A high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is [Oscr]([Delta]t2p+1) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower-upper symmetric Gauss-Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge-Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173-261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411-435), which would only require a time step proportional to the spatial step. Copyright © 2009 John Wiley & Sons, Ltd.
miércoles, 16 de septiembre de 2009
Implicit time integration of hyperbolic conservation laws via discontinuous Galerkin methods
A high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is [Oscr]([Delta]t2p+1) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower-upper symmetric Gauss-Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge-Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173-261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411-435), which would only require a time step proportional to the spatial step. Copyright © 2009 John Wiley & Sons, Ltd.
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