Cone of non-linear dynamical system and group preserving schemes

Liu, Chein‐Shan

doi:10.1016/s0020-7462(00)00069-x

Cited by 162 publications

(82 citation statements)

References 14 publications

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“…Corresponding to the first augmented Lie-group symmetry found by Liu (2001), the above Lie-group symmetry is the second augmented Lie-group symmetry in the Minkowski space M n+1 for Equation (1).…”

Section: The Second Augmented Lie-group Symmetrymentioning

confidence: 94%

“…Because the above Lie-group S O o (n, 1) symmetry of Equation (7) is realized in the Minkowski space for the augmented state vectors, this kind symmetry may be termed the augmented Lie-group symmetry, and the Liegroup symmetry found by Liu (2001) is the first augmented Lie-group symmetry for Equation (1).…”

Section: The First Augmented Lie-group Symmetrymentioning

confidence: 99%

“…In order to distinct the present method from the group-preserving scheme (GPS) developed by Liu (2001), we may call the new algorithm to be the second GPS, simply denoted by GPS2, which is summarized as follows.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In Section 2 we review the first augmented Lie-group S O o (n, 1) symmetry derived by Liu (2001), and introduce the group-preserving scheme (GPS). Section 3 explores the concepts from the Jordan algebra and the Jordan dynamics to help us for finding the Lie-group DS O(n) symmetry of nonlinear ODEs.…”

Section: Introductionmentioning

confidence: 99%

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The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

Liu¹,

Jhao²

2014

JMR

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Liu (2001) derived the first augmented Lie-group S O o (n, 1) symmetry for the nonlinear ordinary differential equations (ODEs):ẋ = f(x, t), and developed the corresponding group-preserving scheme (GPS). However, the earlier formulation did not consider the rotational effect of nonlinear ODEs. In this paper, we derive the second augmented Lie-group S O o (n, 1) symmetry by taking the rotational effect into account. The numerical algorithm exhibits two solutions of the Lie-group G ∈ S O o (n, 1), depending on the sign of f 2 x 2 − 2(f · x) 2 , which means that the algorithm may be switched between two states, depending on x. We give numerical examples to assess the new algorithm GPS2, which upon comparing with the GPS can raise the accuracy about three orders. It is interesting that for the chaotic system the signum function sign( f 2 x 2 − 2(f · x) 2 ) is frequently switched between +1 and −1 in time.

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Section: The Second Augmented Lie-group Symmetrymentioning

confidence: 94%

Section: The First Augmented Lie-group Symmetrymentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

Liu¹,

Jhao²

2014

JMR

Self Cite

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“…Because the last component 1/ √ Q(t) of X is positive, the cone in Eq. (2.7) is a future cone [23]. The cone structure about the residual vector was first pointed out by Liu [24].…”

Section: Introductionmentioning

confidence: 99%

A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

Liu¹

2013

Communications in Numerical Analysis

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Starting from a quadratic invariant manifold in terms of the residual vector r = Bx − b for an n-dimensional ill-posed linear algebraic equations system Bx = b, we derive an ODEs system for x which is equipped with a state feedback controller to enforce the orbit of the state vector x on a specified manifold, whose residual-norm is exponentially decayed. To realize the above idea we develop a very powerful implicit scheme based on the novel GL(n, R) Lie-group method to integrate the resultant differential algebraic equation (DAE). Through numerical tests of inverse problems we find that the present Lie-group DAE algorithm can significantly accelerate the convergence speed, and is robust enough against the random noise.

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A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

Oruç,

Polat

2023

Math Methods in App Sciences

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In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).

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Cone of non-linear dynamical system and group preserving schemes

Cited by 162 publications

References 14 publications

The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

The Second Lie-Group $SO_o(n,1)$ Used to Solve Ordinary Differential Equations

A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

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