ABSTRACT
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 [13]. The classical splitting methods for hyperbolic equations are the alternating direction (ADI) methods, see [67], [149], and [155]. 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 [37], [59], [103], and [133]. In our work, we apply the iterative splitting methods to second-order partial differential equations, see [104]. 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 [193]. 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 [11], [13], [199], [201] and some applications are discussed in [125].
