Decomposition Methods for Hyperbolic Equations
In this chapter, we focus on methods for decoupling multiphysical and multidimensional hyperbolic equations. The main concept is to apply the spatial splitting to the second-order derivative in time. The equations can be performed in their second-order derivatives, which saves more resources, as to convert into a system of ﬁrst-order derivatives, see . The classical splitting methods for hyperbolic equations are the alternating direction (ADI) methods, see , , and . A larger group of splitting methods for hyperbolic equations with respect to the linear acoustic wave equation is known as the local one-dimensional (LOD) methods. This approach splits the multidimensional operators into local onedimensional operators and sweeps implicitly over the directions. The methods are discussed in , , , and . In our work, we apply the iterative splitting methods to second-order partial diﬀerential equations, see . We discuss the stability and consistency of the resulting methods. We discuss strategies for decoupling the diﬀerent operators with respect to the spectrum of the underlying operators, see . We can see similarities to iterative operator-splitting methods for ﬁrst-order partial diﬀerential equations. The theory is given with the sin-and cos-semigroups and the estimates can be done similarly. The functional analysis in this context is performed for the hyperbolic partial diﬀerential equations in , , ,  and some applications are discussed in .