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Summation by Parts Formula for Noncentered Finite Differences

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Summation by Parts Formula for Noncentered Finite Differences ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Z¨urich Swiss Federal Institute of Technology Zurich Summation by Parts Formula for Noncentered Finite Differences R. Bodenmann Research Report No. 95-07 September 1995 Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland Summation by Parts Formula for Noncentered Finite Differences R. Bodenmann Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland Research Report No. 95-07 September 1995 Abstract Boundary conditions for general high order finite difference approximations d to dx are constructed. It is proved that it is always possible to find boundary conditions and a weighted norm in such a way that a summation by parts formula holds. Keywords: high order finite differences, summation by parts formula, energy method, linear stability, initial-boundary value problem 1. Introduction For analysing numerical methods applied to partial differential equa- tions, a priori estimates of the growth rate of the solution are necessary. Such stability estimates ensure that small discretisation errors cannot grow arbitrarily. Analysing convergence behaviour of finite difference approximations to nonlinear problems, it turns out that stability inequalities of the linearisation of the methods have to be proved [1], [8]. Since schemes which are obtained by linearisation have variable coefficients, the en- ergy method turns out to be one of the most effective devices to derive bounds of the growth rate [2], [5], [7]. One of the most important components of the energy method in the continuous case is the integration by parts formula. To get an energy estimate for the discretised problem, the integration by parts formula has to be replaced by its discrete analog, the summation by parts formula. This is accomplished by constructing the difference operator including numerical boundary conditions in an appropriate manner. Kreiss and Scherer proved that in general new scalar products and norms have to be introduced to make possible the existence of differ- ence approximations which fulfil a summation by parts formula. They constructed weighted diagonal or even more general types of norms [3] d and proved the existence of difference approximations to dx of almost antisymmetric type such that the summation by parts formula holds. These results, which are reviewed in the next section, can also be found in [9] together with some examples. In Section 3, the theory is developed for more general difference operators, which do not need to be antisymmetric. It is pointed out in Section 4 that the results from Section 3 are optimal, even though they are weaker than the ones which can be proved in the antisymmetric case. An algorithm to compute proper boundary conditions for the diag- onal norm case is presented in Section 5 and some of its results are displayed in Section 6. 2. Review of the Antisymmetric Case d The goal is to construct difference approximations Q to dx on the interval [0, 1], which are of arbitrarily high order in the interior as well as on the boundaries of the interval and which fulfil a summation by 1 2 parts formula: (u, Qv)h = −(Qu, v)h + uJ vJ − u0v0 for any two grid functions u and v and for a certain discrete scalar product (·, ·)h. Obviously it is enough to investigate the halfline problem. So let us consider the halfline [0, ∞), which we divide into intervals of length h>0. Denote the gridpoints by xj = jh, j =0, 1,... and let uj = ∞ 2 u(xj) be a scalar grid function in l2, i.e. h j=0 |uj| < ∞. A discrete scalar product and a norm are defined by ! ∞ I I (u, v)h = <u, Hv >h +h ujvj "j=r r−1 ∞ (1) = h hi,juivj + h ujvj, i,j=0 j=r 2 " " #u#h =(u, u)h. I T T Here u =(u0,u1, . , ur−1) and H = H > 0 is a symmetric, posi- tive definite (r × r)-matrix. We want to construct matrices H and a d difference approximation Q to dx , such that the summation by parts formula (u, Qv)h = −(Qu, v)h − u0v0 (2) holds. The following theorems are proved in [3], [6], and [9]. Theorem 2.1 (Diagonal Norms). There exist scalar products (1), with H diagonal, and difference operators Q of accuracy p at the boundary and 2p in the interior, 1 ≤ p ≤ 4, such that the summation by parts formula (2) holds. Theorem 2.2 (Full Norms). There exist scalar products (1) and dif- ference operators Q of accuracy 2p − 1 at the boundary and 2p in the interior, p>0, such that the summation by parts formula (2) holds. The proofs are constructive and examples are computed in [9]. The simplest example for the diagonal norm case is given by −11 −0.500.5 hQ = −0.500.5 . .. .. .. and H = diag(0.5, 1, 1,...). 3 3. Noncentered Differences We now want to construct pairs of finite difference operators (Q−,Q+) and scalar products (1) such that the summation by parts formula (u, Q−v)h = −(Q+u, v)h − u0v0 (3) holds for any scalar grid functions u and v. Since u and v are considered as infinite vectors, Q∓ are represented as infinite matrices. Assume that Q∓ have the form (Q∓)11 (Q∓)12 hQ∓ = T , ) C∓ D∓ * with q0,0 q0,1 ··· q0,r−1 . (Q∓)11 = . , qr−1,0 qr−1,1 ··· qr−1,r−1 ∓ q0,r ··· q0,m 0 ··· . (Q∓)12 = . , qr−1,r ··· qr−1,m 0 ··· ∓ α0 α1 ··· αR 00··· 00··· α α α ··· α 0 ··· 00··· −1 0 1 R . .. .. .. .. .. . D− = , α−L ··· α−1 α0 α1 ··· αR 00··· 0 α ··· α α α ··· α 0 ··· −L −1 0 1 R . . .. .. .. .. .. .. .. . T D+ = −D−, 0 C∓ 0 ··· C∓ = , ) C∓ 0 ··· * α−L 00+ ··· 0 α α −L+1 −L 0 ··· 0 C = . .. .. , − . α−2 α−3 ··· α−L 0 + α α ··· α α −1 −2 −L+1 −L 4 αR 00··· 0 α α R−1 R 0 ··· 0 C = − . .. .. , + . α2 α3 ··· αR 0 + α α ··· α α 1 2 R−1 R 0 0 C− and C+ are (r − L) × L and (r − R) × R matrices respectively with only zeroes. Theorem 3.1. The operators Q− and Q+ satisfy the relation (3) if and only if they can be written as H−1A −H−1C H−1B −H−1C hQ = + , hQ = − , − CT D + CT D ) − − * ) + + * (4) where A and B are (r × r)-matrices of the form A = A1 + A2, B = B1 + B2 with a11 0 ··· 0 b11 0 ··· 0 00··· 0 00··· 0 A1 = . ,B1 = . ,a11+b11 = −1, . . 00··· 0 00··· 0 T and A2 + B2 =0. Proof. We can formally write (3) as I I I II −u0v0 = <u ,H(Q−)11v > + <u,H(Q−)12v > II T I II II I I + <u ,C−v > + <u ,D−v > + < (Q+)11u , Hv > II I T I II II II + < (Q+)12u , Hv > + <C+u ,v > + <D+u ,v > and by making use of the symmetry of H I I I II −u0v0 = <u ,H(Q+)11v > + <u,H(Q+)12v > II T I II II I I + <u ,C+v > + <u ,D+v > + < (Q−)11u , Hv > II I T I II II II + < (Q−)12u , Hv > + <C−u ,v > + <D−u ,v >, I T II T where u =(u0,u1, . , ur−1) and u =(ur,ur+1,...) . For D∓ we II II II II have the relation <u ,D−v > + <D+u ,v > = 0. Letting vI = 0, we get I II T I II I II <u ,H(Q−)12v > + <C+u ,v > = <u , (H(Q−)12+C+)v > =0 and I II T I II I II <u ,H(Q+)12v > + <C−u ,v > = <u , (H(Q+)12+C−)v > =0 5 for all vectors uI and vII. Therefore we have −1 −1 (Q−)12 = −H C+ and (Q+)12 = −H C−. Only the relation I T I <u, (H(Q−)11 +(Q+)11H)v > = −u0v0 remains. Therefore H(Q−)11 = A and H(Q+)11 = B must be of the required form. Let h = 1 and denote by rl 0l r l el l ( − 1) l w = ,e=(−1) ,f= 1 ,l=0, 1,... l f l . l ) l * . . . 1l l 0 the discretisation of (x − r) , with the conventions 0 =1, and e−1 =0. The following lemma characterises the accuracy of Q∓ in the interior. Lemma 3.2. −1 T d The operators h (C∓,D∓) approximate dx with order q at the points xj,j= r, r +1,... if and only if R p 1,p=1, ανν = (5) , 0,p=0, 2, 3, . , q. ν"=−L Accuracy conditions for Q∓ at boundary points are given by the next lemma. Lemma 3.3. −1 d The operators h ((Q∓)11, (Q∓)12) approximate dx with order τ at the points xj,j=0, 1, . , r − 1 if and only if lel−1 =(Q∓)11el +(Q∓)12fl,l=0, 1,..., τ. (6) We now want to calculate the coefficients of Q∓ at the boundary points. To do so, we first rewrite (6). By (4) we get − − A2el = gl ,gl = lHel−1 − A1el + C+fl,l=0, 1,..., τ, B e g+,g+ lHe B e C f ,l , ,..., τ. 2 l = l l = l−1 − 1 l + − l =0 1 (7) T Since A2 + B2 =0, the following compatibility conditions must hold − + <el,gm > + <em,gl > =0, 0 ≤ m, l ≤ τ. (8) If these conditions are fulfilled, the system (7) can be resolved as it is expressed in Lemma 3.4.
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