Accuracy of classical conservation laws for Hamiltonian PDEs under Runge-Kutta discretizations
The authors consider the numerical behaviour of conservation laws of Hamiltonian partial differential equations in \((1+1)\) dimensions written in multi-symplectic form. They apply both for the spatial and the temporal discretisation Runge-Kutta methods and show that under certain assumptions on the coefficients of the methods a discrete local norm square conservation law holds. Then they study the accuracy of the methods providing bounds for the approximation of functions evaluated on the solution of the Hamiltonian system. Based on these results, they finally derive uniform error estimates first for the local and then also for the global discrete energy and momentum conservation laws.
scientific article; zbMATH DE number 5550619