On numerical simulation of tsunami run‐up

Abstract

The improved adaptive grid method is presented for the numerical modelling of propagation of long surface waves in large water areas and their interaction with coasts. Adaptive grid methodThe numerical modelling of propagation of long surface waves in large water areas and their interaction with coasts is done in the framework of the non-dispersive shallow water model. The adaptive grid method [1] is improved and applied to the problems on tsunami run-up on coast. The detailed description of the method is given in [1], and the current paper presents a brief summary of the improvements with the detailed description of the algorithm and numerical results to follow in [2].The following improvements are implemented for the predictor-corrector scheme for 1D shallow water equations on moving grids [1].A new formula is used for the computation of the right-hand side of the corrector step. In [1] an additional equation was used on the predictor step for computing a total depth in integer nodes. Now only two equations are solved on the predictor step, reducing significantly the computation time. All advantages of the original predictor-corrector scheme [1] remain, in particular the preservation a state of rest on a fixed grid.In [1] the modified formula for scheme parameters was used, obtained using the entropy fix on the basis of the differential approximation, guaranteeing the non-negativity of approximation viscosity in the vicinity of the sonic point of depression wave. This formula controlled the switching from one scheme to another using the analysis of the difference derivative of the solution vector, i.e. the derivatives of total depth and total impulse. Therefore, the bottom profile could have a strong influence on a choice of a scheme. Nevertheless, the numerous simulations of surface waves over irregular bottom profiles have shown that a scheme switcher should analyze not the variation of total depth (H = η + h, η is free surface elevation, b is bottom profile), but the variation of free surface elevation and the derivative of fluid velocity. Thus, the formula for scheme parameters is updated accordingly.When constructing the conservative schemes on moving grids, the formulas for grid node velocities must be consistent with the formulas for cell areas. The special approximation of these variables guarantees the satisfaction of the difference analogue of the geometric conservation law [3], [4], which is the necessary condition of the consistency of formulas. In [1] the scheme conservation property for the nonlinear scalar equation on moving grids was satisfied with the help of the appropriate choice of the scheme parameter. The current work uses the same approach for the 1D shallow water equations, thus, actually providing a method for automatic construction of conservative schemes. For example, using the appropriate scheme parameters, the conservative upwind scheme on moving grid can be obtained and written in the equivalent divergent and non-divergent forms. An advantage of this scheme is the preservation of...