The application of Runge-Kutta schemes to singular initial value problems
نویسندگان
چکیده
منابع مشابه
The Application of Runge-Kutta Schemes to Singular Initial Value Problems
A theory for explicit Runge-Kutta schemes applied to the initial value problem for a first-order system of differential equations with a singularity of the first kind is developed. It is shown that, in general, the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems.
متن کاملRunge-Kutta Pairs for Scalar Autonomous Initial Value Problems
We present the equations of condition up to sixth order for Runge-Kutta (RK) methods, when integrating scalar autonomous problems. Two RK pairs of orders 5(4) are derived. The first at a cost of only five stages per step, while the other having an extremely small principal truncation error. Numerical tests show the superiority of the new pairs over traditional ones.
متن کاملAvoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems
A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies ...
متن کاملOrder and Stiffness of the Gauss Runge-kutta Method for Initial-boundary Value Problems
Abstract. Existing analysis shows that when the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation with s Gaussian nodes is applied to ordinary differential equation initial value problems, the discretization has order 2s (super-convergent) [8]. However, for time-dependent partial differential equations (PDEs) with boundary conditions, super-convergence is only observe...
متن کاملBlock Runge-Kutta Methods for the Numerical Integration of Initial Value Problems in Ordinary Differential Equations
Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of v...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1985
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1985-0771033-0