As we shall see this will produce a diagonal mass

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Unformatted text preview: a need to use lumping. Thus, we select the approximation Uj (x t) of u(x t) on the mapping of (xj;1 xj ) to the canonical element as p X Uj ( t) = ckj (t)Pk ( ) (10.2.2a) k=0 where ckj (t) is an m-vector and Pk ( ) is the Legendre polynomial of degree k in . Recall (cf. Section 2.5), that the Legendre polynomials satisfy the orthogonality relation Z1 2i Pi( )Pj ( )d = 2i +j 1 ij 0 (10.2.2b) ;1 are normalized as Pi (1) = 1 i0 (10.2.2c) and satisfy the symmetry relation Pi( ) = (;1)i Pi(; ) i 0: (10.2.2d) The rst six Legendre polynomials are P0( ) = 1 P1( ) = 2 3 ; P2( ) = 3 2; 1 P3( ) = 5 2 3 4 2 5 3 P4( ) = 35 ; 30 + 3 P5( ) = 63 ; 70 + 15 : (10.2.3) 2 8 These polynomials are illustrated in Figure 10.2.1). Additional information appears in Section 2.5 and Abromowitz and Stegen 1]. Substituting (10.2.2a) into (10.2.1c), testing against Pi( ), and using (10.2.2b-d) yields _ hj cij + f (U(x t)) ; (;1)if (U(x t)) = Z 1 dPi( ) f (U ( t))d j j ;1 j 2i + 1 ;1 d i = 1 2 ::: p (10.2.4a) where (_) = d( )=dt. Neighboring elements must communicate information to each other and, in this form of the discontinuous Galerkin element method, this is done through the boundary ux 10.2. Discontinuous Galerkin Methods 21 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 10.2.1: Legendre polynomials of degrees p = 0 1 : : : 5. terms. The usual practice is to replace the boundary ux terms f (U(xk t), k = j ; 1 j , by a numerical ux function f (U(xk t) F(Uk (xk t)) Uk+1(xk t)) (10.2.4b) that depends on the approximate solutions Uk and Uk+1 on the two elements sharing the vertex at xk . Cockburn and Shu 12] present several possible numerical ux functions. Perhaps, the simplest is the average (10.2.5a) F(Uk (xk t)) Uk+1(xk t)) = f (Uk (xk t)) +2f (Uk+1(xk t)) : Based on our work with convection-di usion problems in Section 9.5, we might expect that some upwind considerations might be worthwhile. This happens to be somewhat involved for nonlinear vector systems. We'll...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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