36 we obtain 1 mcn1 cn t k2cn cn1 z

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Unformatted text preview: c(n+1); ; cn;) + t K(2cn+ + c(n+1); ) = t (2ln + l(n+1);) M( c + 2 6 6 (n+1); ; cn+ + t K(cn+ + 2c(n+1);) = t (ln + 2l(n+1);): Mc 2 6 6 This pair of equations must be solved simultaneously for the two unknown solution vectors cn+ and c(n+1); . This is an implicit Runge-Kutta method. Problems 1. Consider the Galerkin method in time with a continuous basis as represented by (9.3.1). Assume that the solution c(t) is approximated by the linear function (9.3.2a-c) on (tn tn+1) as in Example 9.3.1, but do not assume that the test space w(t) is linear in time. 1.1. Specifying w( ) = !( ) 1 1 : : : 1]T and assuming that M and K are independent ot t, show that (9.3.1a) is the weighted average scheme M + tK]cn+1 = M ; (1 ; ) tK]cn + t (1 ; )ln + ln+1] with R !( )N ( )d n =;R : !( )d 1 1 1 1 +1 ;1 When di erent trial and test spaces are used, the Galerkin method is called a Petrov-Galerkin method. 1.2. The entire e ect of the test function !(t) is isolated in the weighting factor . Furthermore, no integration by parts was performed, so that !(t) need not be continuous. Show that the choices of !(t) listed in Table 9.3.1 correspond to the cited methods. 2. The discontinuous Galerkin method may be derived by simultaneously discretizing a partial di erential system in space and time on (t ; n; t(n+1); ). This form may have advantages when solving problems with rapid dynamics since the mesh may be either moved or regenerated without concern for maintaining continuity 18 Parabolic Problems Scheme ! Forward Euler (9.2.10b) (1 + ) 0 Crank-Nicolson (9.2.12b) ( ) 1/2 Crank-Nicolson (9.2.12b) 1 1/2 Backward Euler (9.2.11b) (1 ; ) 1 1 Galerkin (9.3.3) Nn+1( ) 2/3 Table 9.3.1: Test functions ! and corresponding methods for the nite element solution of (9.2.4a) with a linear trial function. between time steps. Using (9.2.2a) as a model spatial nite element formulation, assume that test functions v(x y t) are continuous but that trial functions u(x y t) have jump discontinuities at tn. Assume Dirichlet boundary data and show that the space-time discontinuous Galerkin form of the problem is (v ut)ST + (v( tn) u( tn+) ; u( tn;)) + AST (v u) = (v f )ST 8v 2 H01( (tn+ t(n+1); )) where and (v u)ST = Zt ; (n+1) ZZ tn+ vudxdydt AST (v u) = (vx pux)ST + (vy puy )ST + (v qu)ST : In this form, the nite element problem is solved on the three-dimensional strips (tn; t(n+1); ), n = 0 1 : : : . 9.4 Convergence and Stability In this section, we will study some theoretical properties of the discrete methods that were introduced in Sections 9.2 and 9.3. Every nite di erence or nite element scheme for time integration should have three properties: 1. Consistency: the discrete system should be a good approximation of the di erential equation. 2. Convergence: the solution of the discrete system should be a good approximation of the solution of the di erential equation. 3. Stability: the solution of the discrete system should not be sensitive to small perturbations in the data. 9.4. Convergence and Stability 19 Somewh...
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