On those ordinary differential equations that are solved exactly by the improved Euler method

Hans Jakob Rivertz

Displaying similar documents to “On those ordinary differential equations that are solved exactly by the improved Euler method”

Implicit Runge-Kutta methods for transferable differential-algebraic equations

M. Arnold (1994)

Banach Center Publications

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The numerical solution of transferable differential-algebraic equations (DAE’s) by implicit Runge-Kutta methods (IRK) is studied. If the matrix of coefficients of an IRK is non-singular then the arising systems of nonlinear equations are uniquely solvable. These methods are proved to be stable if an additional contractivity condition is satisfied. For transferable DAE’s with smooth solution we get convergence of order $m i n (k_{E}, k_{I} + 1)$ , where $k_{E}$ is the classical order of the IRK and $k_{I}$ is the stage order....

Explicit two-step Runge-Kutta methods

Zdzisław Jackiewicz, Rosemary Anne Renaut, Marino Zennaro (1995)

Applications of Mathematics

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The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order $p \leq 5$ the minimal number of stages for explicit TSRK method of order $p$ is equal to the minimal number of stages for explicit Runge-Kutta method of order $p - 1$ . Numerical results are presented...

Fifth-order Runge-Kutta with higher order derivative approximations.

Goeken, David, Johnson, Olin (1999)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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Zero Dissipative DIRKN Pairs of Order 5(4) for Solving Special Second Order IVPs

S. O. Imoni, M. N. O. Ikhile (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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For initial value problem (IVPs) in ordinary second order differential equations of the special form $y^{''} = f (x, y)$ possessing oscillating solutions, diagonally implicit Runge–Kutta–Nystrom (DIRKN) formula-pairs of orders 5(4) in 5-stages are derived in this paper. The method is zero dissipative, thus it possesses a non-empty interval of periodicity. Some numerical results are presented to show the applicability of the new method compared with existing Runge–Kutta (RK) method applied to the problem...

Order conditions for partitioned Runge-Kutta methods

Zdzisław Jackiewicz, Rossana Vermiglio (2000)

Applications of Mathematics

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We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.