( ) 3= f(x; t); 11

12 @ο N 5 (x; ( ) 3= f(x; t) N 5 (x; ( ) 3= f(x; t) ; uniformly. We now take N (x; t) =(1 x ) ο N 5 (x; t); (4.3) then N is a olynomial of degree at most N 1 and vanishes at both boundary oints x = ±1 together with its rst and second x derivatives. Moreover, it can be easily veried that N (x; t) f(x; t); N (x; t) f x (x; t); x N (x; t) f t (x; t); (4.4) t uniformly. We again rst show that the boundary term does not cause a roblem: Lemma 4.1. (1 + x)tn(x) N(x; t) dx = ; ()TN(x) N(x; t) dx = : (4.5) Proof: We only rove the rst equality. Zero boundary values of N (x; t) and its rst x derivative imly (1 + x)tn(x) N(x; t) N (x; t) dx = T N (x) = = : T N (x) (1 + (x; t) x dx (1 + x)+ N(x; t) dx The last equality is due to the fact N (x; t) (1 + x)+ N(x; t) N(x; t) (1 + x) + (1 + x)( )ο N 5 (x; t) is a olynomial of degree at most N, hence is orthogonal to T N (x) with the weight 1 1 x. 1

13 With (4.5), we can now multily (4.1) by N(x;t) 1 x, integrate over x, and integrate by arts for the second term to obtain N (x; N (x; t) dx = ffl()s N s N (x; t) We now estimate the right hand side of (4.6): N (x; t) " x ( J N f(u N (x; t)) dx # s u N (x; t) dx: (4.6) Lemma 4.. lim N1 ffl() s N s N (x; t) " 1 # s u N (x; t) dx = : (4.7) Also, the quantity under the limit sign is uniformly bounded with resect to t. Proof: Integrating by arts s times, and noticing that the boundary terms are always because of the fact that N (x; t) vanishes at the boundaries and because of the factor, we obtain N (x; t) " 1 # s u N (x; t) dx = " u N (x; t) 1 s x N(x; t) dx: (4.8) Recalling the denition of ο N 5 (x; t) in(4.),we have ο N 5 (x; t) = N 5 X ^ο k (t)t k (x) where ^ο k (t) are the collocation Chebyshev coefcients of the C 3 function ( ) 3= f(x; t), hence Now, by the relationshi between N and ο N 5 in (4.3): N 5 X N (x; t) = ( ) j^ο k (t)j» C k 3 : (4.9) ^ο k (t)t k (x) X = 1 N 5 4 (1 T (x)) ^ο k (t)t k (x) 13

14 = 1 4 = 1 16 X N N 5 X ^ο k (t)t k (x) N X N 5 X N 5 X ^ο k (t)(t k+ (x)+t k (x)) ^ο k (t)(t k+4 (x)+t k (x)+t k 4 (x)) ^οk 4 (t) 4^ο k (t)+6^ο k (t) 4^ο k+ (t)+^ο k+4 (t) T k (x) ^ k (t)t k (x) where we take the convention that ^ο k (t) = for k < or k > N 5. This, together with (4.9), clearly imlies We now use the equality " 1 j ^ k (t)j» C k 3 : (4.1) # s N(x; t) = and the integral-quadrature equivalence: " u N (x; t) 1 s x N(x; t) dx = NX j= X N () s k s ^k (t)t k (x) (4.11) w j u N (x j 1 # s N(x; t)1 A where x j and w j are the nodes and weights of the Chebyshev Gauss-Lobatto quadrature formula, because the integrand " # 1 u N (x; s x N(x; t) is a olynomial of degree at most N 1. This, together with the uniform boundedness of u N (x j ;t) and (4.8), (4.1) and (4.11), imlies (4.7) and the uniform boundedness of the quantity under the limit sign with resect to t. We thus only have to deal with the left hand side of (4.6). We integrate (4.6) in t, integrate by arts for the rst term, and use Lemma 4. to obtain: Z T N (x; t) N (x; t) u N (x; t) + J N f(u N (x; t)) t 14 u N (x; ) N(x; ) x dx dt x=xj dx = o(1): (4.1)

15 Again, the difculty is that we have only assumed the uniform boundedness of u N (x j ;t), hence of f(u N (x j ;t)), not the uniform boundedness of either u N (x; t) or J N f(u N (x; t)). We again get around this by switching between integrals and quadrature summations: N (x; t) N (x; t) u N (x; t) + J N f(u N (x; t)) = NX j= w N (x; t) un (x j ;t) t t x=xj + f(u N (x j ;t)) because we can easily verify that the integrand N (x; t) u N (x; t) + J N f(u N (x; t)) t N (x; t) N (x; t) x dx x 1 x x=xj A (4.13) is a olynomial of degree at most N 1. Recalling the denition of the function v x (x; t) in (1.) and that of f x (x; t) etc. in (.1), we can use (4.13) to rewrite (4.1) as x 1 x N (x; t) N (x; t) v x (x; t) + f(v x (x; t)) A dx dt where Z T t (x)v x (x; ) N (x; ) (x) = w j ; x j» x<x j+1 : x j+ j x x dx = o(1) (4.14) Clearly, (x) is uniformly bounded and converges to 1 as x. By assumtion, v x (x; t) converges to u(x; t) boundedly a.e. Also, (4.4) guarantees the uniform convergence of the N related terms to the right limits. Using the dominant convergence theorem, taking the limit as x in (4.14), we obtain (1.3). This roves that u(x; t) is a weak solution of (1.1), i.e. we have roved the following Proosition 4.3. If the function v x (x; t) dened in (1.), obtained from the solution of the Chebyshev collocation scheme (4.1) at the Chebyshev Gauss-Lobatto oints x j, converges almost everywhere to a function u(x; t), then u(x; t) isaweak solution to (1.1). 15

16 5 Multi-Domain Legendre Methods In this section we will discuss stable and conservative interface boundary conditions for the multi-domain Legendre method alied to equation (1.1). We assume that the domain» x» 1 is divided into two domains, and for the sake of simlicity we assume that the interface oint is x =. We will denote by u N (x; t) the numerical aroximation in» x» andby v N (x; t) the solution at» x» 1. The multi-domain Legendre method is given N I I N f(u N) = B(u N (;t)) + 1 Q I (x) h f + (u N (;t)) f + (v N (;t)) i + Q I (x) h f (u N (;t)) f (v N (;t)) i + SV (u N ); (5.1) I II N f(v N) = 3 Q II (x) h f + (v N (;t)) f + (u N (;t)) i + 4 Q II (x) h f (v N (;t)) f (u N (;t)) i +SV (v N )+B(v N (1;t)): (5.) Equation (5.1) holds in the interval» x», and (5.) holds in» x» 1. I I N f(u N) interolates f(u N ) at the zeroes ο j of the olynomial xq I and I II N f(v N) interolates f(v N ) at the zeroes j of the olynomial xq II, where Q I (x) = (1 + x)p N(x +1) P N(1) ; Q II (x) = ()P N(x 1) : PN() The sectral viscosities SV (u N ) and SV (v N ) are of the form " # s SV (u N ) = x(x +1) u N s N ; (5.3) " # s SV (v N ) = x() v N s N : (5.4) At this oint we stress that the results of this section are valid only for this form of sectral viscosity and not for the others discussed in Section 3. The reason for that will be evident in the roof. Finally the boundary oerators B at the ends of the interval» x» 1 are left unsecied for now. 16

17 We will also denote the scalar roduct (; q) N = P N j= T (ο j )q(ο j ) j if (x) and q(x) are dened in [; ] and (; q) N = P N j= T ( j )q( j ) j if (x) andq(x) are dened in [; 1], and j are the weights in the Gauss-Lobatto Legendre quadrature formula. Note that if q is a olynomial of degree at most N 1 dened in [; ] then (; q) N =(; q) = A similar formula holds in the interval [; 1]. Z T (x)q(x)dx: Our aim in this section is to show that the choice of the arameters i, i = 1; 4 that leads to linear stability is sufcient for roving conservation, i.e. if the numerical solution u N (x; t);v N (x; t) converges boundedly a.e to functions u(x; t);v(x; t), then the solution w dened by w(x; t) =u(x; t) if» x» and w(x; t) =v(x; t) if» x» 1converges to the weak solution of (1.1). We will discuss rst the stability of (5.1)-(5.). We state Proosition 5.1. The boundary oerators are dissiative, i.e. (u N ;B(u N (;t))) N + 1 ut N(;t)Au n (;t)» ; (5.5) (v N ;B(v N (1;t))) N 1 vt N(;t)Av N (1;t)» : (5.6) Proosition 5.1 imlies that the boundary treatment at the end-oints of the interval is stable. Examle of such oerators are given in [4]. We are ready to state the stability theorem for the linear constant coefcient case. In this case there is no need for the sectral viscosity terms and we will ignore them. We assume that f = Au where A is symmetric in equation (1.1) and in the same way f + = A + u, f = A u where the eigenvalues of A + are nonnegative and those of A are nonositive. Theorem 5.. Let u N, v N be the solutions of (5.1)-(5.). Dene E(t) =(u N (x; t);u N (x; t)) N +(v N (x; t);v N (x; t)) N ; 17

18 then E(t)» E() rovided that 1» 1 ; 1 ; 3» 1 ; 4 1 ; (5.7) 1 3 = 1 ; 4 = 1 : Proof: The roof follows from multilying (5.1), (5.) by u T N ;vt N and taking the scalar roduct. We use Proosition 5.1 and the following notation u = u N (;t); v = v N (;t); ff ± = u A ± u ; ± = v A ± v ; fl ± = u A ± v to get 1 d dt E(t)» ( 1 1 )ff + ( )fl + +( ) + +( 1 )ff ( + 4 )fl +( ) (5.8) The conditions stated above for the i 's guarantee that the right hand side of (5.8) is nonositive and the roof is comleted. Remark: The Discontinuous Galerkin method alied to this roblem leads to the uwinding choice 1 = 4 =, = 3 = 1. Another attractive choice that involves no slitting of the fluxes is 1 = = 3 = 4 = 1 : We turnnow to the main urose of this section, namely the roof of convergence in the nonlinear case of the numerical solution to the correct entroy solution. We rst show that the sectral suerviscosity terms do not create any roblems: consider a comactly suorted (in [-1,1]) test function Ψ(x; t)inc m [; 1], m is to be secied later. 18

19 Lemma 5.3. Let f N (x; t) and N (x; t) be the Legendre interolation olynomials of Ψ(x; t) intheintervals [; ] (with collocation oints ο j ), and [; 1] (with collocation oints j ), resectively. Then lim (f N;SV(u N )) = ; (5.9) N1 lim ( N;SV(v N )) = : (5.1) N1 where the sectral suerviscosities are dened in (5.3) and (5.4). Proof: Since u N is a olynomial of degree N it can be reresented as Therefore from (5.3) u N (x) = SV (u N ) = ffl ()s+1 N s = ffl 1 NX N s Also the test function f N can be reresented as f N = NX ^u k;n P k (x +1): " x(x +1) u N NX k s (k +1) s^u k;n P k (x +1): ^f k;n P k (x +1): From the orthogonality of the Legendre olynomials it follows that j(f N ;SV(u N ))j = ffl 1 NX N s We can choose m large enough such that ^f k;n ^u k;n k s (k +1) s (P k (x +1);P k (x + 1)): j ^f k;n ^u k;n j» C k 3 and since ^u k and (P k (x +1);P k (x + 1)) are bounded, the roof is established. The roof for (5.1) is similar. 19

20 It is self evident that the form of the sectral viscosity SV is crucial. In fact the factor is necessary in the roof. Note that (f N ;SV(u N )) = (SV (f N );u N )=(SV (f N );u N ) N (5.11) We basically roved that the rst argument in the scalar roduct in the right hand side of (5.11) tends to zero whereas the second argument is bounded. The relation (5.11) is not true for other forms of the sectral viscosity where the factor does not aear. Lemma 5.4. Let i satisfy (5.8), then (f N N ) N (f(u N N ) N +( N N ) N (f(v N N = (f N ;SV(u N ))+( N ;SV(v N )) (5.1) ) N Proof: Taking the scalar roduct of equation (5.1) with f N and (5.) with N and denoting by f(;t)=f, for all the quantities one gets (f N N ) N +(f N I N f(u N) ) N = 1 f f + (u ) f + (v ) + f f (u ) f (v ) +(f N ;SV(u N )) N ; (5.13) ( N N ) N +( N II N f(v N) ) N = 3 f + (v ) f + (u ) 4 f (v ) f (u ) +( N ;SV(v n )) N ): (5.14) Here we used the fact that f(;t)=(1;t)=. Now, (f N N f(u N) ) N = (f N N f(u N) ) = f f(u ) (I I N f(u N ) = f f(u ) (f(u N N ) N (5.15)

21 and by the same token ( N N f(v N) ) N = f f(v ) (f(v N N ) N : (5.16) Using (5.15)-(5.16) in (5.13)-(5.14) one gets f N N (f(u N N ) N + N N (f(v N N ) N = f + (u ) f + (v ) [ 1 3 1] f + f (u ) f (v ) [ 4 1] f +(f N ;SV(u N ))+( N ;SV(v N )) : Taking (5.8) into account, the lemma is roven. We integrate now (5.1) with resect to time to get Z T ( u N N f(u N N N + v N N = (u N (t =);f N (t = )) (v N (t =); N (t = )) : f(v N N We now use Lemma 3.3 in Section 3 to convert u N, v N to u x v x, which are dened in (1.) as the iecewise olynomials having the values of u N, v N at the grid oints. Combining then with Lemma 5.3 it follows: Theorem 5.5. Let u N and v N be the multi-domain Legendre aroximation (5.1)-(5.) to (1.1). Assume that the functions u x and v x dened in (1.) converge boundedly a.e., then the limit function is a weak solution of (1.1). References N ) dt [1] C. Canuto, M.Y. Hussaini, A. Quarternoni and T. A. Zang, Sectral Methods in Fluid Dynamics, Sringer-Verlag, [] M. Carenter, D. Gottlieb and S. Abarbanel, Stable and accurate boundary treatments for comact, high order nite difference schemes, Alied Numerical Mathematics, v1 (1993),

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