Efficient implementation of geometric integrators for separable Hamiltonian problems

We here investigate the efficient implementation of the energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced for the numerical solution of Hamiltonian problems. In this note, we describe an iterative procedure, based on a triangular splitting, for solving the generated discrete problems, when the problem at hand is separable.Comment: 4 page

Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation

Conservation laws are among the most fundamental geometric properties of a given partial differential equation. However, standard finite difference approximations rarely preserve more than a single conservation law. A novel symbolic-numerical approach, introduced in [1], exploits the fact that divergences belong to the kernel of the Euler operator to construct schemes that preserve multiple conservation laws. However, this approach is limited by the complexity of the symbolic computations, whose cost is high even when the nonlinearity in the PDE is only quadratic. Some key simplifications, making the symbolic computations tractable, have been introduced in [2]. We apply this simplified strategy to the modified Korteweg-de Vries equation, having a cubic nonlinearity, to construct new bespoke finite-difference schemes that preserve the local conservation laws of the mass and of the energy

Energy conservation issues in the numerical solution of the semilinear wave equation

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur

Recent Advances in the Numerical Solution of Hamiltonian Partial Differential Equations

In this paper, we study recent results in the numerical solution of Hamiltonian partial differential equations (PDEs), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional (which derives from a proper space semi-discretization), confers more robustness to the numerical solution of such problems

Solving the nonlinear Schrödinger equation using energy conserving Hamiltonian Boundary Value Methods

In this paper we study the use of energy-conserving methods, in the class of Hamiltonian Boundary Value Methods, for the numerical solution of the nonlinear Schrödinger equation

A New Technique for Preserving Conservation Laws

This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation

Locally conservative finite difference schemes for the Modified KdV equation

Finite diffrence schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the Modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature

Simple bespoke preservation of two conservation laws

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic-numeric approach feasible. To illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared to others in the literature

Line Integral solution of Hamiltonian PDEs

In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach