Related papers: Functional-preserving predictor-corrector multider…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs).…
Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh--Abe…
Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For…
Numerical integration methods are central to the study of self-gravitating systems, particularly those comprised of many bodies or otherwise beyond the reach of analytical methods. Predictor-corrector schemes, both multi-step methods and…
We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when…
We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories…
In this work, we present a novel class of parallelizable high-order time integration schemes for the approximate solution of additive ODEs. The methods achieve high order through a combination of a suitable quadrature formula involving…
This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme…
In this paper we design high-order positivity-preserving approximation schemes for an integro-differential model describing photochemical reactions. Specifically, we introduce and analyze three classes of dynamically consistent methods,…
We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of…
In this paper, a class of high-order compact finite difference Hermite scheme is presented for the simulation of double-diffusive convection. To maintain linear stability, the convective fluxes are split into positive and negative parts,…
The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational…
In this work, we construct novel discretizations for the unsteady convection-diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unknowns…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e. numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is…