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Unformatted text preview: ordinary di erential equations 24, 19] and hyperbolic 5, 6, 7, 8, 12, 11, 13, 16], parabolic 14, 15], and elliptic 4, 3, 28] partial di erential equations. A recent proceedings contains a complete and current survey of the method and its applications 10]. The discontinuous Galerkin method has a number of advantages relative to traditional nite element methods when used to discretize hyperbolic problems. We have already noted that it has the potential of sharply representing discontinuities. The piecewise continuous trial and test spaces make it unnecessary to impose interelement continuity. There is also a simple communication pattern between elements that makes it useful for parallel computation. We'll begin by describing the method for conservation laws (10.1.1) in one spatial dimension. In doing this, we present a simple construction due to Cockburn and Shu 12] rather than the (more standard) approach 19] used in Section 9.3 for time integration. Using a method of lines formulation, let us divide the spatial region into elements (xj;1 xj ), j = 1 2 : : : N , and construct a local Galerkin problem on Element (xj;1 xj ) in the usual manner by multiplying (10.1.1a) by a test function v and integrating to obtain Zx vT ut + f (u)x ]dx = 0: (10.2.1a) j x ;1 j The loading term b(x t u) in (10.1.1a) causes no conceptual or practical di culties and we have neglected it to simplify the presentation. Following the usual procedure, let us map (xj;1 xj ) to the canonical element (;1 1) using the linear transformation x = 1 ; xj;1 + 1 + xj : (10.2.1b) 2 2 Then, after integrating the ux term in (10.2.1a) by parts, we obtain hj Z 1 vT u d + vT f (u)j1 = Z 1 vT f (u)d (10.2.1c) ;1 2 ;1 t ;1 where hj = xj ; xj;1: (10.2.1d) 20 Hyperbolic Problems Without a need to maintain interelement continuity, there are several options available for selecting a nite element basis. Let us choose one based on Legendre polynomials. As we shall see, this will produce a diagonal mass matrix without...
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