On efficient fourth-order Runge-Kutta-Central schemes for modeling tsunami propagation and run-up prediction

Abstract

In our work, one-dimensional shallow water equations are proposed to model the phenomena of tsunami propagation and run-up. This model is known to be reliable to make predictions on maximum run-up along the shorelines above the mean sea level induced by the tsunami waves. 1D shallow water equations in a depth-averaged manner can be derived from incompressible Navier-Stokes equations by the simplification of lateral velocity (z-direction) using boundary conditions imposed on both free surface and bottom topography. In our current work, we assume there is no motion in the y-direction. Further, 1D shallow water equations in a conservative form are solved using the fourth-order Runge-Kutta method for time and second-order central difference schemes in space. We observed that the resulting high-order method in time is efficient, fairly accurate for their smaller stencil sizes, and has a slightly better stability property than that of the conventional forward Euler in time. This high-order method allows us to take larger mesh size and time steps while still producing small discretization errors. Our method was implemented using a low-level C++ programming language which is essential to incorporate larger domains (consecutively larger degrees of freedom) and to render fast computations. Using a conceptual test problem, we validate the numerical error, reproduce the theoretical convergence rate in both time and space while observing its computational efficiency. We also verified our model using two test problems – one for wave propagation and another is for the run-up. Our numerical models reproduced the expected outcomes when a tsunami wave travels in the ocean just before it arrives at the shoreline in terms of a solitary wave and finally collapses.