In this paper we study the efficiency of Strong Stability Preserving (SSP) Runge–Kutta methods that can be implemented with a low number registers using their Shu–Osher representation. SSP have been studied in literature and stepsize restrictions ensure numerical monotonicity found. However, for some problems, observed are larger than theoretical ones. Aiming at obtaining additional properties ...

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions stochastic driven by processes that are rougher than a Brownian motion. use Taylor series representation (B-series) both the numerical scheme and solution of equation in order determine conditions guarantee desired local error underlying method. Subsequently, we prove global given rate. In add...

This paper investigates numerical methods for direct decoupled sensitivity and discrete adjoint sensitivity analysis of stiff systems based on implicit Runge Kutta schemes. Efficient implementations of tangent linear and adjoint schemes are discussed for two families of methods: fully implicit three-stage Runge Kutta and singly diagonally-implicit Runge Kutta. High computational efficiency is a...

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of v...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in [9, 10]. The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher’s a...

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems. Mathematics Subject Classification. 65P10, 65L06. Received May 21, 2008. Published online July 8, 2009. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods p...