Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficient...

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combines adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual of the equation, and the Runge-Kutta method is supplied via its Butcher’s table. Ar...

Three new Runge–Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of...

in this paper, a class of semi-implicit two-stage stochastic runge-kutta methods (srks) of strong global order one, with minimum principal error constants are given. these methods are applied to solve itô stochastic differential equations (sdes) with a wiener process. the efficiency of this method with respect to explicit two-stage itô runge-kutta methods (irks), it method, milstien method, sem...

In this paper we develop numerical techniques of order 2, 4 and 6 for the solution of a fourth order linear equation. A priori error bound is obtained for the fourth order method to prove the convergence of the finite difference scheme. A sufficient condition guaranteeing the uniqueness of the solution of the boundary value problem is also given. Numerical illustrations are tabulated and result...

In this article we proposed three explicit Improved Runge-Kutta (IRK) methods for solving first-order ordinary differential equations. These methods are two-step in nature and require lower number of stages compared to the classical RungeKutta method. Therefore the new scheme is computationally more efficient at achieving the same order of local accuracy. The order conditions of the new methods...

In this paper, we use the spectral collocation method based on Chebyshev polynomials for spatial derivatives and fourth order Runge-Kutta (RK) method for time integration to solve the generalized Zakharov equation (GZE). Firstly, theory of application of Chebyshev spectral collocation method on the GZE is presented. This method yields a system of ordinary differential equations (ODEs). Secondly...

An optimized explicit low-storage fourth-order Runge–Kutta algorithm is proposed in the present work for time integration. Dispersion and dissipation of the scheme are minimized in the Fourier space over a large range of frequencies for linear operators while enforcing a wide stability range. The scheme remains of order four with nonlinear operators thanks to the low-storage algorithm. Linear a...

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues...

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we a...