Approximate solution of the wave equation using the Multigrid Method

Resumo

In order to obtain the solution of the wave equation, the problem is discretized using the Finite Difference
Method (FDM). The discretization of the temporal direction is performed using the Time-Stepping Method. Thus,
it is a system of equations that involves spatial variables and is solved at every step of the time, which in real
problems, is usually a big variation. Using a conventional solver, such as Gauss-Seidel, to obtain the solution of
such systems, convergence factors are very close to the unit, which often implies a very high computational time.
To accelerate the convergence and decrease the computational time, it was proposed to use the Multigrid Method
to obtain the solution of the system of equations. Multigrid was initially proposed for elliptical problems, but it
has been used successfully in some parabolic and hyperbolic problems. This work deals especially with waves in
strings (1D), where a new approach is proposed to solve this class of problems.