3 research outputs found
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
Symplectic Schemes for Linear Stochastic Schrödinger Equations with Variable Coefficients
This paper proposes a kind of symplectic schemes for linear Schrödinger equations with variable coefficients and a stochastic perturbation term by using compact schemes in space. The numerical stability property of the schemes is analyzed. The schemes preserve a discrete charge conservation law. They also follow a discrete energy transforming formula. The numerical experiments verify our analysis