Stability and numerical dispersion properties of the distributional finite-difference method

Numerical methods for solving the wave equation have been widely adopted in seismology to simulate earthquakes and other seismic phenomena, with the goal of better understanding the underlying physical processes. Commonly used approaches, such as finite-difference methods and spectral element methods, have been extensively studied in terms of their numerical properties, computational cost, and accuracy in modelling the Earth’s response to seismic phenomena.

The recently introduced distributional finite-difference method (DFDM) is a novel approach to seismic wave modelling, aiming to combine advantages of established methods to achieve high accuracy at reduced computational cost. While recent studies have shown the accuracy of DFDM by comparing it to existing methods, a comprehensive investigation of the numerical properties governing its cost and accuracy in the context of solving the wave equation has not yet been carried out.

In this contribution, we focus on two key numerical properties: stability and numerical dispersion. We assess the trade-offs between time-step restrictions and accuracy, and introduce a cost metric to quantify the computational effort required to achieve a prescribed dispersion error threshold. Our results show that, although DFDM has more restrictive stability bounds, it provides superior dispersion performance at lower spatial resolutions compared to conventional methods. This makes DFDM particularly attractive for applications with stringent memory constraints, such as global-scale simulations or full-waveform inversion. These results provide practical guidance for selecting numerical methods in large-scale, physics-based wave simulations.