Runge–Kutta–Nyström methods with maximized stability domain in structural dynamics

Lunk, Christoph; Simeon, Bernd

doi:10.1016/j.apnum.2004.09.003

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…A seventh-order explicit symplectic Nyström algorithm was obtained by Calvo and Sanz-Serna (1993). Some optimized symplectic Nyström algorithms have also been developed (Blanes and Moan 2002;Lunk and Simen 2005). Some optimized symplectic Nyström algorithms have also been developed (Blanes and Moan 2002;Lunk and Simen 2005).…”

Section: Nyström Methodsmentioning

confidence: 99%

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Lax‐Wendroff and Nyström methods for seismic modelling

Chen

2009

Geophysical Prospecting

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Lax‐Wendroff and Nyström methods are numerical algorithms of temporal approximations for solving differential equations. These methods provide efficient algorithms for high‐accuracy seismic modeling. In the context of spatial pseudospectral discretizations, I explore these two kinds of methods in a comparative way. Their stability and dispersion relation are discussed in detail. Comparison between the fourth‐order Lax‐Wendroff method and a fourth‐order Nyström method shows that the Nyström method has smaller stability limit but has a better dispersion relation, which is closer to the sixth‐order Lax‐Wendroff method. The structure‐preserving property of these methods is also revealed. The Lax‐Wendroff methods are a second‐order symplectic algorithm, which is independent of the order of the methods. This result is useful for understanding the error growth of Lax‐Wendroff methods. Numerical experiments based on the scalar wave equation are performed to test the presented schemes and demonstrate the advantages of the symplectic methods over the nonsymplectic ones.

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Section: Nyström Methodsmentioning

confidence: 99%

“…Tsitouras (1999) developed a tenthorder explicit symplectic Nyström algorithm. Some optimized symplectic Nyström algorithms have also been developed (Blanes and Moan 2002;Lunk and Simen 2005).…”

Section: Nyström Methodsmentioning

confidence: 99%

Lax‐Wendroff and Nyström methods for seismic modelling

Chen

2009

Geophysical Prospecting

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“…Theoretically, a numerical method for Hamiltonian dynamical systems can be called a symplectic algorithm if the resulting numerical solution is also a symplectic mapping. Some symplectic algorithms for partial differential equations have been developed and used, such as Lax–Wendroff methods (Lax & Wendroff 1960; Carcione et al 2002) and Nyström methods (Qin & Zhu 1991; Okunbor & Skeel 1992; Calvo & Sanz‐Serna 1993; Hairer et al 1993; Tsitouras 1999; Blanes & Moan 2002; Lunk & Simen 2005). Chen (2009) discussed the structure‐preserving property of Lax–Wendroff and Nyström methods in detail.…”

Section: Introductionmentioning

confidence: 99%

Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method

Wang

et al. 2012

Geophysical Journal International

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S U M M A R YIn this paper, we introduce the so-called symplectic discrete singular convolution differentiator (SDSCD) method for structure-preserving modelling of elastic waves. In the method presented, physical space is discretized by the DSCD, whereas an explicit third-order symplectic scheme is used for the time discretization. This approach uses optimization and truncation to form a localized operator. This preserves the fine structure of the wavefield in complex media and avoids non-causal interaction when parameter discontinuities are present in the medium. Theoretically, the approach presented is a structure-preserving algorithm. Also, some numerical experiments are shown in this paper. Elastic wavefield modelling experiments on a laterally heterogeneous medium with high parameter contrasts demonstrate the superior performance of the SDSCD for suppression of numerical dispersion. Long-term computational experiments exhibit the remarkable capability of the approach presented for long-time simulations. Promising numerical results suggest the SDSCD is suitable for high-precision and long-time numerical simulations, as it has structure-preserving property and it can suppress effectively numerical dispersion when coarse grids are used.

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“…A numerical algorithm for Hamiltonian dynamical systems can be termed a symplectic scheme if the resulting solution is also a symplectic mapping. Some symplectic schemes for partial differential equations have been proposed and applied, such as Lax-Wendroff algorithms [1,2] and Nyström algorithms [3][4][5][6][7][8][9]. Chen [10] minutely discussed the structure-preserving property of these algorithms.…”

Section: Introductionmentioning

confidence: 99%

A symplectic method for structure-preserving modelling of damped acoustic waves

Liu

et al. 2015

Proc. R. Soc. A.

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In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

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Runge–Kutta–Nyström methods with maximized stability domain in structural dynamics

Cited by 5 publications

References 9 publications

Lax‐Wendroff and Nyström methods for seismic modelling

Lax‐Wendroff and Nyström methods for seismic modelling

Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method

A symplectic method for structure-preserving modelling of damped acoustic waves

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