Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.

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“…(2) Semi-implicit methods in which the implicitness is only restricted to the drift coefficient [11,13,14];…”

Section: Introductionmentioning

confidence: 99%

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

Wang

Ding

2020

Mathematics

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“…With the rapid growth of science and industry in the 20th century, the numerical methods that solve differential equations attract much attentions in different new fields. [1][2][3][4][5] Some of the recent methods include differential transform methods, [6][7][8][9] spectral Galerkin methods, [10][11][12] wavelet methods, 9,[13][14][15][16][17][18] collocation methods, [19][20][21][22][23][24] Legendre methods, [25][26][27] and some other numerical methods for ordinary differential equations. [28][29][30][31][32][33][34][35][36] We consider the following model problem:…”

Section: Introductionmentioning

confidence: 99%

“…Differential equations reflect the correlation between functions and function derivatives. With the rapid growth of science and industry in the 20th century, the numerical methods that solve differential equations attract much attentions in different new fields . Some of the recent methods include differential transform methods, spectral Galerkin methods, wavelet methods, collocation methods, Legendre methods, and some other numerical methods for ordinary differential equations …”

Section: Introductionmentioning

confidence: 99%

A collocation method for differential equations with one‐point Taylor initial conditions

Ghabarpoor

Kajani

Allame

2019

Math Methods in App Sciences

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This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

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“…Recently, there are many works to study the numerical solution for stochastic ordinary differential equations (SDEs). Euler-Maruyama method, Milstein method and Runge-Kutta method were proposed to solve SDEs numerically, see [3,4,13,16,17,19,20,31] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Spectral Element Methods for Stochastic Differential Equations with Additive Noise

Zhang¹

2018

JMS

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In this paper, we propose numerical schemes for stochastic differential equations driven by white noise and colored noise, respectively. For this purpose, we first discretize the white noise and colored noise, and give their regularity estimates. Then we use spectral element methods to solve the corresponding stochastic differential equations numerically. The approximation errors are derived, and the numerical results demonstrate high accuracy of the proposed schemes.

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Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

Cited by 6 publications

References 19 publications

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

A collocation method for differential equations with one‐point Taylor initial conditions

Spectral Element Methods for Stochastic Differential Equations with Additive Noise

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