1,720,967 research outputs found
Explicit four stage fourth order Runge-Kutta methods for preserving quadratic conservation laws
No full textThis paper concerns the study of explicit 4-stage fourth order Runge–Kutta methods which preserve quadratic conservation laws when their weights are made solution dependent. Application to orthogonal and Hamiltonian linear systems and numerical tests are also reported
Variable Step-Size Techniques in Continuous Runge-Kutta Methods for Isospectral Dynamical Systems
No full textIn this paper we consider numerical methods for the dynamical system L′ = [B(L), L], L(0) = L0, (*) where L0 is a n × n symmetric matrix, [B(L), L] is the commutator of B(L) and L, and B(L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L(t) preserves the eigenvalues of L0, for t ≥ 0. The matrix B(L) characterizes the flow, and for special B(·), the solution matrix L(t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (*) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by EdGLRKs) consist in solving the system (*) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (*), in order to convert the approximation of L(t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class EdGLRKs we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be show
Piecewise interpolants on matrix lie groups
No full textAbstractHere we consider a numerical procedure to interpolate on matrix Lie groups. By using the exponential map and its (1, 1) diagonal Padé approximant, piecewice interpolants may be derived. The approach based on the Padé map has the advantage that the computation of exponentials and logarithms of matrices are reduced. We show that the updating technique proposed by Enright in [1] may be applied when a dense output is required. The application to the numerical solution of a system ODEs on matrix group and to a classical interpolation problem are reported
Numerical methods for dynamical systems in the Lorentz group
No full textIn recent years several numerical methods have been developed to integrate matrix differential systems whose solutions remain on a certain Lie group throughout the evolution. In this paper we describe some numerical methods derived for the solution of dynamical systems in the Lorentz quadratic group. This group has been extensively studied in past expecially by physiscists since some differential systems of great importance in relativity evolve in this group. Numerical tests will show the performance of the numerical methods described
Variable step-size techniques in continuous Runge-Kutta methods for isospectral dynamical systems
No full textIn this paper we consider numerical methods for the dynamical system L′ = [B(L), L], L(0) = L0, (*) where L0 is a n × n symmetric matrix, [B(L), L] is the commutator of B(L) and L, and B(L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L(t) preserves the eigenvalues of L0, for t ≥ 0. The matrix B(L) characterizes the flow, and for special B(·), the solution matrix L(t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (*) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by EdGLRKs) consist in solving the system (*) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (*), in order to convert the approximation of L(t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class EdGLRKs we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be shown
A Newton Type Method for Solving Nonlinear Equations on Quadratic Matrix Groups
No full textIn this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F(Y)=0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert
Runge Kutta type methods for isodynamical matrix flows: applications to balanced realization
No full textRecently several numerical methods have been proposed for solving isospectral problems which are matrix differential systems whose solutions preserve the spectrum during the evolution. In this paper we consider matrix differential systems called isodynantical flows in which only a component of the matrix solution preserves the eigenvalues during the evolution and we propose procedures for their numerical solution. Applications of such numerical procedures may be found in systems theory, in particular in balancing realization problems. Several numerical tests will be reported
Semi-explicit time-stepping methods for dynamical systems with complementarity constraints
No full textIn this paper we consider numerical methods for dynamical systems with complementary conditions. The dynamics of these systems are characterized by possible jumps in the solution caused by events which separate continuous phases of the state vector. These events occur on the basis of certain inequality constraints similar to those appearing in the context of dynamic optimization. The numerical techniques used up till now, were based on the implicit backward Euler method, while we explore one-step procedures based on semi-explicit schemes
Numerical methods for dynamical systems in the Lorentz group
No full textIn recent years several numerical methods have been developed to integrate matrix differential systems whose solutions remain on a certain Lie group throughout the evolution. In this paper we describe some numerical methods derived for the solution of dynamical systems in the Lorentz quadratic group. This group has been extensively studied in past expecially by physiscists since some differential systems of great importance in relativity evolve in this group. Numerical tests will show the performance of the numerical methods described
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