Unconditionally stable time-domain elastic wave simulations by the alternative direction implicit method

The stability of the conventional staggered-grid finite-difference time-domain (FDTD) method for elastic wave simulations is limited by the Courant condition and material heterogeneity. Its computational efficiency is significantly hampered when the mesh size is much smaller than a wavelength (for geometric modeling accuracy) and/or with a high impedance contrast. An unconditionally stable alternating direction implicit (ADI) method is proposed to overcome the conditional stability. It is based on additively splitting the Crank Nicholson (CN) operator into two sub-operators and subsequently solving the CN scheme in two sub-steps. In each sub-step, a tri-diagonal matrix is formed based on one of the sub-operators and the associated field variables are solved implicitly with O(N) computational complexity. The rest of the field variables are solved explicitly. Due to its semi-implicit nature, it can be proved that the ADI method is unconditionally stable regardless of the time step size. The numerical disper...