Fatode is a fortran library for the integration of ordinary differential equations with direct and adjoint sensitivity analysis capabilities. The paper describes the capabilities, implementation, code organization, and usage of this package. Fatode implements four families of methods – explicit Runge-Kutta for nonstiff problems and fully implicit Runge-Kutta, singly diagonally implicit Runge-Ku...

Strong stability preserving (SSP) time discretizations were developed for use with the spatial discretization of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified time-step restriction. We turn to implicit Runge–Kutta methods to alleviate ...

Implicit integration schemes for ODEs, such as Runge-Kutta and Runge-Kutta-Nyström methods, are widely used in mathematics and engineering to numerically solve ordinary differential equations. Every integration method requires one to choose a step-size, h, for the integration. If h is too large or too small the efficiency of an implicit scheme is relatively low. As every implicit integration sc...

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 ...

Abstract In this work, we study the A [ ? ] – stability of additive methods Runge- Kutta kind orders ranging from 2 up to 4 that will be applied for determining some stiff nonlinear system ODEs. Moreover, find function Runge-Kutta method and type order 2,3, 4. Where ( A,B 1 ) is A-stable semi-implicit explicit. Furthermore, term managed by while no treated explicit Runge method. Those are suita...

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...

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),...