Solution of the Wave-type PDE by Numerical Damping Control Multistep Methods

The second order Ordinary Differential Equation (ODE) system obtained after semidiscretizing the wave-type partial differential equation (PDE) with the finite element method (FEM) shows strong numerical stiffness. Its resolution requires the use of numerical methods with good stability properties and controlled numerical dissipation in the high-frequency range. The HHT-α and BDF-α methods are second order precision, unconditionally stable and able to dissipate high-modes for some values of the parameters. The finite element method has been applied to the one-dimensional linear wave-type PDE and to a non-linear version of a string of a guitar. The ODE systems obtained after applying FEM are solved by these two methods, proving that both are able to dissipate the high-modes.