Usually the straightforward generalization of explicit Runge{Kutta methods for ordinary diierential equations to half-explicit methods for diierential-algebraic systems of index 2 results in methods of order q 2 ((8]). The construction of higher order methods is simpliied substantially by a slight modiication of the method combined with an improved strategy for the computation of the algebraic ...

Abstract: This paper presents some numerical methods for Allen-Cahn equation using different time stepping and space discretization methods with non-periodic boundary conditions. In space the equation is discretized by Chebyshev spectral method, while in time the exponential time differencing fourth-order Runge-Kutta (ETDRK4) and implicit-explicit scheme is used. For comparison we also use the ...

Last time, we investigated the fourth-order Runge-Kutta method. We saw that the computations involved in performing this approximation were less than ideal. To create more computationally viable methods, we introduced multistep methods, in which the approximation at a given point is obtained using only the values of the differential equation to be approximated and the approximation itself at pr...

In this paper, the coupled mode theory is used to analyze apodized fiber Bragg gratings (FBGs). Since the profile of gratings varies with the propagation distance, the coupled mode equations (CMEs) of apodized FBGs are solved by the fourth-order Runge-Kutta method (RKM) and piecewise-uniform approach (PUA). We present two discretization techniques of PUA to analyze the apodization profile of gr...

This article looks at the steady flow of Micropolar fluid over a stretching surface with heat transfer in the presence of Newtonian heating. The relevant partial differential equations have been reduced to ordinary differential equations. The reduced ordinary differential equation system has been numerically solved by Runge-Kutta-Fehlberg fourth-fifth order method. Influence of different involv...

A Runge–Kutta method takes small time steps, to approximate the solution to an initial value problem. How accurate is this approximation? If the error is asymptotically proportional to hp, where h is the stepsize, the Runge–Kutta method is said to have “order” p. To find p, write the exact solution, after a single time-step, as a Taylor series, and compare with the Taylor series for the approxi...

This paper is concerned with time-stepping numerical methods for computing sti semi-discrete systems of ordinary di erential equations for transient hypersonic ows with thermo-chemical nonequilibrium. The sti ness of the equations is mainly caused by the viscous ux terms across the boundary layers and by the source terms modeling nite-rate thermo-chemical processes. Implicit methods are needed ...

We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit secondand third-order Runge–Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous a...

Abstract In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourthorder 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standar...

The optimization of some W-methods [7] for the time integration of time-dependent PDEs in several spatial variables is considered. In [2, Theorem 1] several three-parametric families of three-stage W-methods for the integration of IVPs in ODEs were studied. Besides, the optimization of several specific methods for PDEs when the Approximate Matrix Factorization Splitting (AMF) [3, 4] is used to ...