This study utilised positive Lyapunov exponents’ criteria to develop chaos diagram on the parameters space of 4-dimensional harmonically excited vibration absorber control Duffing’s Oscillator. Relevant simulations were effected by choice combination of constant step Runge-Kutta methods and Grahm Schmidt Orthogonal rules. Simulations of 4-dimensional hyper-chaotic models of modified Lorenz and ...

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

In this work, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both...

No Runge-Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the Averaged Vector Field (AVF) method can be interpreted as a Runge-Kutta method whose weights bi and abscissae ci represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge-Kutta scheme...

A connection between the algebra of rooted trees used in renormalization theory and Runge-Kutta methods is pointed out. Butcher’s group and B-series are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally B-series are used to solve a class of non-linear partial differential equations.

The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.