Exponentially Fitted Trapezoidal Scheme for a Stochastic Oscillator

Yin, Xiuling

doi:10.4208/jcm.1612-m2016-0677

Cited by 3 publications

(4 citation statements)

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“…To inherit the symplecticity of this stochastic oscillator, different kinds of symplectic methods have been constructed (see [7,9,14,22,23,24,28] and references therein).…”

Section: Introductionmentioning

confidence: 99%

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

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It is well known that symplectic methods have been rigorously shown to be superior to non-symplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we focus on the superiority of stochastic symplectic methods applied to a linear stochastic oscillator, from the perspective of large deviations principle (LDP). Based on the Gärtner-Ellis theorem, we first study the LDPs of the mean position and the mean velocity for both the exact solution of the stochastic oscillator and its numerical approximations. Then, by giving the conditions which make numerical methods have first order convergence in mean-square sense, we prove that symplectic methods asymptotically preserve these two LDPs, in the sense that the modified rate functions of symplectic methods converge to the rate functions of exact solution. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the "hitting probability" of the mean position and mean velocity of the original system. However, it is shown that non-symplectic methods do not asymptotically preserve these two LDPs by using the tail estimation of Gaussian random variables. To the best of our knowledge, this is the first result about using LDP to show the superiority of stochastic symplectic methods compared with non-symplectic ones.

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“…To inherit the symplecticity of this stochastic oscillator, different kinds of symplectic methods have been constructed (see [7,9,14,22,23,24,28] and references therein).…”

Section: Introductionmentioning

confidence: 99%

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The exponentially fitted Runge-Kutta method is a kind of efficient numerical tool to simulate ordinary differential equation with oscillatory solutions [27][28][29][30]. For example, exponentially…”

Section: Exponentially Fitted Trapezoidal Schemementioning

confidence: 99%

“…fitted trapezoidal scheme is a special exponentially fitted Runge-Kutta method. Yin et al [30] applied exponentially fitted trapezoidal scheme to simulate a kind of stochastic oscillator system and obtained efficient and stable numerical results. For initial value problem of ordinary differential equation u 0 = f(t, u), u(t 0 ) = u 0 , exponentially fitted trapezoidal scheme has the following formula…”

Section: Plos Onementioning

confidence: 99%

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions

et al. 2021

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Aiming at conservative Maxwell equations with periodic oscillatory solutions, we adopt exponentially fitted trapezoidal scheme to approximate the temporal and spatial derivatives. The scheme is a multisymplectic scheme. Under periodic boundary condition, the scheme satisfies two discrete energy conservation laws. The scheme also preserves two discrete divergences. To reduce computation cost, we split the original Maxwell equations into three local one-dimension (LOD) Maxwell equations. Then exponentially fitted trapezoidal scheme, applied to the resulted LOD equations, generates LOD multisymplectic scheme. We prove the unconditional stability and convergence of the LOD multisymplectic scheme. Convergence of numerical dispersion relation is also analyzed. At last, we present two numerical examples with periodic oscillatory solutions to confirm the theoretical analysis. Numerical results indicate that the LOD multisymplectic scheme is efficient, stable and conservative in solving conservative Maxwell equations with oscillatory solutions. In addition, to one-dimension Maxwell equations, we apply least square method and LOD multisymplectic scheme to fit the electric permittivity by using exact solution disturbed with small random errors as measured data. Numerical results of parameter inversion fit well with measured data, which shows that least square method combined with LOD multisymplectic scheme is efficient to estimate the model parameter under small random disturbance.

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“…Stochastic Hamiltonian systems, as a kind of structure-preserving stochastic differential equations, are used to simulate dynamic systems under stochastic dissipative disturbance [25][26][27][28][29][30]. If the disturbances are seen as white noise, stochastic Hamiltonian systems can be written as stochastic differential equations driven by Wiener processes.…”

Section: Introductionmentioning

confidence: 99%

Symplectic-Structure-Preserving Uncertain Differential Equations

Yin

Gao

et al. 2021

Symmetry

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Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

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Exponentially Fitted Trapezoidal Scheme for a Stochastic Oscillator

Cited by 3 publications

References 0 publications

The probabilistic superiority of stochastic symplectic methods via large deviations principles

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions

Symplectic-Structure-Preserving Uncertain Differential Equations

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