Numerical Algorithms

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on ...

Considering validated Runge-Kutta schemes requires the computation of local truncation error. Computation of such error is expensive in term of complexity, thus few techniques have been proposed to compute or at least estimate it. For example, ...

In this paper, we introduce a new method based on a library of chemical reactions for constructing a system of ordinary differential equations from stochastic simulations arising from an agent-based model. The advantage of this approach is that ...

A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set $\mathrm{\Lambda }$ with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable ...$\mathrm{}$$\mathcal{}$${\mathbb{}}^{\mathrm{}}$$\mathcal{}$$\mathcal{}$

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially ...${}^{}\mathrm{}\mathrm{}\mathrm{}$${}^{\mathrm{}}$

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving ...

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by ...

Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecular ...

Recently, a stability theory has been developed to study the linear stability of modified Patankar–Runge–Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the ...$\mathit{}$$$$$$\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}$$\mathrm{}$

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-...

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge–Kutta methods and continuous-stage Runge–Kutta methods ...

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant ...

The ordinary differential equation (ODE) models of optimization methods allow for concise proofs of convergence rates through discussions based on Lyapunov functions. The weak discrete gradient (wDG) framework discretizes ODEs while preserving the ...

Sixty years ago, Butcher (Butcher Math. Soc. 3, 185–201 1963) characterized a natural tabulation of the order conditions for Runge–Kutta methods of order p as an isomorphism from the set of rooted trees having up to p nodes and provided examples ...

It is common for mathematical models of physical systems to possess qualitative properties such as positivity, monotonicity, or conservation of underlying physical behavior. When these models consist of differential equations, it is also common ...