A modified Boris-like integration, in which the spatial coordinate is the independent variable, is derived. This spatial-Boris integration method is useful for beam simulations, in which the independent variable is often the distance along the beam. The new integration method is second order accurate, requires only one force calculation per particle per step, and preserves conserved quantities ...

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...

Much attention has been paid in the literature to total-variation-diminishing (TVD) numerical processes in the solution of nonlinear hyperbolic differential equations. For special Runge– Kutta methods, conditions on the stepsize were derived that are sufficient for the TVD property; see, e.g., Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439–471] and Gottlieb and Shu [Math. Comp., 67 (1998),...

The main purpose of this paper is to review the work on Runge-Kutta methods at the University of Toronto during the period 1963 to the present (1996). To provide some background, brief mention is also made of related work on the numerical solution of ordinary diierential equations, but, with just a few exceptions, speciic references are given only if the referenced material has a direct bearing...

This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution differential equations (FDEs). We construct explicit and implicit FORK FDEs by using Caputo generalized Taylor series formula. Due to dependence derivatives on fixed base point, in proposed method, we had modify right-hand side given equation all steps methods. Some coeffici...

Time-stepping methods that guarantee to avoid spurious fixed points are said to be regular. For fixed stepsize Runge-Kutta formulas, this concept has been well studied. Here, the theory of regularity is extended to the case of embedded Runge-Kutta pairs used in variable stepsize mode with local error control. First, the limiting case of a zero error tolerance is considered. A recursive regulari...

Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi‐ step integration methods. The coefficients are reverse engineered based on samples from a target function and its derivative used for training. The Runge‐Kutta schemes are tra...