A numerical method for solving the nonlinear Fermi-Pasta-Ulam problem (Q2874172)

The paper deals with the finite difference approximation of the nonlinear Fermi-Pasta-Ulam problem (FPU) NEWLINE\[NEWLINE\frac{\partial^2 u(x,t)}{\partial t^2}=\bigg[1+\epsilon \frac{\partial u(x,t)}{\partial x}\bigg]\frac{\partial^2 u(x,t)}{\partial x^2}-\gamma \frac{\partial u(x,t)}{\partial t}-m^2 u(x,t),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)=\phi(x), \; \frac{\partial u(x,0)}{\partial t}=\psi(x), \; 0 \leq x \leq L,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,t)=u(L,t)=0, \; 0 < t \leq T. NEWLINE\]NEWLINE The proposed finite difference scheme inherits the energy conservation property from the FPU differential problem. The conservation of the difference scheme as well as its solvability, convergence with the order \({\mathcal {O}} (\tau^2+h^2)\) and stability are proven. Numerical experiments are given which verify the convergence order and show the functionality of the algorithms.