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 A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Chengjian Zhang
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Integrodifferential Equations ◽  
Stability Criteria ◽  
Nonlinear Functional ◽  
Runge Kutta ◽  
Theoretical Results

This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.

scholarly journals Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

2017 ◽  
Vol 17 (3) ◽  
pp. 479-498 ◽  
Author(s):  
Raphael Kruse ◽  
Yue Wu
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Quadrature Rule ◽  
Time Variable ◽  
Important Ingredient ◽  
Riemann Sum ◽  
Runge Kutta ◽  
Theoretical Results

AbstractThis paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the {L^{p}(\Omega;{\mathbb{R}}^{d})}-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

Exponential Additive Runge-Kutta Methods for Semi-Linear Differential Equations

2017 ◽  
Vol 7 (2) ◽  
pp. 286-305
Author(s):  
Jingjun Zhao ◽  
Teng Long ◽  
Yang Xu
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Order Conditions ◽  
Linear Differential Equations ◽  
Stability Properties ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes ◽  
Runge Kutta ◽  
Linear Differential ◽  
Theoretical Results

AbstractExponential additive Runge-Kutta methods for solving semi-linear equations are discussed. Related order conditions and stability properties for both explicit and implicit schemes are developed, according to the dimension of the coefficients in the linear terms. Several examples illustrate our theoretical results.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

scholarly journals Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kasim Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Order Conditions ◽  
Numerical Results ◽  
Type Method ◽  
First Order ◽  
New Methods ◽  
Runge Kutta ◽  
Fifth Order

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

scholarly journals Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

2021 ◽  
Author(s):  
Simon Becher ◽  
Gunar Matthies
Keyword(s):  
Numerical Experiments ◽  
Discontinuous Galerkin Methods ◽  
Time Discretization ◽  
Quadrature Formulas ◽  
Galerkin Methods ◽  
Initial Value ◽  
Discretization Methods ◽  
Continuous Galerkin ◽  
Higher Regularity ◽  
Theoretical Results

AbstractWe present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.

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