On order conditions for modified Patankar–Runge–Kutta schemes

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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Order conditions of Runge-Kutta methods for index 2 systems

Lecture Notes in Mathematics - The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods ◽

10.1007/bfb0093952 ◽

1989 ◽

pp. 55-70 ◽

Cited By ~ 1

Author(s):

Ernst Hairer ◽

Michel Roche ◽

Christian Lubich

Keyword(s):

Order Conditions ◽

Index 2 ◽

Runge Kutta

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

Mathematical Problems in Engineering ◽

10.1155/2015/893763 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 3

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.

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