Symmetric Splitting Methods for Wave Equations: First Approaches

Abstract

This article consider symmetric splitting methods for solving wave equations. We propose novel splitting methods, which are unconditionally stable, and apply diagonalization techniques to gain efficient solvers. The benefits to standard methods are discussed and we apply the methods to multidimensional wave equations. We are motivated by fast exponential computations based on matrix decompositions. We reformulate wave equations as system of coupled initial value boundary problems by discretizing only in space and use fast exponential of the resulting matrix. Therefore, we could solve fast systems of Cauchy initial value boundary problems. This process results in symmetric splitting of higher dimensional partial differential equations (PDEs), which are decoupled into a system of symmetric first-order ordinary differential equations (ODEs).