Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

Abstract

SUMMARYThe Riemann solver is the fundamental building block in the Godunov-type formulation of many nonlinear fluid-flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth-order term. For the nonlinear shallow-water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite-volume model with a Godunov-type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two-dimensional dam-break problems, thereby verifying the proposed Riemann solver for general implementation.

Extreme tsunami inundation in Hawai‘i from Aleutian–Alaska subduction zone earthquakes

“…The staggered finite difference model builds on the nonlinear shallow-water equations with a momentum conservation scheme to approximate breaking waves as bores or hydraulic jumps as in a finite volume model (e.g., Wei et al 2006;Wu and Cheung 2008). The code accommodates up to five levels of two-way nested grids to describe processes of different time and spatial scales from the open ocean to the coast.…”

On the interaction of deep water waves and exponential shear currents

“…So, according to the definitions (22) and (23), the high-order deformation equation (32) is linear with constant coefficients, subject to the linear boundary conditions (33) and (34) on the known boundary ψ = 0. Such kind of linear boundary-value problem in a fixed domain is easy to solve, as shown in the coming subsection.…”

“…Among them, the so-called homotopy analysis method (HAM) [17,18,19,20,21,22] is rather attractive, which has been successfully applied to many aspects of nonlinear problems [23,24,25,26,27,28,29,30,31,32,33]. In the field of the wave propagation, Liao and Cheung [19] used the homotopy analysis method to give high-order convergent series solutions for the dispersion relationship of the deep water waves.…”

Solution of Nonlinear Fractional Differential Equation (NFDE) by Four Different Approximate Methods

“…Several numerical methods for solving fractional differential equations have bee introduced lately [1].The homotopy analysis method (HAM), initially proposed by Liao [2], is a powerful method to solve the nonlinear problems. A new analytical approach that can be applied to solve nonlinear fractional differential equations(NFDE) is to employ the Homotopy Analysis Method (HAM) [3][4][5][6].An account of recent development of HAM was given by Liao [7].Cang et al [8] solved nonlinear Riccati differential equations of fractional order using HAM. Hashim et al [9] employed HAM to solve fractional initial value problems(FIVPs) for ODEs.…”