Theoretical Time Evolution of Numerical Errors When Using Floating Point Numbers in Shallow-Water Models

Abstract

We carried out a theoretical investigation of the impact of the numerical errors caused by using floating point numbers (FPNs) in simulations, such as rounding errors. Under the presupposition that model variables can be written as the linear sum of the true value and the numerical error, equations governing the time evolution of numerical errors due to FPNs (FPN errors) are obtained by considering the total errors of the results of simulations of shallow-water models and estimating the errors incurred by using FPNs with varying precision. We can use the time evolution equations to estimate the behavior of the FPN errors, then confirm these estimations by carrying out numerical simulations. In a geostrophic wind balance state, the FPN error oscillates and gradually increases in proportion to the square root of the number of time steps, like a random walk. We found that the error introduced by using FPNs can be considered as stochastic forcing. In a state of barotropic instability, the FPN error initially evolves as stochastic forcing, as in the case of the geostrophic wind balance state. However, it then begins to increase exponentially, like a barotropic instability wave. These numerical results are obtained by using a staggered-grid arrangement and stable time-integration method to retain near-neutral numerical stability in the simulations. The FPN error tends to behave as theoretically predicted if the numerical stability is close to neutral.