K e y w o r d s O r d i n a r y differential equations, Initial value problems, Runge-Kut ta methods, Stiffness detection, Symbolic computation, Computer algebra systems, Computer generation of numerical methods. 1. I N T R O D U C T I O N A framework for explicit Runge-Kutta methods is being implemented as part of an ongoing overhaul of MATHEMATICA~S differential equation solver NDSolve. One o...

We study Runge{Kutta methods for the integration of ordinary diierential equations and their retention of algebraic invariants. As a general rule, we derive two conditions for the retention of such invariants. The rst is a condition on the co-eecients of the methods, the second is a pair of partial diierential equations that otherwise must be obeyed by the invariant. The cases related to the re...

The Kinetic PreProcessor (KPP) is a widely used software environment which generates Fortran90, Fortran77, Matlab, or C code for the simulation of chemical kinetic systems. High computational efficiency is attained by exploiting the sparsity pattern of the Jacobian and Hessian. In this paper we report on the implementation of two new families of stiff numerical integrators in the new version 2....

Citation A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge–Kutta pairs. We consider orders four through twelve, including both serial and parallel implementations. We cast extrapolation and defer...

A Runge–Kutta method takes small time steps, to approximate the solution to an initial value problem. How accurate is this approximation? If the error is asymptotically proportional to hp, where h is the stepsize, the Runge–Kutta method is said to have “order” p. To find p, write the exact solution, after a single time-step, as a Taylor series, and compare with the Taylor series for the approxi...

We now begin the definition and construction of Runge-Kutta methods. These one–step methods are essentially always stable, but designing Runge– Kutta methods which are consistent to high order can be difficult. This theory is presented in Sec. We have already seen several examples of Runge–Kutta methods: explicit and implicit Euler, the implicit midpoint rule, the explicit midpoint rule with Eu...

Fatode is a fortran library for the integration of ordinary differential equations with direct and adjoint sensitivity analysis capabilities. The paper describes the capabilities, implementation, code organization, and usage of this package. Fatode implements four families of methods – explicit Runge-Kutta for nonstiff problems and fully implicit Runge-Kutta, singly diagonally implicit Runge-Ku...

Strong stability preserving (SSP) time discretizations were developed for use with the spatial discretization of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified time-step restriction. We turn to implicit Runge–Kutta methods to alleviate ...

Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, h, for the integration. If h is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration sc...