Modeling a discrete system by eigenvalue decomposition

Carlos A. Cunha Filho

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ABSTRACT

Numerical solutions of the wave equation require the discretization of the spatial and temporal dimensions, which creates numerical artifacts such as dispersion. If the physical system is already spatially discrete, the exact solution will be truly dispersive. The intrinsically dispersive character of these systems are a result of the constraints in the energy flux between neighbor cells of the system. Using the eigenvalue decomposition of the spatial operator I solve, exactly, the equations of motion associated with 1 and 2-D transverse isotropic spatially discrete systems. The method is unconditionally stable and requires no time discretization, but the cost is prohibitively high for practical uses. The intrinsically dispersive character of these systems are a result of the constraints in the energy flux between neighbor cells of the system. To overcome the high cost of the eigenvalue decomposition method, I have implemented an approximate solution using a time recursive scheme in a parallel architecture, with results that show good agreement with the exact solution.