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

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 equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.

In this paper, we report on 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.

In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.