Use of analytic solutions in the statement of difference boundary conditions on a movable shoreline

Abstract

Solutions to initial boundary value problems are constructed for the shallow water equations in the form of series locally convergent in the neighbourhood of a movable water-land boundary for an arbitrary bottom relief. The motion law and the velocity of this boundary are determined for various wave-shore interaction modes. The obtained results of analytic study of the solutions are used for the development of approximations of boundary conditions on the movable shoreline. Test problems are numerically solved using an explicit predictor-corrector scheme of the second order of approximation on adaptive grids retracing the position of the water-land boundary. The results of these calculations are presented.

On Numerical Methods for Solving Run-Up Problems. Comparative Analysis of Numerical Algorithms and Numerical Results.

“…These values, obtained using the TVD+SPH method, are significantly larger than the results from the work [4] for practically all considered amplitudes. Yet, they are close to the results obtained using the methodology from the work [6], which, however, happen to be somewhat smaller for the largest amplitudes. Let us note, that if for the absolute values of the run-up values R the monotonic increase takes place, then the corresponding relative values R/A increase first (up to the ampli- (1, 4), the methods from [6] (2, 5) and [4] (3, 6); the domain of applicability of the analytical solution [13] is marked by grey color.…”

“…For small amplitudes (nearly up to A = 1 m) the values, obtained by the TVD+SPH method, increase monotonically and practically coincides with the results from [4], which starts to decrease sharply and tend to zero for the largest amplitude. On the contrary, the results, obtained by the TVD+SPH algorithm, continue the stable monotonic increase and stay smaller than the results from [6] for the whole range of amplitude variation. The analysis of the graphs with the maximal values |R|/A in the run-down phase points in the first place to the preserving anomalous tendency of the results from [4] to zero for A > 1 m and their closeness to the results, obtained by the TVD+SPH algorithm, up to the amplitude A = 1 m. The behavior of the TVD+SPH results are qualitatively close to the distribution computed by the methodology from [6], which however is positioned significantly higher.…”

“…On the contrary, the results, obtained by the TVD+SPH algorithm, continue the stable monotonic increase and stay smaller than the results from [6] for the whole range of amplitude variation. The analysis of the graphs with the maximal values |R|/A in the run-down phase points in the first place to the preserving anomalous tendency of the results from [4] to zero for A > 1 m and their closeness to the results, obtained by the TVD+SPH algorithm, up to the amplitude A = 1 m. The behavior of the TVD+SPH results are qualitatively close to the distribution computed by the methodology from [6], which however is positioned significantly higher. The results obtained for the smaller initial amplitudes, as in the run-up phase, demonstrate the presence of the extremum, which is slightly shifted to the direction of the smallest values of A.…”

“…Further we compare our results with those of the work [4]. Our results were obtained by the TVD+SPH method and the algorithm, based of the definition of the waterfront point using the analytical solutions of the shallow water equations for the formulation of the difference boundary conditions on a moving waterfront [6]. The values from [4] were obtained partially from its text and partially by the digitization of the graphs presented there.…”

“…The second method, conventionally named as "exact", is presented in [6,7]. It uses a moving grid, and the analytical solution of the problem in a small vicinity of a waterfront point is applied to compute its position and velocity.…”