Lagrangian blocks on Eulerian mesh for shallow-water wave computations

Waves in shallow water are computed by moving blocks of water in the direction of the flow by the Lagrangian method. Mass and momentum in the blocks are re-distributed onto the fixed Eulerian mesh to minimize distortion at each increment of time. This Lagrangian blocks on Eulerian mesh (LBEM) method is introduced as an alternative to the classical finite volume methods. The method is positive-depth definite and free of numerical oscillation. A large number of numerical computations are carried out to evaluate the accuracy and the computational stability of the LBEM method. The results of 6 publications in journals and 9 papers in conference proceedings are compiled to produce the total of 8 chapters and 4 appendices of this manuscript-based thesis. Computations using the LBEM method have been conducted for one-dimensional and two-dimensional waves in shallow water. These include the computations of the two-dimensional standing waves in a parabolic bowl, the propagation of oblique shock waves in a square basin, the shoaling of solitary waves and periodic waves over levee, the dam-break waves over dike, and the flood routing through an idealized city. These computational problems have been selected because they have either analytical solutions or available experimental data for comparison with the computations. When exact solutions are available, block refinements are conducted to show the convergence of the LBEM computations toward the exact solutions. The goals of these computations are to show how the LBEM method (i) can track the dry-and-wet water interface without using interface treatment, (ii) can capture shock wave without using any flux or slope limiter, and (iii) can calculate the intermittent flow such as the wave overtopping the levee. Although the tracking of interfaces, the capture of shock waves and the calculation of intermittent overflow are difficult problems for the classical methods, the block advection can compute these problems with absolute stability to produce accurate numerical solutions convergent to the exact solutions.