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12
Discrete mechanics and variational integrators
 Acta Numer
, 2001
"... This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the disc ..."
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Cited by 146 (28 self)
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This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
Symplectic specialized partitioned additive RungeKutta methods for conservative systems with holonomic constraints
 Dept. of Math., Univ. of Iowa, USA
"... Abstract. We consider a general class of systems of overdetermined differentialalgebraic equations (ODAEs). We are particularly interested in extending the application of the symplectic Gauss methods to Hamiltonian and Lagrangian systems with holonomic constraints. For the numerical approximation t ..."
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Cited by 6 (3 self)
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Abstract. We consider a general class of systems of overdetermined differentialalgebraic equations (ODAEs). We are particularly interested in extending the application of the symplectic Gauss methods to Hamiltonian and Lagrangian systems with holonomic constraints. For the numerical approximation to the solution to these ODAEs, we present specialized partitioned additive Runge– Kutta (SPARK) methods, and in particular the new class of (s, s)Gauss–Lobatto SPARK methods. These methods not only preserve the constraints, symmetry, symplecticness of the flow, and variational nature of the trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but they also have an optimal order of convergence 2s.
Error estimation and control for ODEs
 J. of Scientific Computing
, 2005
"... This article is about the numerical solution of initial value problems for systems of ordinary differential equations (ODEs). At first these problems were solved with a fixed method and constant step size, but nowadays the generalpurpose codes vary the step size, and possibly the method, as the int ..."
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Cited by 5 (0 self)
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This article is about the numerical solution of initial value problems for systems of ordinary differential equations (ODEs). At first these problems were solved with a fixed method and constant step size, but nowadays the generalpurpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article.
Inexact Simplified Newton Iterations For Implicit RungeKutta Methods.
 SIAM J. Numer. Anal
, 2000
"... . We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major difficulty and computational bottleneck in the implementation of fully implicit RungeKutta (IRK) methods resides in the numerical solution of the resulting systems of nonlinear equations. To solve ..."
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Cited by 3 (2 self)
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. We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major difficulty and computational bottleneck in the implementation of fully implicit RungeKutta (IRK) methods resides in the numerical solution of the resulting systems of nonlinear equations. To solve those systems we show that the use of inexact simplified Newton methods is efficient. Linear systems of the simplified Newton method are solved approximately with a preconditioned linear iterative method. Sufficient conditions ensuring local convergence of the inexact simplified Newton method for general nonlinear equations are given. The preconditioner that we use is based on the Wtransformation of the RK coefficients and on the blockLU decomposition of the simplified Jacobian after Wtransformation. A new code based on those techniques, SPARK3, is shown to be effective on two problems, the first one is a linear convectiondiffusion problem and the second one a reactiondiffusion problem...
Preconditioning and parallel implementation of implicit Runge–Kutta methods
, 2001
"... Abstract. A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with ..."
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Cited by 2 (2 self)
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Abstract. A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually be solved approximately with a preconditioned linear iterative method. In this article we present a truly parallelizable preconditioner to the approximate Jacobian matrix. Its decomposition cost for a sequential or parallel implementation can be made equivalent to the cost corresponding to the implicit Euler method. The application of the preconditioner to a vector consists of three steps: two steps involve the solution of a linear system with the same blockdiagonal matrix and one step involves a matrixvector product. The preconditioner is asymptotically correct for the Dahlquist test equation. Some free parameters of the preconditioner can be determined in order to optimize certain properties of the preconditioned approximate Jacobian matrix. Key words. GMRES, implicit RungeKutta methods, inexact modified Newton iterations, linear iterative methods, nonlinear equations, ordinary differential equations, parallelism, preconditioning, stiffness AMS subject classifications. 65F10, 65H10, 65L05, 65L06, 65Y05 1. Introduction. We
Iterative solution of SPARK methods applied to DAEs
, 2002
"... this article a broad class of systems of implicit differentialalgebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned add ..."
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this article a broad class of systems of implicit differentialalgebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive RungeKutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIABCC # D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented
unknown title
, 2005
"... www.elsevier.com/locate/cam Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models � ..."
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www.elsevier.com/locate/cam Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models �
PRECONDITIONING OF IMPLICIT RUNGEKUTTA METHODS
"... Abstract. A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with ..."
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Abstract. A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually be solved approximately with a preconditioned linear iterative method. In this article we present a truly parallelizable preconditioner to the approximate Jacobian matrix. Its decomposition cost for a sequential or parallel implementation can be made equivalent to the cost corresponding to the implicit Euler method. The application of the preconditioner to a vector consists of three steps: two steps involve the solution of a linear system with the same blockdiagonal matrix and one step involves a matrixvector product. The preconditioner is asymptotically correct for the Dahlquist test equation. Some free parameters of the preconditioner can be determined in order to optimize certain properties of the preconditioned approximate Jacobian matrix. Key words. GMRES, implicit RungeKutta methods, inexact modified Newton iterations, linear iterative methods, nonlinear equations, ordinary differential equations, parallelism, preconditioning, stiffness AMS subject classifications. 65F10, 65H10, 65L05, 65L06, 65Y05 This paper is dedicated to Ian Gladwell on the occasion of his retirement. 1. Introduction. We
DIFFERENTIALALGEBRAIC EQUATIONS BY LOBATTO RUNGEKUTTA METHODS ∗
"... We consider the numerical solution of systems of index 2 implicit differentialalgebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIABCC ∗D methods are included. We show superconvergence of optimal order 2s − 2forthesstage ..."
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We consider the numerical solution of systems of index 2 implicit differentialalgebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIABCC ∗D methods are included. We show superconvergence of optimal order 2s − 2forthesstage Lobatto families provided the constraints are treated in a particular way which strongly relies on specific properties of the SPARK coefficients. Moreover, reversibility properties of the flow can still be preserved provided certain SPARK coefficients are symmetric.
Numer. Math. (2004) 97: 635–666 Digital Object Identifier (DOI) 10.1007/s0021100405189
"... Approximate compositions of a near identity map by multirevolution RungeKutta methods ⋆ ..."
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Approximate compositions of a near identity map by multirevolution RungeKutta methods ⋆