Energy conservation issues in the numerical solution of the semilinear wave equation
For researchers in numerical analysis and computational physics, this work provides a method to maintain energy conservation in simulations of Hamiltonian PDEs, though it is an incremental extension of existing HBVMs to a specific class of problems.
The paper addresses energy conservation in numerical solutions of the semilinear wave equation, showing that energy-conserving properties of the continuous problem can be preserved in fully discretized solutions using Hamiltonian Boundary Value Methods (HBVMs).
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 Schrödinger equation.