[HDS08-15] Derivation of Time Difference Equation using Continuous Expression in Nonlinear Long Wave Equation and Evaluation of its Discretization Error
Keywords:Tsunami numerical simulation, Nonlinear long wave equation, Finite difference formula
Next, these computational stability in the local inertial equation has already been evaluated in previous studies [Tanaka et al., 2017], suggesting that Simple-Implicit is more stable.
Therefore, in this paper, we evaluated time discretization error between Simple-Implicit and Combined-Implicit about under various conditions. For this purpose, we have created a new tsunami calculation code. It is a 1D nonlinear long-wave equation that can calculate Simple-Implicit and Combined-Implicit at the same time, and can handle virtual terrain and virtual initial waveform. We did not perform run-up calculations to eliminate errors due to run-up calculation conditions, and using radiation boundary condition by characteristic curve.
In order to evaluate the errors of these methods, a value that serves as a criterion is necessary, but no analytical solution has yet been found for the nonlinear long-wave equation. Also, it cannot be evaluated with observation values because it contains various other errors. For that reason, we use a characteristic that regardless of the time difference method, if the integration time dt is shortened as much as possible, each method will produce the same result. We determined that dt was short enough if the value was below a certain criterion, and then set the waveform as a tentative true value. Then, we evaluated the discretization errors of the time difference by comparing with the calculation results at dt which is much larger than its tentative value. dt used in the calculation is usually in the range used in the tsunami numerical calculation. In this case, we compared dt with 1/128 from the original dt.
In this comparison, there are time discretization errors of the gravity term and the advection term, and these errors are often larger than the time discretization error of the friction term. Therefore, it was impossible to compare the results of Simple-Implicit and Combined-Implicit. For that reason, the calculation was performed with the original dt and dt reduced to 1/128, excluding the friction term, and by taking the difference between them, the time discretization error of the advection term and the gravity term was estimated. By subtracting them from the calculation results of Simple-Implicit and Combined-Implicit, only the differences between Simple-Implicit and Combined-Implicit were extracted. The result is Fig.4. In all the calculated cases, it was shown that the error of Simple-Implicit was always small.
In summary, when solved analytical, the form of Simple-Implicit is obtained, and Simple-Implicit has better computational stability than Combined-Implicit and has less discretization error. In other words, this paper showed that Simple-Implicit is a more suitable calculation method for real-time simulations performed for disaster prevention.