Additive Runge Kutta (ARK) methods are investigated for application to the spatially discretized one dimensional convection diffusion reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge Kutta methods are grouped into a single composite method. Then, implicit explicit, N = 2, additive Runge Kutta (ARK2) m...

A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space combined with detailed mechanisms for accommodating the adapting grids ensure that conservation is maintained a...

A new diagonally implicit Runge-Kutta-Nyström (RKN) method is developed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. Presented is a method which is three-stage fourth-order with dispersive order six and 'small' principal local truncation error terms and dissipation constant. The analysis of phase-lag, dissipatio...

we present here the numerical solution of damped forced oscillator problem using haar wavelet and compare the numerical results obtained with some well-known numerical methods such as runge-kutta fourth order classical and taylor series methods. numerical results show that the present haar wavelet method gives more accurate approximations than above said numerical methods.

New Runge–Kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations, by Chebyshev or modified Chebyshev methods, are introduced. These Runge–Kutta methods optimize the time step necessary for stable solutions, while holding dispersion and dissipation fixed. It is found that maximizing disp...

The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient tha...

A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competi...

To integrate large systems of ordinary differential equations (ODEs) with disparate timescales, we present a multirate method with error control that is based on embedded, explicit Runge-Kutta (RK) formulas. The order of accuracy of such methods depends on interpolating certain solution components with a polynomial of sufficiently high degree. By analyzing the method applied to a simple test eq...