Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge–Kutta family

2007 ◽  
Vol 57 (2) ◽  
pp. 147-165 ◽  
Author(s):  
Yoshio Komori
Keyword(s):  
Rooted Tree ◽  
Order Conditions ◽  
Weak Order ◽  
Tree Analysis ◽  
Runge Kutta

scholarly journals Continuous stage stochastic Runge–Kutta methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xuan Xin ◽  
Wendi Qin ◽  
Xiaohua Ding
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Additive Noise ◽  
Rooted Tree ◽  
Order Theory ◽  
Order Conditions ◽  
Quadrature Formulas ◽  
General Order ◽  
Runge Kutta ◽  
Theoretical Results

AbstractIn this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

scholarly journals Order Conditions of Stochastic Runge--Kutta Methods by B-Series

2000 ◽  
Vol 38 (5) ◽  
pp. 1626-1646 ◽  
Author(s):  
K. Burrage ◽  
P. M. Burrage
Keyword(s):  
Order Conditions ◽  
Runge Kutta

scholarly journals Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations

2014 ◽  
Vol 142 (5) ◽  
pp. 2067-2081 ◽  
Author(s):  
Oswald Knoth ◽  
Joerg Wensch
Keyword(s):  
Euler Equations ◽  
Time Integration ◽  
Order Conditions ◽  
Compressible Euler Equations ◽  
Step Size ◽  
Integration Interval ◽  
New Methods ◽  
Runge Kutta Method ◽  
Compressible Euler ◽  
Runge Kutta

Abstract The compressible Euler equations exhibit wave phenomena on different scales. A suitable spatial discretization results in partitioned ordinary differential equations where fast and slow modes are present. Generalized split-explicit methods for the time integration of these problems are presented. The methods combine explicit Runge–Kutta methods for the slow modes and with a free choice integrator for the fast modes. Order conditions for these methods are discussed. Construction principles to develop methods with enlarged stability area are presented. Among the generalized class several new methods are developed and compared to the well-established three-stage low-storage Runge–Kutta method (RK3). The new methods allow a 4 times larger macro step size. They require a smaller integration interval for the fast modes. Further, these methods satisfy the order conditions for order three even for nonlinear equations. Numerical tests on more complex problems than the model equation confirm the enhanced stability properties of these methods.

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