Previously, it has been shown that discretising a multi-Hamiltonian PDE in space and time with partitioned Runge–Kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law. However, these studies use the same partitioning of the variables into two partitions in both space and time. This gives rise to a large number of cases to be consider...

We consider a quasilinear parabolic problem u′(t) = Q ( u(t) ) u(t), u(t0) = u0 ∈ D, where Q(w) : D ⊂ X → X, w ∈ W ⊂ X, is a family of sectorial operators in a Banach space X with fixed domain D. This problem is discretized in time by means of a strongly A(θ)-stable, 0 < θ ≤ π/2, Runge–Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the depe...

The 4-th order Runge-Kutta method in the complex plane is proposed for numerically advancing the solutions of a system of first order differential equations in one external invariant satisfied by the master integrals related to a Feynman graph. Some results obtained for the 2-loop self-mass MI are reviewed. The method offers a reliable and robust approach to the direct and precise numerical eva...

In this paper, we introduce the numerical solution of the system of SEIR nonlinear ordinary differential equations, which are studied the effect of vaccine on the HIV (Human Immunology virus). We obtained the numerical solutions on stable manifolds by Runge-Kutta fourth order method.

In this short note we show how to obtain Lagrangian integrators from known sym-plectic partitioned Runge-Kutta methods.

This is a survey on discretizing delay equations from a geometric–qualitative view– point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center–unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various C j to...

We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form ẏ = J∇H(y) . Such a class of methods contains as particular cases the methods of Miesbach and Pesch [13], and also the family of symplectic Runge-Kutta methods. As in the case of the methods introduced in [13], the new integration methods are constructed by defining a generating functi...

Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by m-dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical B-convergence theory to...

A criterion is established for the convergence of sequences of a general type of Runge-Kutta method. The criterion is expressed in terms of "weak convergence", a property defined in the paper.