An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

1970 ◽  
Vol 15 (2) ◽  
pp. 151-164 ◽  
Author(s):  
Minoru Urabe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
High Order ◽  
High Order Accuracy ◽  
Step Method ◽  
Order Accuracy ◽  
One Step

scholarly journals High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Quan Zhou ◽  
Yabing Sun
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Taylor Expansion ◽  
Backward Stochastic Differential Equations ◽  
High Order ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
One Step

<p style='text-indent:20px;'>In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.</p>

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries

2017 ◽  
Vol 7 (4) ◽  
pp. 714-727 ◽  
Author(s):  
Dongdong He ◽  
Kejia Pan
Keyword(s):  
Differential Equations ◽  
Stream Function ◽  
Stokes Flow ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
High Order ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Compact Difference ◽  
Fifth Order

AbstractIncompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

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