Japan is an island nation that experiences frequent earthquakes. When an earthquake occurs, it is important to forecast its resultant tsunami: its size, location, time of arrival, etc. These forecasts are made using numerical simulations. The initial conditions are very important for numerical simulations, but the small number of tide stations makes it difficult to make highly precise forecasts. The distance between stations is normally several tens of km, and this lowers the precision of the initial data afforded by them. It is therefore common to use data interpolated from the sparse observation data at timet=0. Even so, high-resolution interpolation cannot be expected since the original data is of poor quality. In addition, the interpolated values may not be physically valid because the governing equation may not have been considered when the data were interpolated. We therefore propose a new method of estimating the initial value by using a characteristic equation. In this method, we replace the spatial resolution with time resolution. This results in a high-resolution initial value because the same place is measured more than once. In addition, the characteristic equation is based on the governing equation. Therefore, in this method, an accurate estimation of initial value is considered to be possible. In this paper, we show two applications of this approach, one for a dimensional shallow water wave equation and one for Euler’s equation. The shallow water wave equation is for the tsunami, and the Euler equation is the governing equation of the numerical weather forecast.
On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation