Parallelization of Implicit Finite Difference Methods for Wave Propagation Simulation

This manuscript proposes a general parallel directsolver framework to parallelize and accelerate the implicit numerical simulation of wave propagation. To illustrate the concept, 1D implicit finite difference is employed, and the speedup performance is demonstrated. The theoretical complexity is reduced from ${\mathcal{O}}\left({N{M^\alpha }}\right)$ to ${\mathcal{O}}\left({N{{(M/K)}^\alpha } + N{{(2K)}^\alpha }}\right)$ with α ≈ 2.37, where K,M,N represent the number of parallel processors, element nodes, and time steps, respectively. This parallelization method can also be readily applied to the finite element method and other 2D/3D numerical simulation problems.