Strong stability-preserving (SSP) Runge–Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge–Kutta methods with optimal SSP time-step restrictions, first for the cas...

In this work we introduce a new approach to Dynamical Monte Carlo methods to simulate markovian processes. We apply this approach to formulate and study an epidemic generalized SIRS model. The results are in excellent agreement with the fourth order Runge-Kutta method in a region of deterministic solution. Introducing local stochastic interactions, the Runge-Kutta method is no longer applicable...

The construction of symmetric and symplectic exponentially-fitted Runge-Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions is reconsidered. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new fourth-order integrators are cons...

The construction of new explicit Runge–Kutta methods taking into account, not only their accuracy, but also the preservation of Quadratic Invariants (QIs) is studied. An expression of the error of conservation of a QI by a Runge–Kutta method is given, and a new six–stage formula with classical order four and seventh order of QI–conservation is obtained by choosing their coefficients so that the...

An error analysis of Runge-Kutta convolution quadrature is presented for a class of nonsectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ0 and a polynomial bound O(s 1) there, the stronger polynomial bound O(s2) in convex sectors of the form | arg s| ≤ π/2 − θ < π/2 for θ > 0. The order of convergence of the Runge-Kutt...

The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational efficiency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method; thi...

In this paper, we classified a class of fourth-order partial differential equations (PDEs) to be fourth-order PDE of type I, II, III and IV. The PDE of type IV is solved by using an efficient numerical method. The PDE is first transformed to a system of fourth-order ordinary differential equations (ODEs) using the method of lines, then the resulting system of fourth-order ODEs is solved using d...